From the 2nd Law of Thermodynamics to AC-Conductivity Measures of Interacting Fermions in Disordered Media
J.-B. Bru, W. de Siqueira Pedra
TL;DR
This paper extends the concept of macroscopic AC-conductivity measures to interacting fermions in disordered media, deriving laws from thermodynamics and proposing a Levy process-based model for electrical conductivity.
Contribution
It introduces a novel extension of AC-conductivity measures to interacting fermions, incorporating thermodynamic principles and Levy processes, beyond free fermion models.
Findings
Derived Ohm and Joule's laws in the AC regime for interacting fermions.
Found that AC-conductivity at large frequencies is significantly smaller than classical models predict.
Proposed Levy processes as an effective model for electrical conductivity phenomena.
Abstract
We study the dynamics of interacting lattice fermions with random hopping amplitudes and random static potentials, in presence of time-dependent electromagnetic fields. The interparticle interaction is short-range and translation invariant. Electromagnetic fields are compactly supported in time and space. In the limit of infinite space supports (macroscopic limit) of electromagnetic fields, we derive Ohm and Joule's laws in the AC-regime. An important outcome is the extension to interacting fermions of the notion of macroscopic AC-conductivity measures, known so far only for free fermions with disorder. Such excitation measures result from the 2nd law of thermodynamics and turn out to be L\'{e}vy measures. As compared to the Drude (Lorentz--Sommerfeld) model, widely used in Physics, the quantum many-body problem studied here predicts a much smaller AC-conductivity at large frequencies.…
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Taxonomy
TopicsQuantum chaos and dynamical systems · Theoretical and Computational Physics · Quantum and electron transport phenomena
From the 2nd Law of Thermodynamics to AC–Conductivity Measures of
Interacting Fermions in Disordered Media
J.-B. Bru
W. de Siqueira Pedra
Abstract
We study the dynamics of interacting lattice fermions with random hopping amplitudes and random static potentials, in presence of time–dependent electromagnetic fields. The interparticle interaction is short–range and translation invariant. Electromagnetic fields are compactly supported in time and space. In the limit of infinite space supports (macroscopic limit) of electromagnetic fields, we derive Ohm and Joule’s laws in the AC–regime. An important outcome is the extension to interacting fermions of the notion of macroscopic AC–conductivity measures, known so far only for free fermions with disorder. Such excitation measures result from the 2nd law of thermodynamics and turn out to be Lévy measures. As compared to the Drude (Lorentz–Sommerfeld) model, widely used in Physics, the quantum many–body problem studied here predicts a much smaller AC–conductivity at large frequencies. This indicates (in accordance with experimental results) that the relaxation time of the Drude model, seen as an effective parameter for the conductivity, should be highly frequency–dependent. We conclude by proposing an alternative effective description – using Lévy Processes in Fourier space – of the phenomenon of electrical conductivity.
Contents
1 Introduction
The present paper belongs to a succession of works on electrical conductivity, starting with [BPH1, BPH2, BPH3, BPH4, BP1].
As claimed in the famous paper [So, p. 505], “it must be admitted that there is no entirely rigorous quantum theory of conductivity.” Concerning AC–conductivity, however, in the last years significant mathematical progress has been made. See [KLM, KM1, KM2, BC, BPH1, BPH2, BPH3, BPH4, W] for examples of mathematically rigorous derivations of linear conductivity from first principles of quantum mechanics in the AC–regime. These results indicate a picture of the microscopic origin of Ohm and Joule’s laws which differs from usual explanations coming from the Drude (Lorentz–Sommerfeld) model. This is discussed with more details in [BPH2].
As electrical resistance of conductors may result from the presence of interactions between charge carriers, the main drawback of these studies is their restriction to non–interacting systems.
A first attempt in this direction has been tried in the parallel work [W], which uses like us an algebraic approach to tackle such problems. With regard to interacting systems, explicit constructions of KMS states are obtained in the Ph.D. thesis [W] for a one–dimensional model of interacting fermions with a finite range pair interaction. But, the author studies in [W, Chap. 9] the linear response theory only for non–interacting fermions, keeping in mind possible generalizations to interacting systems.
Therefore, we aim to extend [BPH1, BPH2, BPH3, BPH4] to fermion systems with interactions. As a first step, [BP1] proves all assertions of [BPH1, BPH2] for fermion systems with short range interactions. We perform here the second part of this program by extending the main results of [BPH3] to interacting fermion systems:
- •
Like in [BP1], we investigate some non–autonomous –dynamical system on the CAR –algebra of cubic infinite lattices of any dimension. The (non–autonomous) dynamics is generated by short–range and translation invariant interactions between particles, random static potentials, and also random next neighbor hopping amplitudes in presence of local and time–dependent electromagnetic fields. Disorder is here defined via ergodic distributions of random potentials and hopping amplitudes.
- •
We study the linear response of interacting fermions at thermal equilibrium in disordered media to macroscopic electric fields that are time– and space–dependent. In particular, we obtain Ohm’s law with a (charge) transport coefficient that is a continuous function (of time) naturally called macroscopic conductivity.
- •
The Fourier transform of the conductivity is named macroscopic AC–conductivity measure. The fact that this Fourier transform is indeed a measure follows from the 2nd law of thermodynamics (see Remark 5.4). The latter corresponds here to the Kelvin–Planck statement while avoiding the concept of “cooling” [LY1, p. 49]. In particular, the 2nd law yields the positivity of the heat production for cyclic processes on equilibrium states. The concept of conductivity measure, introduced by Klein, Lenoble and Müller, has already been used in [KLM, KM1, KM2, BPH2, BPH3, BPH4] for the non–interacting case.
- •
We give a comparison of our results and the Drude (Lorentz–Sommerfeld) model, widely used in Physics [So, LTW] to describe the phenomenon of electrical conductivity. See also [BP2] for a historical perspective of this subject. In particular, we show that the Drude model and its refinements (like the Drude–Lorentz and the Lorentz–Sommerfeld models) always overestimate the in–phase conductivity at high frequencies. This indicates that the relaxation time of the Drude model, seen as an effective parameter for the conductivity, should be frequency–dependent, as already observed for instance in [T, NS1, NS2, SE, YRMK]. In fact, it should either vanish or diverge at large frequencies.
- •
We show that the AC–conductivity measure of the system under consideration is always a Lévy measure. An alternative effective description of the phenomenon of linear conductivity by using Lévy Processes in Fourier space is discussed. This is reminiscent of Boltzmann equation for collective oscillatory modes of charge excitations. It was recently shown that Lévy statistics can efficiently describe quantum phenomena like (subrecoil) laser cooling [BBAC]. As far as we know, there is no mathematically rigorous proof of this fact.
Note that also new results not presented in [BPH3] are obtained here, even for non–interacting fermions. For instance, in contrast with [BPH3], the hopping amplitudes are allowed to be non–homogeneous in space.
Like in [BP1], these results are made possible by Lieb–Robinson bounds for multi–commutators. See [BP3] for more details. Indeed, we need to get error terms uniformly bounded with respect to (w.r.t.) the random parameters and the volume of the box where the electromagnetic field lives. This is a crucial step to get valuable information in the limit of macroscopic electromagnetic fields, otherwise the results presented here would loose almost all its interest. To get such error terms, we apply in [BPH1, BPH2, BPH3, BPH4] tree–decay bounds on multi–commutators in the sense of [BPH1, Section 4]. The latter are based on combinatorial results [BPH1, Theorem 4.1] already used before, for instance in [FMU]. Nevertheless, [BPH1, BPH2, BPH3, BPH4] or [FMU] require Bogoliubov automorphisms (see [BR2, Theorem 5.2.5]). In other words, only non–interacting fermion systems can be tackled with such combinatorial arguments (like [BPH1, Theorem 4.1]).
A solution to that issue for the interacting case has only been recently given in [BP3] via Lieb–Robinson bounds for multi–commutators111Only Lieb–Robinson bounds for multi–commutators of order 3 is necessary in our study., which is not an obvious extension of usual Lieb–Robinson bounds known since 1972 [LR]. This is explained in [BP3, Sections 3.3, 4.3] and [BP1]. Note that Lieb–Robinson bounds have also been recognized as an important ingredient in [W] via the so–called strong localization criterion, see [W, Definition 10.1]. Nevertheless, they are not sufficient for our purpose. Indeed, its extensions to multi–commutators turn out to be pivotal in Theorem 7.1, which is used to prove Theorem 5.2. In fact, the Lieb–Robinson bounds for multi–commutators make the present paper much easier from the technical point of view, even for interacting systems. Compare, for instance, Section 7.2 with [BPH3, Section 5.4]. As a consequence, important conceptual issues, like the derivation of AC–conductivity measures222The conductivity measure can, indeed, be seen as a excitation measure related to electric perturbations. Similar constructions can be performed for many classes of pertubations because of the 2nd law. for interacting fermions on lattices by using the 2nd law as a postulate (see Remark 5.4), become more transparent.
As explained in [BP1], note however that Lieb–Robinson bounds for multi–commutators requires short–range interactions. Our setting includes density–density interactions resulting from the second quantization of two–body interactions defined via a real–valued and summable (in a convenient sense) function . For instance, the celebrated Hubbard model (and any other system with finite range interactions) or models with Yukawa–type potentials are all possible choices, but the Coulomb potential is excluded because it is not summable in space. For more details, see [BP1, Section 2.4].
Our main assertions are Theorems 4.1, 4.2, 5.1, 5.2 and 5.6. The paper is organized as follows:
- •
Section 2 is a preliminary conceptual review on the notion of thermal equilibrium state in relation to the 2nd law of thermodynamics. In this context, the mathematical results of [PW] are discussed.
- •
Section 3 formulates the mathematical setting used to study charge transport properties of fermions. We define in particular a Banach space of short–range interactions.
- •
Section 4 states Ohm’s law for macroscopic electromagnetic fields as well as Green–Kubo relations for current Duhamel fluctuation increments.
- •
In Section 5 we derive the macroscopic AC–conductivity measure from Joule’s law and the 2nd law of thermodynamics. Its relations with microscopic AC–conductivity measures and the Drude model are discussed. In Section 5.4 we propose a notion of time–reversal symmetry for fermion systems on the lattice in presence of disorder and discuss its consequences for the corresponding charge transport coefficients.
- •
Section 6 proposes an effective description of the phenomenon of linear conductivity by using Lévy Processes.
- •
Section 7 gathers technical proofs on which Sections 4–6 are based. The arguments strongly use the results of [BP1, BP3].
Notation 1.1
To simplify notation, we denote by positive and finite constants. These constants do not need to be the same from one statement to another. A norm on a generic vector space is denoted by and the identity map of by . To avoid ambiguity, scalar products in are sometimes denoted by .
2 2nd Law of Thermodynamics and Thermal States
It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.
[Lord Kelvin, 1851]
See [K]. This is the celebrated 2nd law of thermodynamics, the history of which starts with Carnot’s works in 1824. It is “one of the most perfect laws in physics” [LY1, Section 1] and it has never been faulted by reproducible experiments. As explained in [LY1, LY2], different popular formulations of the same principle have been stated by Clausius, Kelvin (and Planck), and Carathéodory. Our study is based on the Kelvin–Planck statement while avoiding the concept of “cooling” [LY1, p. 49]:
No process is possible, the sole result of which is a change in the energy of a simple system (without changing the work coordinates) and the raising of a weight.
The celebrated formulations of Clausius, Kelvin–Planck and Carathéodory are all about impossible processes and let largely open what is possible. This is useful to define the concept of thermal equilibrium states in a simple way. Note that Lieb and Yngvason’s work [LY1] on the 2nd law is an important structural approach which involves possible processes, instead.
We mathematically implement the Kelvin–Planck principle by using algebraic quantum mechanics like in [PW]. Basically, we use some –algebra , the self–adjoint elements of which are the so–called observables of the physical system. States on the –algebra are, by definition, linear functionals {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}\in\mathcal{X}^{\ast} which are normalized and positive, i.e., {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}(\mathbf{1})=1 and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}(B^{\ast}B)\geq 0 for all . They represent the state of the physical system. In the commutative case of classical physics states are usual probability measures.
To define equilibrium states, [PW] is pivotal because it mathematically implements the Kelvin–Planck physical notion of equilibrium:
Systems in the equilibrium are unable to perform mechanical work in cyclic processes.
Note at this point that the above principle (2nd law) defining equilibrium can possibly be violated.
As explained in [PW, p. 276], the above formulation of the 2nd law of thermodynamics is directly related to the notion of passive states. Indeed, one defines a (unperturbed) dynamics of the system by a strongly continuous one–parameter group {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}\equiv\{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}_{t}\}_{t\in{\mathbb{R}}} of –automorphisms of with (generally unbounded) generator . The latter is a dissipative and closed derivation of . If the state of the system at is {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}\in\mathcal{X}^{\ast}, then it evolves as {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}_{t}={{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}\circ{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}_{t-t_{0}} for any . On this system, one produces “excitations” by perturbing the generator of dynamics with bounded time–dependant symmetric derivations
[TABLE]
for arbitrary differentiable families of self–adjoint elements of . In particular, this defines a strongly continuous two–parameter family \{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}_{t,t_{0}}\}_{t\geq t_{0}} of –automorphisms of as the solution of a non–autonomous evolution equation defined, for any B\in\mathrm{Dom}({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 270\relax}}{\mbox{\boldmath\textstyle\mathchar 270\relax}}{\mbox{\boldmath\scriptstyle\mathchar 270\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 270\relax}}}), by
[TABLE]
The state of the system evolves now as {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}_{t}={{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}\circ{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}_{t,t_{0}} for any .
As explained in [PW, p. 276], the energy exchanged between the external device and the perturbed system at time is equal to
[TABLE]
If L_{t}^{A}\left({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}\right)\geq 0 then work is performed on the system, while L_{t}^{A}\left({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}\right)<0 means that one decreases the energy of the system. A cyclic process of time length is, by definition, a differentiable family of self–adjoint elements of such that for all and . Then, the 2nd law of thermodynamics can be formulated in this mathematical framework as follows (cf. [PW, Definition 1.1]):
Definition 2.1** (2nd law of thermodynamics – Passivity)**
**
Let (\mathcal{X},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}) be a –dynamical system. A state {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}\in\mathcal{X}^{\ast} is passive iff L_{T}^{A}\left({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}\right)\geq 0 for all cyclic processes of any time length .
By [PW, Theorem 2.1], passive states of a dynamical system (\mathcal{X},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}) can be equivalently defined as states satisfying
[TABLE]
for all unitaries both in the domain of definition of the generator of the group and in the connected component of the identity of the group of all unitary elements of with the norm topology. See, e.g., [BR2, Definition 5.3.21]. This last condition is strongly related with internal energy increments and the 1st law of thermodynamics, see, e.g., [BP1, Theorem 3.2].
By [PW, Theorem 1.1], such states are invariant with respect to (w.r.t.) the unperturbed dynamics: any passive state {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}\in\mathcal{X}^{\ast} satisfies
[TABLE]
Physically, it means that the dynamics of the system at equilibrium cannot be observed unless one performs external perturbations to extract some excitation spectrum. This last notion will be discussed in detail in a companion paper and the conductivity measure is one notable example of application.
Moreover, for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}_{0}^{+}, all ({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}})–KMS states {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 293\relax}}{\mbox{\boldmath\textstyle\mathchar 293\relax}}{\mbox{\boldmath\scriptstyle\mathchar 293\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 293\relax}}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}})} are passive, see [PW, Theorem 1.2]. The same holds true for {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}=\infty, that is, for ground states of (\mathcal{X},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}). Any convex combination of passive states is also passive. In particular, for any , {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}_{1},\ldots,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}_{n},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}_{1},\ldots,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}_{n}\in\mathbb{R}^{+} with \Sigma_{j=1}^{n}{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}_{j}=1, the state
[TABLE]
is passive, but it is neither a KMS nor a ground state of (\mathcal{X},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}), in general.
We impose another natural condition related to the physical notion [LY1, Definition p. 55] of thermal equilibrium in thermodynamics that excludes such convex combinations. A minimal requirement for the system to be in thermal equilibrium is indeed that it cannot produce work by interacting with any of its copy. To be more precise, prepare copies (\mathcal{X}^{(1)},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}^{(1)},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}^{(1)}),\ldots,(\mathcal{X}^{(n)},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}^{(n)},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}^{(n)}) of the original system defined by (\mathcal{X},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}) and consider the compound system
[TABLE]
If (\mathcal{X},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}) is at thermal equilibrium, the compound system should also be at equilibrium and it must not be possible to extract any energy from cyclic processes, by the 2nd law of thermodynamics. Therefore, \otimes_{j=1}^{n}{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}} should also be passive for all . Such states are named in the literature completely passive states:
Definition 2.2** (Thermal equilibrium states)**
**
Let (\mathcal{X},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}) be a –dynamical system. A state {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 293\relax}}{\mbox{\boldmath\textstyle\mathchar 293\relax}}{\mbox{\boldmath\scriptstyle\mathchar 293\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 293\relax}}}\in\mathcal{X}^{\ast} is completely passive iff \otimes_{j=1}^{n}{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 293\relax}}{\mbox{\boldmath\textstyle\mathchar 293\relax}}{\mbox{\boldmath\scriptstyle\mathchar 293\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 293\relax}}} is a passive state of (\otimes_{j=1}^{n}\mathcal{X}^{(j)},\otimes_{j=1}^{n}{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}^{(j)}) for all . We name them thermal equilibrium states of (\mathcal{X},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}).
[PW, Theorem 1.4] gives an explicit characterization of thermal equilibrium states:
Theorem 2.3** (Pusz–Woronowicz)**
Let (\mathcal{X},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}) be a –dynamical system. is a thermal equilibrium state of (\mathcal{X},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}) iff it is a ({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}})–KMS state of \left(\mathcal{X},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}\right) for some {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\left[0,\infty\right].
The parameter {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\left[0,\infty\right] is named inverse temperature of the system and is a consequence of the 2nd law of thermodynamics. It is a universal parameter of the (possibly infinite) system. In fact, tunes the value of the internal energy density of the system. Equivalently,* it fixes a time scale* since is a ({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}_{t},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}})–KMS state iff is a ({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}t},1)–KMS state ({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}<\infty). The boundary case {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}=0 corresponds to the –invariant traces, also called chaotic states, whereas ({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}},\infty)–KMS states are by definition ground states. [({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}},-{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}})–KMS states correspond to ({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}})–KMS states with a reversal of time.]
The notion of local (relative) entropy seems to be more natural than the concept of local temperature. Indeed, the 2nd law of thermodynamics as expressed in Definitions 2.1–2.2 is a formal expression of the unavoidable lost while one interacts with an object, which is at equilibrium before the interaction. Entropy is only a quantitative counterpart of this lost. It corresponds to heat production in thermodynamics which we study in the context of electricity theory. The positivity of the heat production, which is the content of the 2nd law of thermodynamics, implies the existence of the AC–conductivity measure. See Section 5.
Remark 2.4** (Dynamics versus thermal equilibrium states)**
Let a state {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 293\relax}}{\mbox{\boldmath\textstyle\mathchar 293\relax}}{\mbox{\boldmath\scriptstyle\mathchar 293\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 293\relax}}}\in\mathcal{X}^{\ast} with GNS representation (\mathcal{H},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}},\Psi). Its normal extension \hat{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 293\relax}}{\mbox{\boldmath\textstyle\mathchar 293\relax}}{\mbox{\boldmath\scriptstyle\mathchar 293\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 293\relax}}}} on {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}(\mathcal{X})^{\prime\prime} is a KMS state for a –weakly continuous one–parameter group {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}\equiv\{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}_{t}\}_{t\in{\mathbb{R}}} of –automorphisms of {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 281\relax}}{\mbox{\boldmath\textstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptstyle\mathchar 281\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 281\relax}}}(\mathcal{X})^{\prime\prime} iff \hat{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 293\relax}}{\mbox{\boldmath\textstyle\mathchar 293\relax}}{\mbox{\boldmath\scriptstyle\mathchar 293\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 293\relax}}}} is faithful. See, e.g., [BR2, p. 85]. In this case, the group is unique. The faithfulness of states is a physically natural property: By definition, an observable exists iff the corresponding physical property can be observed. Therefore, one could fix a state {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 293\relax}}{\mbox{\boldmath\textstyle\mathchar 293\relax}}{\mbox{\boldmath\scriptstyle\mathchar 293\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 293\relax}}}\in\mathcal{X}^{\ast} of the system that must be, by definition, a thermal equilibrium state, i.e., a KMS state. This assumption implicitly imposes the existence of some (unique) dynamics given by a group {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 293\relax}}{\mbox{\boldmath\textstyle\mathchar 293\relax}}{\mbox{\boldmath\scriptstyle\mathchar 293\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 293\relax}}})} and is justified a posteriori via the 2nd law. Constructing KMS states {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 293\relax}}{\mbox{\boldmath\textstyle\mathchar 293\relax}}{\mbox{\boldmath\scriptstyle\mathchar 293\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 293\relax}}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}})} from a given dynamics may be technically more involved. It is however the approach we use because the dynamics is fixed by microscopic interactions between particles.
3 –Dynamical Systems for Interacting Fermions
The mathematical framework used here is exactly the one of [BP1]. It is concisely described below. The only additional information is the exact definition of the probability space modelling disorder.
3.1 Disordered Media within Electromagnetic Fields
Disorder in the crystal is modeled by a random variable with distribution taking values in the measurable space . The probability space is defined as follows:
Let () and
[TABLE]
be the set of non–oriented bonds of the cubic lattice . Then,
[TABLE]
I.e., any element of is a pair {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}=\left({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}_{1},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}_{2}\right)\in\Omega, where {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}_{1} is a function on lattice sites with values in and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}_{2} is a function on bonds with values in the complex closed unit disc .
Let , , be an arbitrary element of the Borel –algebra of the interval w.r.t. the usual metric topology. Define
[TABLE]
i.e., is the –algebra generated by the cylinder sets , where for all but finitely many . In the same way, let
[TABLE]
where , , is the Borel –algebra of the complex closed unit disc w.r.t. the usual metric topology. Then
[TABLE]
The measure is an arbitrary ergodic probability measure on the measurable space : It is invariant under the action
[TABLE]
of the group of translations on and, for any such that {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 287\relax}}{\mbox{\boldmath\textstyle\mathchar 287\relax}}{\mbox{\boldmath\scriptstyle\mathchar 287\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 287\relax}}}_{x}^{(\Omega)}\left(\mathcal{X}\right)=\mathcal{X} for all , one has . Here, for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}=\left({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}_{1},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}_{2}\right)\in\Omega, and with ,
[TABLE]
We denote by the expectation value associated with .
For any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}=\left({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}_{1},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}_{2}\right)\in\Omega, V_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}}\in\mathcal{B}(\ell^{2}(\mathfrak{L})) is by definition the self–adjoint multiplication operator with the function {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}_{1}:\mathfrak{L}\rightarrow[-1,1]. It represents a bounded static potential. To all {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\Omega and strength {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}\in\mathbb{R}_{0}^{+} of hopping disorder, we also associate another self–adjoint operator \Delta_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}}\in\mathcal{B}(\ell^{2}(\mathfrak{L})) describing the hoppings of a single particle in the lattice:
[TABLE]
for any and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 288\relax}}{\mbox{\boldmath\textstyle\mathchar 288\relax}}{\mbox{\boldmath\scriptstyle\mathchar 288\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 288\relax}}}\in\ell^{2}(\mathfrak{L}), with being the canonical orthonormal basis of the Euclidian space . In the case of vanishing hopping disorder {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}=0 (up to a minus sign) \Delta_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},0} is the usual –dimensional discrete Laplacian. Since the hopping amplitudes are complex–valued ({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}_{2} takes values in ), note additionally that random electromagnetic potentials can be implemented in our model.
Then, for any realization {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\Omega of disorder and parameters {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}, the Hamiltonian of a single quantum particle within a bounded static potential is the discrete Schrödinger operator (\Delta_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}}+{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}V_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}}) acting on the Hilbert space . The coupling constants {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+} represent the strength of disorder of respectively the external static potential and hopping amplitudes.
The time–dependent electromagnetic potential is defined by a compactly supported time–dependent vector potential
[TABLE]
where is the set of one–forms333In a strict sense, one should take the dual space of the tangent spaces , . on that take values in . The smoothness of is not essential in the proofs and is only assumed for simplicity.
Remark 3.1
To simplify notation, we identify in the sequel with via the canonical scalar product of .
We use the Weyl gauge (also named temporal gauge) for the electromagnetic field and, as a consequence,
[TABLE]
is the electric field associated with . We also define the integrated electric field (or electric tension) along the oriented bond at time by
[TABLE]
Since is by assumption compactly supported, the corresponding electric field satisfies the AC–condition
[TABLE]
for sufficiently large times . From (9),
[TABLE]
is the time at which the electric field is turned off. In other words, we consider cyclic electromagnetic processes.
To simplify notation and without loss of generality (w.l.o.g.), fermions are spinless and have negative charge. The cases of particles with spin and positively charged particles can be treated by exactly the same methods. Thus, using the (minimal) coupling of to the discrete Laplacian, the discrete magnetic Laplacian is (up to a minus sign) the self–adjoint operator
[TABLE]
defined444Observe that the sign of the coupling between the electromagnetic potential and the laplacian is wrong in [BPH1, Eq. (2.8)]. by
[TABLE]
for all , {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\Omega, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}\in\mathbb{R}_{0}^{+} and . Here, is the scalar product in and is the canonical orthonormal basis \mathfrak{e}_{x}(y)\equiv{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 270\relax}}{\mbox{\boldmath\textstyle\mathchar 270\relax}}{\mbox{\boldmath\scriptstyle\mathchar 270\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 270\relax}}}_{x,y} of . In (11), similar to (8), {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}y+(1-{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}})x and are seen as vectors in . In presence of an electromagnetic field associated to an arbitrary vector potential , the one–particle Hamiltonian (\Delta_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}}+{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}V_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}}) at fixed {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\Omega and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+} is replaced with the time–dependent one
[TABLE]
3.2 Banach Space of Short–Range Interactions
Let be the set of all finite subsets of . For all , is the finite dimensional –algebra generated by and generators satisfying the canonical anti–commutation relations, being some finite set of spins. As just explained above, the spin dependence of is irrelevant in our proofs (up to trivial modifications) and, w.l.o.g., we only consider spinless fermions, i.e., the case .
We denote by the CAR –algebra of the infinite system, that is, the inductive limit of the finite dimensional –algebras . The –algebra of all even elements of is denoted by and
[TABLE]
is the subset of local elements. See [BP1, Section 2.2] for more details. Finally, let \{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 287\relax}}{\mbox{\boldmath\textstyle\mathchar 287\relax}}{\mbox{\boldmath\scriptstyle\mathchar 287\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 287\relax}}}_{x}\}_{x\in\mathfrak{L}} be the family of –automorphisms of uniquely defined by the conditions
[TABLE]
An interaction is a family of even and self–adjoint local elements with . We define Banach spaces of short–range interactions by introducing norms that take into account space decay of interactions. To this end, we use positive–valued and non–increasing decay functions . Like in [BP1], we impose the following conditions on :
- •
Summability on .
[TABLE]
- •
Bounded convolution constant.
[TABLE]
Examples of functions satisfying (14)–(15) are given by
[TABLE]
for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 294\relax}}{\mbox{\boldmath\textstyle\mathchar 294\relax}}{\mbox{\boldmath\scriptstyle\mathchar 294\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 294\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 271\relax}}{\mbox{\boldmath\textstyle\mathchar 271\relax}}{\mbox{\boldmath\scriptstyle\mathchar 271\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 271\relax}}}\in\mathbb{R}^{+}. In all the paper, (14)–(15) are assumed to be satisfied.
Then, the norm of any interaction is defined by
[TABLE]
The real separable Banach space is the space of interactions with . Elements are named short–range interactions.
3.3 Interacting Fermion Systems in Disordered Media
To any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\Omega and strength {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}\in\mathbb{R}_{0}^{+} of hopping disorder, we associate a short–range interaction \Psi^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}})}\in\mathcal{W} defined as follows: Fix an interparticle (IP) interaction . Then,
[TABLE]
whenever for , and \Psi_{\Lambda}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}})}:=\Psi_{\Lambda}^{\mathrm{IP}} otherwise.
Let
[TABLE]
We then assume two additional properties of :
- •
Translation invariance. For all ,
[TABLE]
- •
Polynomial decay. There is a constant {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 294\relax}}{\mbox{\boldmath\textstyle\mathchar 294\relax}}{\mbox{\boldmath\scriptstyle\mathchar 294\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 294\relax}}}>2d and, for all , an absolutely summable sequence such that, for all with ,
[TABLE]
Examples of functions satisfying (14)–(15) and (20) are obviously given by (16), for sufficiently large {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 271\relax}}{\mbox{\boldmath\textstyle\mathchar 271\relax}}{\mbox{\boldmath\scriptstyle\mathchar 271\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 271\relax}}}\in\mathbb{R}^{+} in the polynomial case.
Conditions (14)–(15) and [BP1, Theorem 2.2] ensure the existence of a (non–autonomous) infinite volume dynamics \{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}_{t,s}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}},\mathbf{A})}\}_{s,t\in{\mathbb{R}}}, in presence of electromagnetic fields and static potentials (cf. (12)). Indeed, any realization {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\Omega, disorder strengths {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}, interparticle interaction and electromagnetic potential naturally define a family \{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 270\relax}}{\mbox{\boldmath\textstyle\mathchar 270\relax}}{\mbox{\boldmath\scriptstyle\mathchar 270\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 270\relax}}}_{t}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}},\mathbf{A})}\}_{t\in{\mathbb{R}}} of derivations on the subset of local elements of . Then, \{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}_{t,s}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}},\mathbf{A})}\}_{s,t\in{\mathbb{R}}} is the unique strongly continuous two–parameter family of –automorphisms of satisfying, in the strong sense on the dense domain ,
[TABLE]
See [BP1, Section 2.5] for more details. At , the (unperturbed) dynamics is autonomous and we denote the corresponding group of –automorphisms by
[TABLE]
Then, as explained in Section 2, thermal equilibrium states are defined to be completely passive states, see Definition 2.2. This definition is based on the 2nd law of thermodynamics. By Theorem 2.3, they are ({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}})},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}})–KMS states for some inverse temperature, or time scale, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\left[0,\infty\right]. For simplicity, we exclude the boundary cases {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}=0,+\infty. As discussed in [BP1, Section 2.6], the set of ({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}})},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}})–KMS states is non–empty for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+}, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\Omega and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}. Here, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 293\relax}}{\mbox{\boldmath\textstyle\mathchar 293\relax}}{\mbox{\boldmath\scriptstyle\mathchar 293\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 293\relax}}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}})} denotes one element of this set.
For any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+} and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}, we impose two natural conditions on the map
[TABLE]
from the set to the dual space :
- •
Translation invariance. Recall that \{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 287\relax}}{\mbox{\boldmath\textstyle\mathchar 287\relax}}{\mbox{\boldmath\scriptstyle\mathchar 287\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 287\relax}}}_{x}\}_{x\in\mathfrak{L}} is the family of –automorphisms of uniquely defined by (13). It implements the action of the group of lattice translations on the CAR –algebra . On the set this action is represented by the family \{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 287\relax}}{\mbox{\boldmath\textstyle\mathchar 287\relax}}{\mbox{\boldmath\scriptstyle\mathchar 287\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 287\relax}}}_{x}^{(\Omega)}\}_{x\in\mathfrak{L}}, see (4)–(5). Then, we assume that
[TABLE]
- •
Measurability. Thermal equilibrium states are supposed to be random variables. Hence, for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+} and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}, we assume that the map (22) is measurable w.r.t. to the –algebra on and the Borel –algebra of generated by the weak*∗*–topology. Observe that a similar assumption is also necessary for classical disordered systems at equilibrium, see, e.g., [Bo].
These conditions yield the following definition:
Definition 3.2** (Random invariant states)**
**
Let {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\mapsto{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 293\relax}}{\mbox{\boldmath\textstyle\mathchar 293\relax}}{\mbox{\boldmath\scriptstyle\mathchar 293\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 293\relax}}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}})} be a map from to the set of states on . We say that this map is a random invariant state when it is measurable w.r.t. to and and translation invariant in the above sense.
The map (22) is thus a random invariant state. This implies in particular that, for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+} and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}, the averaged state \bar{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 293\relax}}{\mbox{\boldmath\textstyle\mathchar 293\relax}}{\mbox{\boldmath\scriptstyle\mathchar 293\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 293\relax}}}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}})}\in\mathcal{U}^{\ast} defined by
[TABLE]
is translation invariant, i.e.,
[TABLE]
Recall indeed that is also a translation invariant probability measure. [It is even ergodic.]
The existence of such random invariant equilibrium states is not completely clear in general, similar to the classical case. If the ({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}})},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}})–KMS state is unique and (19) is satisfied, then it turns out that the (unique) map (22) is a random invariant state. Indeed, in this case, the map (22) is even continuous w.r.t. the pointwise convergence in and the weak*∗*–topology of . This can be proven by using [BR2, Proposition 5.3.23.]. Uniqueness of KMS states appears for instance when either \Psi^{\mathrm{IP}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}=0, or at small {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+}, or in dimension . Moreover, by using methods of constructive quantum field theory, one can also verify the existence of such random invariant thermal equilibrium states at arbitrary {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+} and dimension if the interparticle interaction is small enough and (19) holds.
Now, in presence of electromagnetic fields, the time evolution of the state of the system equals
[TABLE]
for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+}, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\Omega, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+} and . Recall here that for all .
Remark 3.3** (Time–dependent states as stochastic processes)**
Under the above assumptions, by using Lieb–Robinson bounds as in [BP3, Lemma 4.3], it is possible to show that the family \{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\mapsto{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}_{t}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}},\mathbf{A})}\}_{t\in\mathbb{R}} defines a stochastic process with values in . More precisely, for any , the map {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\mapsto{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}_{t}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}},\mathbf{A})} is measurable w.r.t. to and . This fact is not essential in the sequel.
4 Macroscopic Ohm’s Law and Green–Kubo Relations
4.1 Macroscopic Charge Transport Coefficients
Fix {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\Omega, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}\in\mathbb{R}_{0}^{+}, and time . For any oriented bond , we define the paramagnetic and diamagnetic current observables I_{\mathbf{x}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}})} and \mathrm{I}_{\mathbf{x}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},\mathbf{A})} respectively by
[TABLE]
and
[TABLE]
If the interparticle interaction is locally gauge invariant, that is, for all ,
[TABLE]
then, in absence of external electromagnetic potentials, I_{\mathbf{x}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}})} is the observable related to the flow of particles from the lattice site to the lattice site or the current from to . \mathrm{I}_{\mathbf{x}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},\mathbf{A})} corresponds to a correction, engendered by the presence of an external electromagnetic potential, to the current I_{\mathbf{x}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}})}. See [BP1, Section 3.2]. Let
[TABLE]
Now, for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+}, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\Omega and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}, we define two important functions associated with these observables:
- (p)
The paramagnetic transport coefficient {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 283\relax}}{\mbox{\boldmath\textstyle\mathchar 283\relax}}{\mbox{\boldmath\scriptstyle\mathchar 283\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 283\relax}}}_{\mathrm{p}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}})}\equiv{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 283\relax}}{\mbox{\boldmath\textstyle\mathchar 283\relax}}{\mbox{\boldmath\scriptstyle\mathchar 283\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 283\relax}}}_{\mathrm{p}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}})} is defined, for any and , by
[TABLE]
- (d)
The diamagnetic transport coefficient {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 283\relax}}{\mbox{\boldmath\textstyle\mathchar 283\relax}}{\mbox{\boldmath\scriptstyle\mathchar 283\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 283\relax}}}_{\mathrm{d}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}})}\equiv{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 283\relax}}{\mbox{\boldmath\textstyle\mathchar 283\relax}}{\mbox{\boldmath\scriptstyle\mathchar 283\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 283\relax}}}_{\mathrm{d}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}})} is defined by
[TABLE]
For boxes (18), we then define the space–averaged paramagnetic transport coefficient
[TABLE]
w.r.t. the canonical orthonormal basis of the Euclidian space by
[TABLE]
for any l,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+}, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\Omega, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}, and . See [BP1, Theorem 3.4, Corollary 3.5] for details on the properties of \Xi_{\mathrm{p},l}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}})}. The space–averaged diamagnetic transport coefficient
[TABLE]
corresponds (w.r.t. ) to the diagonal matrix defined by
[TABLE]
Both random coefficients turn out to be the paramagnetic and diamagnetic (in–phase) conductivities.
We define the deterministic paramagnetic transport coefficient
[TABLE]
by
[TABLE]
for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+}, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}, and . It is well–defined, by Theorem 7.1. Furthermore, the convergence is uniform for times in compact sets. Analogously, we also introduce the deterministic diamagnetic transport coefficient
[TABLE]
defined, for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+} and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}, by
[TABLE]
Indeed, since the map (22) is a random invariant state and is an ergodic measure, we have, for all ,
[TABLE]
Clearly, \{\mathbf{\Xi}_{\mathrm{d}}\}_{k,k}\in[-2({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}+1),2({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}+1)] for any .
By using the Akcoglu–Krengel ergodic theorem we show that the limits of \Xi_{\mathrm{p},l}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}})} and \Xi_{\mathrm{d},l}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}})} converge almost surely to and , respectively.
Theorem 4.1** (Macroscopic charge transport coefficients)**
*Assume (14)–(15), (19) and that the map (22) is a random invariant state (see Definition 3.2). Let {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+} and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}. Then, there is a measurable subset \tilde{\Omega}\equiv\tilde{\Omega}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}})}\subset\Omega of full measure (that is, and ) such that, for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\tilde{\Omega}, one has:
(p) Paramagnetic transport coefficient: For all ,*
[TABLE]
*The above limit is uniform for times on compact sets.
(d) Diamagnetic transport coefficient:*
[TABLE]
Proof: Assertion (p) is proven in a similar way as Theorem 7.9. See Equation (102) and the arguments thereafter. Note only that the pointwise convergence of any equicontinuous family of functions on implies its uniform convergence on compacta. The proof of Assertion (d) is even simpler because there is no time dependency. We omit the details.
4.2 Macroscopic Ohm’s Law
For any and , we consider now the space–rescaled vector potential
[TABLE]
Since Ohm’s law is a linear* *response to electric fields, we also rescale the strength of the electromagnetic potential by a real parameter {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}}\in\mathbb{R} and study the behavior of current densities in the limit {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}}\rightarrow 0.
Exactly like in [BPH2, Section 3] and [BP1, Section 3.3], w.l.o.g. we consider space–homogeneous (though time–dependent) electric fields in the box defined by (18) for . More precisely, let be any (normalized w.r.t. the usual Euclidian norm) vector, and set for all . Then, is defined to be the electromagnetic potential such that the electric field equals at time for all and for and . This choice yields rescaled electromagnetic potentials {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}}\mathbf{\bar{A}}_{l} as defined by (36) for and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}}\in\mathbb{R}.
For any l,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+}, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\Omega, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}}\in\mathbb{R}, , and , the total current density is the sum of three currents defined from (27) and (28):
- (th)
The (thermal) current density
[TABLE]
at thermal equilibrium inside the box is defined, for any , by
[TABLE]
- (p)
The paramagnetic current density is the map
[TABLE]
defined by the space average of the current increment vector inside the box at time , that is, for any ,
[TABLE]
- (d)
The diamagnetic (or ballistic) current density
[TABLE]
is defined analogously, for any and , by
[TABLE]
For more details on the physical interpretation of these currents, see [BPH2, Section 3.4].
By [BP1, Theorem 3.7] and Conditions (14)–(15) and (20), the current densities behave, at small |{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}}| and uniformly w.r.t. the size of the box, linearly w.r.t. : For any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}_{0}\in\mathbb{R}_{0}^{+}, and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}}\in\mathbb{R},
[TABLE]
uniformly for l,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+}, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\Omega, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}\in[0,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}_{0}], {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}, (normalized) and .
The –valued linear coefficients
[TABLE]
of the paramagnetic and diamagnetic current densities, respectively, become deterministic for large boxes. They are directly related to and via Ohm’s law:
Theorem 4.2** (Macroscopic Ohm’s law)**
*Assume (14)–(15), (19)–(20) and that the map (22) is a random invariant state. Let {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+} and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}. Then, there is a measurable subset \tilde{\Omega}\equiv\tilde{\Omega}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}})}\subset\Omega of full measure such that, for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\tilde{\Omega}, , and , the following assertions hold true:
(th) Thermal current density:*
[TABLE]
(p)* Paramagnetic current density:*
[TABLE]
(d)* Diamagnetic current density:*
[TABLE]
Proof: (th) is similar to [BPH3, Corollary 5.7 (th)]. Assertions (p) and (d) are deduced from Theorem 4.1 and Lebesgue’s dominated convergence theorem. Note that the intersection of three measurable sets of full measure has full measure.
Like [BP1, Theorem 3.7], Theorem 4.2 can also be extended to space–inhomogeneous macroscopic electromagnetic fields, that is, for space–rescaled vector potentials (36) with arbitrary .
4.3 Green–Kubo Relations
Because of Theorem 4.2 (p)–(d), and are both charge transport coefficients. Thus, they are also named here paramagnetic and diamagnetic (in–phase) conductivities, respectively. From (34) we can deduce Green–Kubo relations for via current Duhamel fluctuations as follows.
Fix in all the subsection {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+} and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}. The Duhamel two–point function (\cdot,\cdot)_{\sim}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}})} is defined by
[TABLE]
for any and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\Omega. See for instance [BPH2, Section A] and references therein for more details. For any and , set
[TABLE]
We name it the *fluctuation observable *of the element in the box . Recall that \{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 287\relax}}{\mbox{\boldmath\textstyle\mathchar 287\relax}}{\mbox{\boldmath\scriptstyle\mathchar 287\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 287\relax}}}_{x}\}_{x\in\mathfrak{L}} implements the action of the group of lattice translations on the CAR –algebra , see (13).
Then, by [BPH2, Eq. (103)] together with (34), one obtains Green–Kubo relations for the paramagnetic (in–phase) conductivity: For any and ,
[TABLE]
with the current observable I_{(x,y)}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}})} defined by (27). The right hand side (r.h.s.) of the above equation is a current Duhamel fluctuation increment. If Conditions (14)–(15) and (19) hold and the map (22) is a random invariant state, then the above limit always exists (and is thus finite), by Theorem 7.1.
Note however that, possibly,
[TABLE]
for some . In other words, it is not a priori clear whether the interacting quantum system has finite current Duhamel fluctuations or not. When it is finite (and so are both terms in the r.h.s. of (41)), similar to [BPH4, Section 3], we can construct a Hilbert space of fluctuations, which implies the existence of a finite conductivity measure as a spectral measure. The finiteness of current Duhamel fluctuations is proven in [BPH4, Section 3] for the non–interacting case with random static potentials and space–homogeneous hopping terms. This can also be shown for sufficiently small and disorder strengths {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}, by using methods of constructive quantum field theory.
5 AC–Conductivity Measure From Joule’s Law
Similar to [BPH3, Section 4.3], our derivation of a macroscopic (in–phase) AC–conductivity measure is based on the 2nd law of thermodynamics. It dovetails with the celebrated Joule’s law of (classical) electricity theory. To this end we start by introducing energy increment densities, in particular the heat production density.
5.1 Energy Increment Densities
The internal energy observable H_{L}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}})}\in\mathcal{U}^{+}\cap\mathcal{U}_{\Lambda} of the interacting fermion system for the box (18) is defined by
[TABLE]
for {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}=({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}_{1},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}_{2})\in\Omega, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+} and . When the electromagnetic field is switched on, i.e., for , the total energy observable for the box that includes the region where the electromagnetic field does not vanish equals
[TABLE]
where, for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\Omega, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}\in\mathbb{R}_{0}^{+}, and ,
[TABLE]
is the electromagnetic* *potential energy observable.
Like in [BP1, Sections 3.1, 3.4], we now define four sorts of energy increments associated with the fermion system for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+}, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\Omega, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+} and :
- ()
The internal energy increment \mathbf{S}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},\mathbf{A})}\equiv\mathbf{S}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}},\mathbf{A})} is the map from to defined by
[TABLE]
Under Conditions (14)–(15) and (20), this map has non–negative finite value and is the heat production because of [BP1, Theorem 3.2].
- ()
The electromagnetic potential energy increment \mathbf{P}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},\mathbf{A})}\equiv\mathbf{P}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}},\mathbf{A})} is the map from to defined by
[TABLE]
- (p)
The paramagnetic energy increment \mathfrak{J}_{\mathrm{p}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},\mathbf{A})}\equiv\mathfrak{I}_{\mathrm{p}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}},\mathbf{A})} is the map from to defined by
[TABLE]
- (d)
The diamagnetic energy increment \mathfrak{I}_{\mathrm{d}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},\mathbf{A})}\equiv\mathfrak{I}_{\mathrm{d}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}},\mathbf{A})} is the map from to defined by
[TABLE]
See [BPH2] for more discussions on the physical interpretation of these energies. Note that the limits described in () and (p) exist at all times. Indeed, the total energy increment
[TABLE]
is shown in [BP1, Theorem 3.2 (ii)] to be the work performed by the electric field and is given in the limit by an expression like (1), which, on the other hand, equals
[TABLE]
Under Conditions (14)–(15) and (20), all increment energies defined above are of order \mathcal{O}\left({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}}^{2}l^{d}\right), as , by [BP1, Theorem 3.8]. Indeed, because of possibly non–vanishing thermal currents, the energy increments \mathbf{P}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},\mathbf{A})} and \mathfrak{I}_{\mathrm{d}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},\mathbf{A})} are rather \mathcal{O}\left(\left|{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}}\right|l^{d}\right) at small . As a consequence, for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+}, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\Omega, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+} and , we define four energy densities:
- ()
The heat production (or internal energy increment) density \mathbf{s}\equiv\mathbf{s}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}},\mathbf{A})} is the map from to defined by
[TABLE]
- ()
The (electromagnetic) potential energy (increment) density \mathbf{p}\equiv\mathbf{p}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}},\mathbf{A})} is the map from to defined by
[TABLE]
- (p)
The paramagnetic energy (increment) density \mathfrak{i}_{\mathrm{p}}\equiv\mathfrak{i}_{\mathrm{p}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}},\mathbf{A})} is the map from to defined by
[TABLE]
- (d)
The diamagnetic energy (increment) density \mathfrak{i}_{\mathrm{d}}\equiv\mathfrak{i}_{\mathrm{d}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}},\mathbf{A})} the map from to defined by
[TABLE]
On a measurable subset of full measure, all energy (increment) densities become deterministic functions that are derived in the next subsection. We explain this in the next subsection.
5.2 Macroscopic Joule’s Law
Similar to the heuristics presented in [BPH3, Section 4.2], we expect from Theorem 4.2 that, for {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+}, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+} and any (possibly space inhomogeneous) electromagnetic potential , the electric field yields space–dependent paramagnetic and diamagnetic current linear response coefficients respectively equal to
[TABLE]
at any position and time . These current linear response coefficients yield two electric work or energy densities produced by the paramagnetic and diamagnetic currents. This fact is proven in the following theorem:
Theorem 5.1** (Macroscopic Joule’s law)**
*Assume (14)–(15), (19)–(20) and that the map (22) is a random invariant state. Let {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+} and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}. Then, there is a measurable subset \tilde{\Omega}\equiv\tilde{\Omega}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}})}\subset\Omega of full measure such that, for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\tilde{\Omega}, and :
(p) Paramagnetic energy density:*
[TABLE]
(d)* Diamagnetic energy density:*
[TABLE]
(Q)* Heat production density:*
[TABLE]
(P)* Electromagnetic potential energy density:*
[TABLE]
Proof: The proof is very similar to the proof of [BPH3, Theorem 4.1]. It is a consequence of the Akcoglu–Krengel ergodic theorem, Lieb–Robinson bounds [BP3, Theorem 3.6 (iv)] and [BP1, Theorem 3.8]. For the detailed proof of Assertion (p), see Theorem 7.9. We omit the details for Assertions (d), (Q) and (P).
For more discussions on this subject, see [BPH3, Section 4.2]. In fact, the above result is an extension of [BPH3, Theorem 4.1] to fermion systems with interactions.
5.3 AC–Conductivity Measure
At {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+} and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}, the paramagnetic transport coefficient \mathbf{\Xi}_{\mathrm{p}}\equiv\mathbf{\Xi}_{\mathrm{p}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}})} is a well–defined –valued function of time. See (34). It is also named here paramagnetic (in–phase) conductivity, because of Theorem 4.2.
The positivity of the heat production (Theorem 5.1), i.e., the 2nd law of thermodynamics, implies that the symmetric part of is conditionally positive definite or, equivalently [SSV, Proposition 4.4], negative definite in the sense of Schoenberg. Observe that the symmetric part of is only conditionally positive definite, and not positive definite, because of the AC–condition (9) on external electric fields. Therefore, similar to [SSV, Theorem 4.12] for complex–valued negative definite functions (in the sense of Schoenberg), there is a Lévy–Khintchine representation of the symmetric part of the (continuous) paramagnetic (in–phase) conductivity . The corresponding Lévy measure {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}_{\mathrm{AC}} is the AC–conductivity measure we are looking for. Note that the measure {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}^{2}{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}_{\mathrm{AC}}\left(\mathrm{d}{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}\right) on is a priori not a finite measure. However, if Conditions (14)–(15) and (19)–(20) hold and the map (22) is a random invariant state, then such a property holds true because , by Theorem 7.1.
Indeed, for any , define its symmetric and antisymmetric parts, w.r.t. to the canonical scalar product of , respectively by
[TABLE]
Here, stands for the transpose of the operator (w.r.t. the canonical scalar product of ). Then we have:
Theorem 5.2** (Lévy–Khintchine representation of )**
Assume (14)–(15), (19)–(20) and that the map (22) is a random invariant state. For any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+} and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}, there is a unique finite and symmetric –valued measure \mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}}\equiv\mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}})} on such that, for any ,
[TABLE]
* stands for the set of positive linear operators on , i.e., symmetric operators w.r.t. to the canonical scalar product of with positive eigenvalues.*
Proof: For all {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 295\relax}}{\mbox{\boldmath\textstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 295\relax}}}\in C_{0}^{\infty}(\mathbb{R};\mathbb{R}^{d}), observe that its derivative {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 295\relax}}{\mbox{\boldmath\textstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 295\relax}}}^{\prime}\in C_{0}^{\infty}(\mathbb{R};\mathbb{R}^{d}) satisfies
[TABLE]
As a consequence, we infer from Theorem 5.1 (p) and the equality
[TABLE]
that, for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 295\relax}}{\mbox{\boldmath\textstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 295\relax}}}\in C_{0}^{\infty}(\mathbb{R};\mathbb{R}^{d}),
[TABLE]
Note that (52) is a simple consequence of the stationarity of KMS states. By Theorem 7.1, if (14)–(15) and (19)–(20) hold and the map (22) is a random invariant state, then . By integration by parts, it follows from (53) that
[TABLE]
[TABLE]
for any . Therefore, is a weakly positive definite continuous map that is symmetric w.r.t. time reversal. Moreover, for any , is (by definition) a symmetric operator w.r.t. the canonical scalar product of . Then, we can apply Corollary 7.11 with to deduce the existence of a unique finite and symmetric –valued measure on such that
[TABLE]
Observe that . Therefore, by integrating this last expression twice, we then obtain that
[TABLE]
Since is a a finite measure on , we can apply twice the Fubini (–Tonelli) theorem to deduce that
[TABLE]
All integrals are of course well–defined because \sin\left({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}\right)=\mathcal{O}({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}) and 1-\cos\left({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}\right)=\mathcal{O}({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}^{2}), as {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}\rightarrow 0.
From Theorem 5.1 it is easy to see that the restriction of {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}^{-2}\mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}}\left(\mathrm{d}{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}\right) on quantifies the heat production per unit volume due to the component of frequency {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}\in\mathbb{R}\backslash\{0\} of the electric field in accordance with Joule’s law in the AC–regime. By (53), note at this point that the antisymmetric component of the paramagnetic conductivity does not contribute to heat production. Therefore, we define this measure to be the (in–phase) AC–conductivity measure:
Definition 5.3** (AC–conductivity measure)**
**
We name {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}_{\mathrm{AC}}\equiv{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}_{\mathrm{AC}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}})}, the restriction of {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}^{-2}\mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}}\left(\mathrm{d}{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}\right) to , the (in–phase) *AC–conductivity measure. *
Remark 5.4** (AC–conductivity measure from the 2nd law)**
AC–Conductivity measures are obtained here for thermal equilibrium states at strictly positive temperatures, that are, ({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}})},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}})–KMS states with {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in(0,\infty). See Theorem 2.3 and Section 3.3. The use of KMS states is however not strictly necessary to get such measures: Theorem 5.2 also holds for passive states {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 293\relax}}{\mbox{\boldmath\textstyle\mathchar 293\relax}}{\mbox{\boldmath\scriptstyle\mathchar 293\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 293\relax}}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}})}, provided the map {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\mapsto{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 293\relax}}{\mbox{\boldmath\textstyle\mathchar 293\relax}}{\mbox{\boldmath\scriptstyle\mathchar 293\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 293\relax}}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}})} is a random invariant state (Definition 3.2). In other words, AC–conductivity measures result from the 2nd law, only. This will be discussed in more detail in a review article in preparation. In fact, in the present paper, we have only considered KMS states to stick to [BP1] where heat productions \mathbf{Q}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},\mathbf{A})} are considered and known to be well–defined for KMS states, see [BP1, Definition 3.1, Theorem 3.2].
The AC–conductivity measure does not vanish in general, see, e.g., [BPH4, Theorem 4.7]. Moreover, in the non–interacting case, we show in [BPH4, Theorem 4.1] that \mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}}\left(\left\{0\right\}\right)=0 and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}_{\mathrm{AC}} is a finite measure on :
[TABLE]
In particular, the measure \mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}}([-{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}]) is \mathcal{O}({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}^{2}) in the limit {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}\rightarrow 0^{+}. These properties are directly related with the finiteness of current Duhamel fluctuations in the limit of large space scales, which is not clear in presence of interactions, see (42) and discussion thereafter.
At high frequencies, by finiteness of the positive measure , the AC–conductivity measure satisfies
[TABLE]
The same property of course holds for negative frequencies, by symmetry of (w.r.t. ). We can compare this property with the corresponding one of the celebrated Drude model.
Indeed, the (in–phase) AC–conductivity measure obtained from the Drude model is absolutely continuous w.r.t. the Lebesgue measure with the function
[TABLE]
being the corresponding Radon–Nikodym derivative. Here, the relaxation time is related to the mean time interval between two collisions of a charged carrier with defects in the crystal. This function is the Fourier transform of the in–phase conductivity
[TABLE]
where is some strictly positive constant. See for instance [BPH4, Section 1] for more discussions.
At high frequencies, Drude’s approach heavily overestimates the AC–conductivity measure {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}_{\mathrm{AC}} obtained from the more realistic model studied here. Indeed, we can infer from (56) that, in the limit {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}\rightarrow\infty of high frequencies,
[TABLE]
whereas, by (57), the corresponding quantity for the Drude model diverges in the same limit:
[TABLE]
The same behavior as for the Drude model holds for the AC–conductivity measure obtained from the Lorentz–Drude model.
Hence, the asymptotics (58) motivates the use of the relaxation time as an effective –dependent parameter of the Drude model, i.e., one replaces with \mathrm{T}({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}) in (57), as observed for instance in [T]. Indeed, with this Ansatz and the asymptotics (58), either \mathrm{T}({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}) vanishes faster than {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}^{-3} or it diverges faster than , as {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}\rightarrow\infty. Note that experimental measurements seem to indicate that
[TABLE]
in some metals. See for instance [T] for one experimental evidence of this fact and [NS1, NS2, SE, YRMK] for theoretical studies.
The concept of relaxation time or mean free path [So] (of electrons) in the Drude model and its extensions is very intuitive. However, the microscopic interpretation of this classical notion is difficult, in particular if one has to take as a –dependent parameter. Quoting meanwhile [LTW, p. 24]:
Physicists had to wait for the discovery of quantum mechanics to understand why electrons apparently do not scatter from ions that occupy regular lattice sites: The wave character of an electron causes the electron to diffract from an ideal crystal. Resistance appears only when electrons scatter from imperfections in the crystal. With that quantum mechanical revision, the Drude model can still be used, but in the new picture an electron is envisaged as zigzagging between impurities.
Indeed, the average length an electron travels before it seems to collide with an ion or defects in the crystal is experimentally measured in metals to be about two order of magnitude larger than the lattice constant. [Note however that defects in our model are allowed to appear on all lattice sites via the probability measure , see Section 3.1.]
The high frequency asymptotics of the (in–phase) AC–conductivity discussed above makes explicit further problems with this classical picture. Observe moreover that if the interparticle interaction has stronger polynomial decay than in the assumptions of Theorem 5.2, then the asymptotics (58) can be improved by replacing {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}^{2} with {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}^{k} for an integer . To show this, one uses Lieb–Robinson bounds for multi–commutators [BP3, Theorems 3.8–3.9] of order to get . See also Remark 7.2. However, we expect the model to physically break down for frequencies corresponding to wavelengths (of light) of the order of the lattice spacing. For usual materials, it would dovetail with the frequency range of hard X–rays.
Similar to [BP1, Corollary 3.5], we deduce now general properties of the paramagnetic conductivity from Theorem 5.2:
Corollary 5.5** (Properties of )**
*Assume all conditions of Theorem 5.2 and let {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+} and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}. Then, and the following holds:
(i) Time–reversal symmetry of : and*
[TABLE]
(ii)* Negativity of :*
[TABLE]
(iii)* Cesàro mean of : If \mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}}\left(\left\{0\right\}\right)=0 and \|{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}_{\mathrm{AC}}\|_{\mathcal{B}(\mathbb{R}^{d})}\left(\mathbb{R}\backslash\{0\}\right)<\infty then*
[TABLE]
Proof: (i)–(iii) are direct consequences of Theorem 5.2, the Fubini (–Tonelli) theorem and Lebesgue’s dominated convergence theorem.
Assuming (14)–(15), note that, for any l,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+}, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\Omega and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}, there exists555{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}_{\mathrm{p},l}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}})} is a finite measure because we take KMS states. For passive states, we only have the existence of finite volume AC–conductivity measures, similar to Theorem 5.2 and Definition 5.3 for . a (generally non–zero) symmetric and finite –valued measure {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}_{\mathrm{p},l}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}})}\equiv{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}_{\mathrm{p},l}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}})} on such that
[TABLE]
Away from {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}=0 and as the finite microscopic conductivity measure {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}_{\mathrm{p},l}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}})} converges in the weak*∗*–topology to the macroscopic AC–conductivity measure {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}_{\mathrm{AC}}:
Theorem 5.6** (From microscopic to macroscopic AC–conductivity measures)**
*Assume Conditions (14)–(15), (19), (20) with {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 294\relax}}{\mbox{\boldmath\textstyle\mathchar 294\relax}}{\mbox{\boldmath\scriptstyle\mathchar 294\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 294\relax}}}>3d, and that the map (22) is a random invariant state. Let {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+} and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}. Then, and there is a measurable subset \tilde{\Omega}\equiv\tilde{\Omega}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}})}\subset\Omega of full measure such that, for all {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\tilde{\Omega}:
(i) Tightness: The sequence \{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}_{\mathrm{p},l}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}})}\}_{l\in\mathbb{R}^{2}} of finite measures is tight.
(ii) Weak∗–convergence away from {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}=0: For any and any bounded continuous function on with ,*
[TABLE]
Proof: Fix {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+} and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}. Under assumptions of the theorem, and there is a measurable subset \tilde{\Omega}\equiv\tilde{\Omega}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}})}\subset\Omega of full measure such that
[TABLE]
The proof is omitted as the arguments are very similar to those proving Theorem 4.1 (p). Note only that Condition (20) with {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 294\relax}}{\mbox{\boldmath\textstyle\mathchar 294\relax}}{\mbox{\boldmath\scriptstyle\mathchar 294\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 294\relax}}}>3d is imposed to obtain Lieb–Robinson bounds for multi–commutators [BP3, Theorems 3.8–3.9] of order four. This is needed to obtain the equicontinuity of the family
[TABLE]
of functions of time. See for instance Remark 7.2, the proofs of Theorems 7.1 and 7.9.
Meanwhile, for , we apply twice the Fubini (–Tonelli) theorem to deduce that
[TABLE]
with \mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}}_{l}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}})}:={{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 279\relax}}{\mbox{\boldmath\textstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptstyle\mathchar 279\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 279\relax}}}^{2}{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}_{\mathrm{p},l}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}})}. Observe from (54) and (60)–(61) that \mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}}_{l}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}})} is a finite measure and
[TABLE]
Now, take any vector . Let \mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}}_{l,\vec{w}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}})} and \mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}}_{\vec{w}} be the measures on respectively defined, for any Borel set , by
[TABLE]
Assume w.l.o.g. that \mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}}_{\vec{w}}(\mathbb{R})>0. Then, by combining (54) and (59)–(62) with (Theorem 7.1) and [D, Theorems 3.2.3 and 3.3.6], we deduce that, on the subset of full measure, the sequence \{\mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}}_{l,\vec{w}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}})}\}_{l\in\mathbb{R}^{2}} is tight and converges in the weak*∗*–topology to \mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}}_{\vec{w}}, as . By Definition 5.3, this implies Assertion (ii) for . Its extension to arbitrary vectors is a consequence of the polarization identity, see, e.g., (107). Assertion (i) easily follows from the tightness of \{\mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}}_{l,\vec{w}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}})}\}_{l\in\mathbb{R}^{2}} for and the polarization identity.
5.4 Time–Reversal Invariance of Random Equilibrium States
In this subsection we define time–reversal invariance of random fermion systems and derive its consequences on conductivity. We do not define this symmetry of random systems as the almost surely time–reversal invariance. But instead, we give a weaker, and hence more general, notion of “time–reversal invariance in average”. This is done in the same spirit of what we do above to introduce translation invariance for disordered systems at thermal equilibrium. See, for instance, Definition 3.2. In fact, by doing this, we allow for a large class of random magnetic potentials.
Let be a –algebra with unity and assume the existence of a map with the following properties:
- •
is antilinear and continuous.
- •
and .
- •
for all .
- •
for all .
Such a map is called a time–reversal operation of the –algebra . For (CAR –algebra of the lattice ), there is a natural time–reversal operation , which is uniquely defined by the condition
[TABLE]
See also [BPH2, Section 2.1.4].
For any strongly continuous one–parameter group {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}:=\{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}_{t}\}_{t\in{\mathbb{R}}} of –automorphisms of , the family {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}^{\Theta}:=\{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}_{t}^{\Theta}\}_{t\in{\mathbb{R}}} defined by
[TABLE]
is again a strongly continuous one–parameter group of automorphisms. Similarly, for any state {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}\in\mathcal{X}^{\ast}, the linear functional {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}^{\Theta} defined by
[TABLE]
is again a state. We say that and are time–reversal invariant w.r.t. if they satisfy {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}_{t}^{\Theta}={{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}_{-t} for all and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}^{\Theta}={{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 282\relax}}{\mbox{\boldmath\textstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptstyle\mathchar 282\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 282\relax}}}. If is time–reversal invariant then, for all {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+}, there is at least one time–reversal invariant ({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}})–KMS state {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 293\relax}}{\mbox{\boldmath\textstyle\mathchar 293\relax}}{\mbox{\boldmath\scriptstyle\mathchar 293\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 293\relax}}}\in\mathcal{X}^{\ast}, provided the set of ({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}})–KMS states is not empty. This follows from the convexity of the set of KMS states, see [BPH2, Lemma A.12].
Now, we introduce a notion of time–reversal invariance for the random system considered here. If is an interaction, we call it time–reversal invariant whenever
[TABLE]
For any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}=({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}_{1},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}_{2})\in\Omega, we define \overline{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}}:=({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}_{1},\overline{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}_{2}})\in\Omega, where
[TABLE]
We say that the random state (22) is time–reversal symmetric if, for all {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\Omega,
[TABLE]
Similarly, we call the random dynamic (21) on time–reversal symmetric if, for all {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\Omega,
[TABLE]
It is not difficult to see that, if the interparticle interaction is time–reversal invariant then the (unperturbed) random dynamics {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}})} is time–reversal symmetric in the above sense for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}. Further, we say that the –valued random variable , the distribution of which is given by the probability space , is time–reversal invariant if the map {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\mapsto\overline{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}} is measurable w.r.t. and preserves the measure .
Like in the case of translation invariance, the existence of random invariant thermal equilibrium states which are time–reversal symmetric in the above sense is not clear in general. If the ({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}})},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}})–KMS state is unique and is time–reversal invariant, then the (unique) map (22) is a random state which is time–reversal symmetric. The arguments to prove this are similar to the ones used in the proof of [BPH2, Lemma A.12]. As already discussed, if (19) holds then (22) is, moreover, a random invariant state. See Section 3.3.
Time–reversal invariance implies the following important properties of charge transport coefficients related to the models considered here:
Theorem 5.7** (Consequences of time–reversal symmetry)**
*Assume (14)–(15), (19)–(20), time–reversal invariance of the interparticle interaction and the (–valued) random variable , as well as that the map (22) is a random invariant state which is time–reversal symmetric. Let {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+} and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}. Then, the following assertions hold true:
(th) Vanishing thermal current density:*
[TABLE]
(p)* Vanishing antisymmetric part of the paramagnetic conductivity:*
[TABLE]
Proof: (th)* *directly follows form Theorem 4.2 (th), the equality \mathfrak{T}(I_{\left(e_{k},0\right)}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}})})=-I_{\left(e_{k},0\right)}^{(\overline{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}})}, which is a consequence of (63), {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 293\relax}}{\mbox{\boldmath\textstyle\mathchar 293\relax}}{\mbox{\boldmath\scriptstyle\mathchar 293\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 293\relax}}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}})}(I_{\left(e_{k},0\right)}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}})})\in\mathbb{R}, the time–reversal invariance of the random variable and the time–reversal symmetry of the random state {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 293\relax}}{\mbox{\boldmath\textstyle\mathchar 293\relax}}{\mbox{\boldmath\scriptstyle\mathchar 293\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 293\relax}}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}})}. These facts combined with the time–reversal symmetry of {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}})}, which follows from the assumptions on , and the stationarity of KMS states imply (p).
6 Epilogue: AC–Conductivity and Lévy Processes
By Theorem 5.2, charge transport properties of interacting fermions in disordered media are governed by a Lévy measure. This suggests an alternative effective description of the phenomenon of linear conductivity by using Lévy Processes in Fourier space. It is a very interesting mathematical result since Lévy statistics turn out to efficiently describe quantum phenomena. Indeed, quantum Monte-Carlo methods have already permitted to observe that certain quantum processes obey Lévy statistics. Moreover, a relation between quantum systems and (classical) stochastic processes has also been experimentally observed: For instance, in quantum optics, the (subrecoil) cooling process of atom in presence of laser radiation can be modeled by a Lévy process [BBAC] with so–called quantum jumps in momentum space (w.r.t. space variables). This gives very good agreements with experimental measurements, see [BBAC, Chap. 8]. However, as far as we know, there is no rigorous derivation of this fact from quantum mechanics. Thus, this section is written to propose an approach to that issue and suggest a Lévy processes that could be behind the phenomenon of linear conductivity.
For simplicity, we assume that the paramagnetic conductivity is of the form \mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 283\relax}}{\mbox{\boldmath\textstyle\mathchar 283\relax}}{\mbox{\boldmath\scriptstyle\mathchar 283\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 283\relax}}}}_{\mathrm{p}}\mathbf{1}_{\mathbb{R}^{d}} with \mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 283\relax}}{\mbox{\boldmath\textstyle\mathchar 283\relax}}{\mbox{\boldmath\scriptstyle\mathchar 283\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 283\relax}}}}_{\mathrm{p}} being a real–valued function of time. In particular, and, by (52), \mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 283\relax}}{\mbox{\boldmath\textstyle\mathchar 283\relax}}{\mbox{\boldmath\scriptstyle\mathchar 283\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 283\relax}}}}_{\mathrm{p}}(t)=\mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 283\relax}}{\mbox{\boldmath\textstyle\mathchar 283\relax}}{\mbox{\boldmath\scriptstyle\mathchar 283\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 283\relax}}}}_{\mathrm{p}}(-t) for any with \mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 283\relax}}{\mbox{\boldmath\textstyle\mathchar 283\relax}}{\mbox{\boldmath\scriptstyle\mathchar 283\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 283\relax}}}}_{\mathrm{p}}(0)=0. This property of holds true, for instance, if the random variables \left\{\left({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}_{1}\left(x\right),{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}_{2}\left(b\right)\right)\right\}_{x\in\mathfrak{L},b\in\mathfrak{b}} are independently and identically distributed and the interparticle interaction has the form
[TABLE]
whenever for , and when . Here, is a real–valued function such that
[TABLE]
See [BPH3, Lemma 5.23] for more details.
In this case, by Theorem 5.2, for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+} and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}, there is a unique finite and symmetric –valued measure on such that, for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 267\relax}}{\mbox{\boldmath\textstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptstyle\mathchar 267\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 267\relax}}}\in\mathbb{R},
[TABLE]
where \mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}}\left(\left\{0\right\}\right)=D_{\left\{0\right\}}\mathbf{1}_{\mathbb{R}^{d}} with and
[TABLE]
Equations (64)–(65) correspond to the Lévy–Khintchine representation of the function \mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 283\relax}}{\mbox{\boldmath\textstyle\mathchar 283\relax}}{\mbox{\boldmath\scriptstyle\mathchar 283\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 283\relax}}}}_{\mathrm{p}}. Observe that {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}_{\mathrm{AC}}=\mathfrak{m}_{\mathrm{AC}}\mathbf{1}_{\mathbb{R}^{d}} and by a slight abuse of terminology, we also name the AC–conductivity measure.
Therefore, by [Ky, Theorem 2.1.], there is a probability space on which a –valued Lévy process with characteristic exponent \mathbf{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 283\relax}}{\mbox{\boldmath\textstyle\mathchar 283\relax}}{\mbox{\boldmath\scriptstyle\mathchar 283\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 283\relax}}}}_{\mathrm{p}} (up to a minus sign) exists. More explicitly,
[TABLE]
with being the expectation value associated with the probability measure . In this context, is called the Lévy measure of . It describes the jumps of . In other words, similar to laser cooling [BBAC], such a Lévy Process describes quantum jumps in Fourier space (but w.r.t. time coordinates instead of position coordinates as in sub-recoil laser cooling). For a comprehensive account on Lévy processes, see for instance [B, Ky] and references therein.
By (64), has no drift but a diffusion component when . There is also a Poisson random measure (see, e.g., [Ky, Definition 2.3.]) distributed on
[TABLE]
being the Borel –algebra of , with characteristic measure (or intensity) such that
[TABLE]
Here, is the associated martingale measure
[TABLE]
and is a Brownian motion. The second term in the r.h.s. of (66) is a compound Poisson process with rate and jump distribution
[TABLE]
provided . The third term in the r.h.s. of (66) is another Lévy process, which is a square integrable martingale on the same probability space. It is the uniform limit {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 290\relax}}{\mbox{\boldmath\textstyle\mathchar 290\relax}}{\mbox{\boldmath\scriptstyle\mathchar 290\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 290\relax}}}\rightarrow 0^{+} (along an appropriate deterministic subsequence) on compacta of the compound Poisson process with drift
[TABLE]
The limit Lévy process can also be seen as a superposition of an infinite number of compound Poisson processes with drift, see for instance [Ky, Section 2.5].
When
[TABLE]
is a compound Poisson process with rate and jump distribution
[TABLE]
See [Ky, Lemma 2.13]. In particular, the AC–conductivity measure describes the jump structure of the symmetric Lévy process in the frequency domain .
As an example, we can take the AC–conductivity measure obtained from the Drude model. This measure is absolutely continuous w.r.t. the Lebesgue measure with Radon–Nikodym density {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}_{\mathrm{T}} defined by (57). Recall that the relaxation time is the mean time interval between two collisions of a charged carrier with defects in the crystal. For all , the measure of the full set equals \|{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}_{\mathrm{T}}\|_{1}=D. In particular, the mean time between frequency jumps does not depend on in this new classical process. In the limit of perfect isolator {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}_{\mathrm{T}}\rightarrow 0 uniformly on while in the limit of perfect conductor {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}_{\mathrm{T}}\rightarrow 0 uniformly on \mathbb{R}\backslash[-{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 290\relax}}{\mbox{\boldmath\textstyle\mathchar 290\relax}}{\mbox{\boldmath\scriptstyle\mathchar 290\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 290\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 290\relax}}{\mbox{\boldmath\textstyle\mathchar 290\relax}}{\mbox{\boldmath\scriptstyle\mathchar 290\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 290\relax}}}] for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 290\relax}}{\mbox{\boldmath\textstyle\mathchar 290\relax}}{\mbox{\boldmath\scriptstyle\mathchar 290\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 290\relax}}}>0. Hence, by a similar expression to (68) for the Drude model and because of (57), the probability of large (frequency) jumps increases in the limit (isolator limit), but decreases when (conductor limit). The stochastic process gives an alternative classical picture to electrical conduction.
7 Technical Proofs
7.1 Study of the Paramagnetic Conductivity
Lieb–Robinson bounds and their extensions [BP3] to multi–commutators are here pivotal mathematical tools.
For any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}_{0},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}\in[0,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}_{0}], {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\Omega, , and with disjoint sets ,
[TABLE]
This is the usual Lieb–Robinson bound. See, e.g., [BP1, Theorem 2.1 (iii)]. Here, the real constant D_{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}_{0}} is defined, for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}_{0}\in\mathbb{R}_{0}^{+}, by
[TABLE]
See Sections 3.1 and 3.3. As a consequence, the paramagnetic transport coefficient {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 283\relax}}{\mbox{\boldmath\textstyle\mathchar 283\relax}}{\mbox{\boldmath\scriptstyle\mathchar 283\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 283\relax}}}_{\mathrm{p}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}})} defined by (30) satisfies
[TABLE]
for and
[TABLE]
with . This inequality implies the existence of the macroscopic paramagnetic conductivity defined by (34) with its first derivative. The existence and continuity of its second derivative follow from Lieb–Robinson bounds for multi–commutators [BP3, Theorems 3.8–3.9] of order three:
Theorem 7.1** (Paramagnetic conductivity)**
Assume (14)–(15), (19) and that the map (22) is a random invariant state. Let {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+} and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}. Then, there is such that, uniformly for times on compacta,
[TABLE]
Moreover, if (20) also holds, then and, uniformly for times on compacta,
[TABLE]
Proof: The three limits are proven in the same way. The first two only need (7.1), which follows from usual Lieb–Robinson bounds. By contrast, the last limit requires Lieb–Robinson bounds for multi–commutators [BP3, Theorems 3.8–3.9] of order three and is thus technically more difficult than the other ones. As a consequence, we focus on the limit of \partial_{t}^{2}\mathbb{E}[\Xi_{\mathrm{p},l}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}})}\left(t\right)], as , and we omit the details for the first two.
Fix {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+}, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}_{0},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}\in[0,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}_{0}], and . By [BP1, Theorem 2.1 (i)], \{{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 284\relax}}{\mbox{\boldmath\textstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptstyle\mathchar 284\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 284\relax}}}_{t}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}})}\}_{t\in{\mathbb{R}}} is a –group of –automorphisms with generator {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 270\relax}}{\mbox{\boldmath\textstyle\mathchar 270\relax}}{\mbox{\boldmath\scriptstyle\mathchar 270\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 270\relax}}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}})}. We thus compute from Equations (30) and (32) that
[TABLE]
Then, since the map (22) is a random invariant state and is an ergodic measure while (19) holds, one computes that
[TABLE]
with
[TABLE]
For any , the map x\mapsto{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 280\relax}}{\mbox{\boldmath\textstyle\mathchar 280\relax}}{\mbox{\boldmath\scriptstyle\mathchar 280\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 280\relax}}}_{l}\left(x\right) on has finite support and, for any ,
[TABLE]
Paramagnetic current observables (27) are obviously local elements, i.e.,\ I_{\mathbf{x}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}})}\in\mathcal{U}_{0} for any , while from [BP1, Theorem 2.1 (ii)]
[TABLE]
for any . Therefore, we get that
[TABLE]
The most delicate term in this equation is the last one. In fact, for all and , define the set
[TABLE]
while . By using (20), {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 294\relax}}{\mbox{\boldmath\textstyle\mathchar 294\relax}}{\mbox{\boldmath\scriptstyle\mathchar 294\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 294\relax}}}>2d, and
[TABLE]
together with Lieb–Robinson bounds for multi–commutators of order three [BP3, Corollary 3.10] (tree–decay bounds), one gets that, for {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\Omega, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}_{0},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}\in[0,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}_{0}], and ,
[TABLE]
Here, the positive constant does not depend on {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\Omega, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}_{0},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}\in[0,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}_{0}] and . Note that
[TABLE]
is a consequence of (20) and . The same kind of inequality holds for the 1st term in the r.h.s. of (74). Then, using Lebesgue’s dominated convergence theorem, one gets from (72)–(73) that the map
[TABLE]
converges uniformly on compacta, as , to a continuous function .
Remark 7.2** (Conductivity and space decays of interactions)**
**
Under stronger assumptions like in the case of exponential decays of interactions, much stronger results can be deduced from Lieb–Robinson bounds for multi–commutators. In particular, under assumptions of [BP3, Theorem 4.6] in the autonomous case, one verifies that is a Gevrey map of order . In particular, for , is in this case a real analytic map. Recall that is the space dimension of the lattice .
7.2 Study of the Paramagnetic Energy Increment
The aim of this subsection is to derive the paramagnetic energy density defined by (46). This is achieved in various lemmata which then yield two theorems and one corollary. The derivation ends with Theorem 7.9, which serves as springboard to obtain Theorem 5.1.
First, by assuming (14)–(15) and (20), [BP1, Theorem 3.8 (p)] says that, for any l,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+}, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\Omega, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}_{0},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}\in[0,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}_{0}], {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}}\in\mathbb{R}, and ,
[TABLE]
where, for any ,
[TABLE]
The subleading term in the r.h.s. of (75) is order \mathcal{O}({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}}^{3}l^{d}), uniformly for {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+}, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\Omega, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}\in[0,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}_{0}], {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+} and . Here,
[TABLE]
is the set of oriented bonds of nearest neighbors. Note also that the integral in (75) can be exchanged with the (finite) sum (76) because .
The first important result of the present subsection is a proof that the random variable \mathbf{X}_{l}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}})} almost surely converges to a constant function, as . See Corollary 7.8. To prove this, Condition (20) is not anymore necessary. Then, Lebesgue’s dominated convergence theorem yields the paramagnetic energy increment \mathfrak{I}_{\mathrm{p}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}}\mathbf{A}_{l})}\left(t\right) in the limit ({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}},l^{-1})\rightarrow(0,0), see Theorem 7.9.
We use the same strategy of proof as the one of [BPH3, Section 5.4] for the non–interacting case with homogeneous hopping terms. However, in spite of interactions, we strongly simplify the corresponding technical arguments by using Lieb–Robinson bounds. In particular, we do not anymore need complex times. But like in [BPH3, Section 5.4], the (compact) support of the vector potential at is divided in small regions to use the piecewise–constant approximation of the smooth electric field . To do this, we assume w.l.o.g. that, for all ,
[TABLE]
From now on we fix the parameters {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+}, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}_{0},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}\in[0,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}_{0}] and with (78).
Then, for every , we divide the elementary box in boxes of side–length , where
[TABLE]
Explicitly, for any ,
[TABLE]
For any , {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\Omega, and , let
[TABLE]
We show now that the accumulation points of \mathbf{Y}_{l,n}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}})}, as , do not depend on and coincide with those of \mathbf{X}_{l}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}})}:
Lemma 7.3** (Approximation I)**
[TABLE]
uniformly for {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\Omega and .
Proof: We observe from (76), (80) and (81) that
[TABLE]
where, for any with complement ,
[TABLE]
Because , note that
[TABLE]
Therefore, using (14), (7.1), (83) and the fact that for any (cf. (10)), we deduce from Inequality (7.2) that
[TABLE]
for all , {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\Omega, and , where
[TABLE]
and
[TABLE]
Clearly, one has
[TABLE]
Therefore, it remains to prove that vanishes when in order to prove the lemma.
To this end, for any and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 270\relax}}{\mbox{\boldmath\textstyle\mathchar 270\relax}}{\mbox{\boldmath\scriptstyle\mathchar 270\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 270\relax}}}\in[0,1], define two constants:
[TABLE]
Obviously, by (85), for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 270\relax}}{\mbox{\boldmath\textstyle\mathchar 270\relax}}{\mbox{\boldmath\scriptstyle\mathchar 270\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 270\relax}}},l\in\mathbb{R}^{+},
[TABLE]
Recall that , which encodes the short range property of interactions, is a non–increasing function, by assumption. As a consequence, explicit estimates using show that, for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 270\relax}}{\mbox{\boldmath\textstyle\mathchar 270\relax}}{\mbox{\boldmath\scriptstyle\mathchar 270\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 270\relax}}},l\in\mathbb{R}^{+},
[TABLE]
while
[TABLE]
Take {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 270\relax}}{\mbox{\boldmath\textstyle\mathchar 270\relax}}{\mbox{\boldmath\scriptstyle\mathchar 270\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 270\relax}}}=l^{-\frac{(d+1/2)}{d+1}}. Then, by (89), |\mathbf{K}_{l,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 270\relax}}{\mbox{\boldmath\textstyle\mathchar 270\relax}}{\mbox{\boldmath\scriptstyle\mathchar 270\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 270\relax}}}}^{\leq}|=\mathcal{O}(l^{-1/2}) and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 270\relax}}{\mbox{\boldmath\textstyle\mathchar 270\relax}}{\mbox{\boldmath\scriptstyle\mathchar 270\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 270\relax}}}l=l^{\frac{1}{2\left(d+1\right)}}, which combined with (14), (88) and (90) yield
[TABLE]
By (7.2) and (87), we thus arrive at the assertion.
We now consider piecewise–constant approximations of the (smooth) electric field (7), that is,
[TABLE]
For any , let be any fixed point of the box . Then, we define the function
[TABLE]
for any , {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\Omega, and , where and , see (77). This new function approximates (81) arbitrarily well, as and :
Lemma 7.4** (Approximation II)**
[TABLE]
uniformly for {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\Omega and .
Proof: By taking the canonical orthonormal basis of , we directly infer from (8), (36) and (91) that, for any , , , , and ,
[TABLE]
where
[TABLE]
In particular, since , there is a finite constant not depending on , , and such that
[TABLE]
Therefore, using (14), (7.1), (83), (93) and the fact that for any (cf. (10)), like in (7.2), we deduce from (81) and (7.2) that
[TABLE]
This upper bound implies the lemma.
By taking the canonical orthonormal basis of and setting for each , we rewrite the function (7.2) as
[TABLE]
for any , {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\Omega, and , where, for , {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\Omega, , and ,
[TABLE]
Notice that, as compared to (7.2), we have added in (7.2) terms related to on the boundary of , but we use the same notation \mathbf{\bar{Y}}_{l,n}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}})} for simplicity. These terms have indeed vanishing contribution in the limit . Here, for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\Omega, , and ,
[TABLE]
see (30). Hence, it remains to analyze the limit of (95), as . But before doing this study, observe that, for all , and , the map
[TABLE]
is bounded and measurable w.r.t. the –algebra , by assumption. Indeed, the map (22) is a random invariant state (Definition 3.2). Recall also that is the expectation value associated with the probability measure , see Section 3.1.
Lemma 7.5** (Infinite volume limit and ergodicity)**
Assume (14)–(15), (19) and that the map (22) is a random invariant state. For any , there is a measurable subset \tilde{\Omega}\left(t\right)\equiv\tilde{\Omega}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}})}\left(t\right)\subset\Omega of full measure such that, for , , and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\tilde{\Omega}\left(t\right),
[TABLE]
Proof: For any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\Omega, , and , let
[TABLE]
By the assumptions of the lemma, this sum is uniformly bounded for all and defines a random variable. Indeed, we infer from (14) and (7.1) that
[TABLE]
We now define an additive process \{\mathfrak{F}_{t,k,q}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}})}\left(\Lambda\right)\}_{\Lambda\in\mathcal{P}_{f}(\mathfrak{L})} by
[TABLE]
for any finite subset with cardinality , see [BPH3, Definition 5.2]666Replace the product measure of [BPH3] with ergodic measures , as defined in Section 3.1.. Indeed, the map {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\mapsto\mathfrak{F}_{t,k,q}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}})}\left(\Lambda\right) is bounded and measurable w.r.t. the –algebra for all . Moreover, by Conditions (19) and (23),
[TABLE]
See Section 3.3, in particular Definition 3.2. For any ,
[TABLE]
because of (98)–(99). Then, by (100) and ergodicity of the measure , for any and , [BPH3, Theorem 5.5]2 applied on the previous additive process holds and one gets the existence of a measurable subset
[TABLE]
of full measure such that, for all {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\hat{\Omega}_{k,q}\left(t\right), and ,
[TABLE]
In the same way one proves Lemma 7.3,
[TABLE]
Using this with (97), (99) and (101), and observing meanwhile from the proof of Theorem 7.1 that
[TABLE]
for all and any , we arrive at the assertion for any realization {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\tilde{\Omega}\left(t\right) with
[TABLE]
[Any countable intersection of measurable sets of full measure has full measure.]
Exactly like in the proof of Lemma 7.5, one shows that, for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+}, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+} and , there is a measurable subset \tilde{\Omega}\left(t\right)\equiv\tilde{\Omega}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}})}\left(t\right)\subset\Omega of full measure such that, for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\tilde{\Omega}\left(t\right),
[TABLE]
This holds under Conditions (14)–(15) and (19), provided that the map (22) is a random invariant state.
Define the deterministic function
[TABLE]
for any . We show next that the function \mathbf{X}_{l}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}})} defined by (76) almost surely converges to \mathbf{X}_{\infty}\equiv\mathbf{X}_{\infty}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}})}, as :
Theorem 7.6** (Infinite volume limit of –integrands – I)**
Assume (14)–(15), (19) and that the map (22) is a random invariant state. Let {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+}, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+} and . Then, there is a measurable subset \tilde{\Omega}\left(s_{1},s_{2}\right)\equiv\tilde{\Omega}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}})}\left(s_{1},s_{2}\right)\subset\Omega of full measure such that, for any and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\tilde{\Omega}\left(s_{1},s_{2}\right),
[TABLE]
Proof: Let {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+}, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}_{0},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}\in[0,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}_{0}] and . Assume w.l.o.g. that (78) holds. Using Lemmata 7.3–7.5 and (7.2)–(95), we obtain the existence of a measurable subset \tilde{\Omega}\left(s_{1},s_{2}\right)\equiv\tilde{\Omega}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}})}\left(s_{1},s_{2}\right)\subset\Omega of full measure such that, for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\tilde{\Omega}\left(s_{1},s_{2}\right),
[TABLE]
The latter implies the theorem because the term within the limit is a Riemann sum and for any , see (91).
To find the energy increment \mathfrak{I}_{\mathrm{p}}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}}\mathbf{A}_{l})}\left(t\right) given by (75) in the limit ({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 273\relax}}{\mbox{\boldmath\textstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptstyle\mathchar 273\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 273\relax}}},l^{-1})\rightarrow(0,0), we use below Lebesgue’s dominated convergence theorem and we thus need to remove the dependency of the measurable subset on , see Theorem 7.6. To achieve this, we first show uniform boundedness and equicontinuity of the function \mathbf{X}_{l}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}})} defined by (76):
Lemma 7.7** (Uniform Boundedness and Equicontinuity of –integrands)**
[TABLE]
of maps from to is uniformly bounded and equicontinuous.
Proof: The uniform boundedness of this collection of maps is an immediate consequence of (7.1) and (83). The arguments are indeed similar to those proving Inequality (7.2): Assume w.l.o.g. that (78) holds. Then, by combining (76) with (7.1) and (83) one gets
[TABLE]
for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+}, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}_{0},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}\in[0,{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}}_{0}] and .
To prove the uniform equicontinuity, we use [BP1, Theorem 3.6], which is also an immediate consequence of (7.1). We omit the details.
Theorem 7.6 and Lemma 7.7 allows us to eliminate the –dependency of the measurable set of Theorem 7.6.
Corollary 7.8** (Infinite volume limit of –integrands – II)**
Assume (14)–(15), (19) and that the map (22) is a random invariant state. Let {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+} and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}. Then, there is a measurable subset \tilde{\Omega}\equiv\tilde{\Omega}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}})}\subset\Omega of full measure such that, for any , and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\tilde{\Omega},
[TABLE]
Proof: Fix {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+} and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}. By Theorem 7.6, for any , there is a measurable subset of full measure such that (104) holds. Let be the intersection of all such subsets . Since this intersection is countable, is measurable and has full measure. By Lemma 7.7 and the density of in , it follows that (104) holds true for any , and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\tilde{\Omega}.
Therefore, because of (75), Lemma 7.7 and Corollary 7.8, we can now use Lebesgue’s dominated convergence theorem to get the paramagnetic energy density defined by (46):
Theorem 7.9** (Paramagnetic energy density)**
Assume (14)–(15), (19)–(20) and that the map (22) is a random invariant state. Let {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}}\in\mathbb{R}^{+} and {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}}\in\mathbb{R}_{0}^{+}. Then, there is a measurable subset \tilde{\Omega}\equiv\tilde{\Omega}^{({{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 268\relax}}{\mbox{\boldmath\textstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptstyle\mathchar 268\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 268\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 291\relax}}{\mbox{\boldmath\textstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptstyle\mathchar 291\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 291\relax}}},{{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 277\relax}}{\mbox{\boldmath\textstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptstyle\mathchar 277\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 277\relax}}})}\subset\Omega of full measure such that, for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 289\relax}}{\mbox{\boldmath\textstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptstyle\mathchar 289\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 289\relax}}}\in\tilde{\Omega}, and ,
[TABLE]
This theorem yields Theorem 5.1 (p).
7.3 Appendix: the Bochner Theorem
For completeness, we give in this appendix a proof of the Bochner theorem for weakly positive definite maps from to . By weakly positive definite –valued map, we mean that, for any {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 295\relax}}{\mbox{\boldmath\textstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptstyle\mathchar 295\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 295\relax}}}\in C_{0}^{\infty}(\mathbb{R};\mathbb{R}^{d}),
[TABLE]
It is a simple consequence of the usual Bochner theorem for weakly positive definite complex–valued functions:
Theorem 7.10** (The Bochner theorem)**
*The following are equivalent:
(i) is a weakly positive definite and continuous function, i.e.,*
[TABLE]
(ii)* There is a unique finite positive measure on such that*
[TABLE]
Proof: See for instance [RS2, Theorem IX.9 and discussion thereafter].
Corollary 7.11** (A Bochner theorem for real matrix–valued maps)**
Let be a weakly positive definite continuous map. If is symmetric w.r.t. the canonical scalar product of for any , then there is a unique finite and symmetric –valued measure {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}_{\Upsilon} on such that
[TABLE]
Proof: First, for any , we define as an operator on by
[TABLE]
For , let be the complex–valued function on defined by
[TABLE]
If is symmetric w.r.t. the canonical scalar product of for any , then is a weakly positive definite and continuous (complex–valued) function. By Theorem 7.10, for any , there is a unique finite positive measure {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}_{\vec{w}} on such that
[TABLE]
Now, we define a –valued measure {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}_{\Upsilon} on by using the polarization identity: For any Borel set , in the canonical orthonormal basis of ,
[TABLE]
By this definition, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}_{\Upsilon}\left(\mathcal{X}\right) is a symmetric operator on (w.r.t. the canonical scalar product). Moreover, one can check that, for all and any Borel set ,
[TABLE]
Indeed, if then, by symmetry of the operator ,
[TABLE]
Hence, from the injectivity of the Fourier transform of finite measures,
[TABLE]
and (108) follows. By positivity of {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}_{\vec{w}}, {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}_{\Upsilon} is a –valued measure on . Moreover, we deduce from (106) that
[TABLE]
If for any , then {{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}_{\Upsilon}\left(\mathcal{X}\right)={{}\mathchoice{\mbox{\boldmath\displaystyle\mathchar 278\relax}}{\mbox{\boldmath\textstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptstyle\mathchar 278\relax}}{\mbox{\boldmath\scriptscriptstyle\mathchar 278\relax}}}_{\Upsilon}\left(-\mathcal{X}\right) for any Borel set and hence,
[TABLE]
By using the symmetry of the operators and
[TABLE]
at any fixed , we arrive at the assertion from (109).
Acknowledgments: This research is supported by the agency FAPESP under Grant 2013/13215-5 as well as by the Basque Government through the grant IT641-13 and the BERC 2014-2017 program and by the Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa accreditation SEV-2013-0323, MTM2014-53850. Finally, we thank very much the referees for their work and interest in the improvement of this paper.
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