Equilibration properties of small quantum systems: further examples

TL;DR

This paper examines the equilibration behavior of small quantum systems through analytical analysis of specific models, revealing how parameters influence the transition from good to poor equilibration.

Contribution

It provides analytical insights into the equilibration properties of small quantum systems, including random Hamiltonians and specific physical models, highlighting parameter-dependent transitions.

Findings

Trace T varies between 1 and N, indicating degrees of equilibration.

Random Hamiltonians show statistical properties of T under symmetry groups.

Parameter changes induce continuous transitions in equilibration quality.

Abstract

It has been proposed to investigate the equilibration properties of a small isolated quantum system by means of the matrix of asymptotic transition probabilities in some preferential basis. The trace $T$ of this matrix measures the degree of equilibration of the system prepared in a typical state of the preferential basis. This quantity may vary between unity (ideal equilibration) and the dimension $N$ of the Hilbert space (no equilibration at all). Here we analyze several examples of simple systems where the behavior of $T$ can be investigated by analytical means. We first study the statistics of $T$ when the Hamiltonian governing the dynamics is random and drawn from a distribution invariant under the group U$(N)$ or O$(N)$. We then investigate a quantum spin $S$ in a tilted magnetic field making an arbitrary angle with the preferred quantization axis, as well as a tight-binding particle on a finite electrified chain. The last two cases provide examples of the interesting situation where varying a system parameter -- such as the tilt angle or the electric field -- through some scaling regime induces a continuous transition from good to bad equilibration properties.

Tables

Table 1. Table 1: Values of the integrals A 𝐴 A , B 𝐵 B , C 𝐶 C and D 𝐷 D for both symmetry classes. For the unitary group U ( N ) 𝑁 (N) (see ( 3.3 )), the one-vector integrals A U subscript 𝐴 U A_{\rm U} , B U subscript 𝐵 U B_{\rm U} and C U subscript 𝐶 U C_{\rm U} are given in  [ 28 , 37 ] while the two-vector integral D U subscript 𝐷 U D_{\rm U} , given in  [ 35 ] , can be checked using the recursive approach of  [ 37 ] . For the orthogonal group O ( N ) 𝑁 (N) (see ( 3.8 )), the one-vector integrals A O subscript 𝐴 O A_{\rm O} , B O subscript 𝐵 O B_{\rm O} and C O subscript 𝐶 O C_{\rm O} are given in  [ 37 ] , and the two-vector integral D O subscript 𝐷 O D_{\rm O} , given in  [ 36 ] , can again be checked using  [ 37 ] .

U$\left(N\right)$	O$\left(N\right)$
$A_U=\frac{2^{}}{N\left(N+1\right)}$	$A_O=\frac{3}{N\left(N+2\right)}$
$B_U=\frac{{24}^{}}{N\left(N+1\right)\left(N+2\right)\left(N+3\right)}$	$B_O=\frac{105}{N\left(N+2\right)\left(N+4\right)\left(N+6\right)}$
$C_U=\frac{4^{}}{N\left(N+1\right)\left(N+2\right)\left(N+3\right)}$	$C_O=\frac{9}{N\left(N+2\right)\left(N+4\right)\left(N+6\right)}$
$D_U=\frac{4^{}}{N^2\left(N-1\right){\left(N+3\right)}_{}}$	$D_O=\frac{9\left(N+3\right)\left(N+5\right)}{N\left(N^2-1\right)\left(N+2\right)\left(N+4\right)\left(N+6\right)}$

Table 2. Table 2: The three models known to exhibit a continuous crossover from good to bad equilibration properties in the regime where N 𝑁 N is large whereas the control parameter is small: definition of the control parameters and of the variables entering the scaling behavior of T 𝑇 T .

model	$N$	control parameter	scaling variable	Ref.
$\begin{array}{c}\text{spin in tilted}\\ \text{magnetic field}\end{array}$	$2S+1$	tilt angle $\theta$	$u=N\theta$	Sec. 4
$\begin{array}{c}\text{particle on}\\ \text{disordered chain}\end{array}$	$\begin{array}{c}\text{number}\\ \text{of sites}\end{array}$	disorder strength $w$	$x=N{w}^{2/3}$	[8]
$\begin{array}{c}\text{particle on}\\ \text{electrified chain}\end{array}$	$\begin{array}{c}\text{number}\\ \text{of sites}\end{array}$	electric field $F$	$\{\begin{array}{c}x=N{F}^{1/3}\text{(Airy)}\\ y=NF\text{(Bessel)}\end{array}$	Sec. 5

Equations

$$T\approx T^{(0)}+\frac{1}{N}\Phi (x),x=N{w}^{2/3}.$$

$$P_{ab}(t)=\sum _{m,n}{\text{rme}}^{\mathrm{i}(E_n-E_m)t}⟨b|m⟩⟨m|a⟩⟨a|n⟩⟨n|b⟩,$$

$$Q_{ab}=\underset{t\to \infty }{\lim }\frac{1}{t}{\int }_0^t{P}_{ab}(t^{\prime })\mathrm{d}{t}^{\prime }.$$

$$Q_{ab}=\sum _n|⟨a|n⟩{|}^2|⟨b|n⟩{|}^2.$$

$$Q=R{R}^T,$$

$$R_{an}=|⟨a|n⟩{|}^2,$$

$$\sum _a{R}_{an}=\sum _n{R}_{an}=\sum _a{Q}_{ab}=\sum _b{Q}_{ab}=1.$$

$$U_{an}=⟨a|n⟩.$$

$$Q_{aa}=\sum _n|⟨a|n⟩{|}^4,$$

$$Q_{aa}=\text{tr}{\omega }_a^2,$$

$${\omega }_a=\sum _n|⟨a|n⟩{|}^2|n⟩⟨n|$$

$$d_a=\frac{1}{Q_{aa}},$$

$$T=\sum _{a,n}|⟨a|n⟩{|}^4=\sum _a{Q}_{aa}=\sum _n{I}_n,$$

$$I_n=\sum _a|⟨a|n⟩{|}^4$$

$$D=\frac{N}{T}$$

$$\frac{1}{D}=\frac{1}{N}\sum _a\frac{1}{d_a}.$$

$$R_{an}=\frac{1}{N}$$

$$Q_{ab}=\frac{1}{N}$$

$$T_{\min }=1,D_{\max }=N.$$

$$R_{an}={\delta }_{an}$$

$$Q_{ab}={\delta }_{ab}$$

$$T_{\max }=N,D_{\min }=1.$$

$$T=\sum _{a,n}{\left(\sum _{i=1}^{{\mu }_n}|⟨a|n,i⟩{|}^2\right)}^2,$$

$$⟨T{⟩}_{\mathrm{U}}$$

$$(\mathrm{var}T{)}_{\mathrm{U}}$$

$$\matrix{A_{\rm U}=\langle|U_{11}|^{4}\rangle_{\rm U},\hfill&B_{\rm U}=\langle|U_{11}|^{8}\rangle_{\rm U},\hfill\cr C_{\rm U}=\langle|U_{11}|^{4}|U_{12}|^{4}\rangle^{\vphantom{X}}_{\rm U},\hfill\qquad&D_{\rm U}=\langle|U_{11}|^{4}|U_{22}|^{4}\rangle_{\rm U},\hfill\cr}$$

$$⟨T{⟩}_{\mathrm{U}}$$

$$(\mathrm{var}T{)}_{\mathrm{U}}$$

$$⟨T{⟩}_{\mathrm{O}}$$

$$(\mathrm{var}T{)}_{\mathrm{O}}$$

$$\matrix{A_{\rm O}=\langle\Omega_{11}^{4}\rangle_{\rm O},\hfill&B_{\rm O}=\langle\Omega_{11}^{8}\rangle_{\rm O},\hfill\cr C_{\rm O}=\langle\Omega_{11}^{4}\Omega_{12}^{4}\rangle^{\vphantom{X}}_{\rm O},\hfill\qquad&D_{\rm O}=\langle\Omega_{11}^{4}\Omega_{22}^{4}\rangle_{\rm O},\hfill\cr}$$

$$⟨T{⟩}_{\mathrm{O}}$$

Full text

1 Introduction

The last decade has witnessed an immense activity around the themes of thermalization and equilibration in isolated quantum systems (see [1, 2, 3, 4, 5, 6, 7] for comprehensive reviews). This renewed interest in an old classic subject of Quantum Mechanics was mostly triggered by progress on the experimental side in cold-atom physics. The physical mechanisms underling equilibration and thermalization are far from obvious. On the one hand, an isolated quantum system evolves by some unitary dynamics, and therefore keeps the memory of its initial state. On the other hand, if the system is large enough, concepts from Statistical Mechanics can be expected to apply at least approximately. Most recent works have dealt with large many-body quantum systems, the key issues including the description of a small subsystem by a thermodynamical ensemble (either microcanonical or canonical), the thermalization properties of single highly-excited states, and the role of conservation laws and of integrability.

An alternative and complementary viewpoint, where the main focus is on the unitary evolution of a small quantum system considered as a whole, has been emphasized in [8]. There, the key quantity is the matrix $Q$ of asymptotic transition probabilities in some preferential basis. The trace $T$ of this matrix is also the sum of the inverse participation ratios of all eigenstates of the Hamiltonian in the same basis. This quantity has been put forward as a measure of the degree of equilibration of the full system if launched from a typical basis state; the larger $T$, the poorer the equilibration.

The above line of thought has been illustrated by means of a detailed study of a single tight-binding particle on a finite segment of $N$ sites in a random potential [8]. In the absence of disorder, the quantity $T$ exhibits a finite value $T=T^{(0)}\approx 3/2$, testifying good equilibration. In the localized regime, $T\approx N{\overline{Q}}$ grows linearly with the sample size, with a finite mean asymptotic return probability ${\overline{Q}}$, testifying poor equilibration. The outcome of most physical interest concerns the regime of a weak disorder, where the disorder strength $w$ is much smaller than the bandwidth. In this regime, $T$ obeys a universal scaling law of the form

[TABLE]

This finite-size scaling law interpolates between both above regimes. In the ballistic regime, where $x$ is small, the first correction due to disorder scales as $\Phi(x)\approx 13x^{3}/540$. When $x$ is large, the quadratic growth $\Phi(x)\approx Ax^{2}$, with $A\approx 0.21$, describes the crossover to the localized regime. The mean return probability accordingly scales as ${\overline{Q}}\approx Aw^{4/3}$ at weak disorder. The crossover exponent 2/3 entering the variable $x$ is dictated by the well-known anomalous scaling of the localization length in the vicinity of band edges. Typical equilibration properties of a single particle in a weak random potential, as testified by the scaling behavior of $T$, are therefore governed by the few most localized band-edge states. This is a not an artifact of considering a global quantity such as $T$, albeit a genuine physical effect: a typical initial state will generically have a non-zero overlap with the anomalously localized band-edge states, and the very slow relaxation of the latter overlap will cause the relatively poor equilibration properties of such a generic initial state.

The behavior of $T$ in a clean quantum many-body system has then been considered in [9], where the statistics of inverse participation ratios in the XXZ spin chain with anisotropy parameter $\Delta$ has been investigated numerically. There, the quantity $T$ was found to grow linearly with the system size ($T\sim L$) in the gapless phase of the model ($|\Delta|<1$), and exponentially ($T\sim\exp(aL)$) in the gapped phase ($|\Delta|>1$). In the particular case of the XX spin chain ($\Delta=0$), which can be mapped onto a free-fermion problem, the inverse participation ratios of some individual eigenstates have been investigated in [10]. The amount of technical effort needed to derive the partial results presented there attests the high level of difficulty of the analysis of seemingly simple quantities such as $T$, even in the simplest of integrable models.

In this paper we analyse in detail several other examples of simple quantum systems for which the behavior of $T$ can be investigated by analytical means. Section 2 provides a reminder of the transition matrix formalism proposed in [8], emphasizing the role and the meaning of $T$. In section 3 we investigate the statistics of $T$ corresponding to a random Hamiltonian whose distribution is invariant under the group U$(N)$ or O$(N)$, thus keeping in line with the random matrix theory approach to quantum chaos. Section 4 deals with a single quantum spin $S$ submitted to a tilted magnetic field making an angle $\theta$ with respect to the quantization axis defining the preferred basis. Finally, the case of a tight-binding particle on a finite electrified chain is considered in section 5. This situation is somehow a deterministic analogue of the case considered in [8], where the ladder of Wannier-Stark states replaces Anderson localized states. It is however richer, as it features two successive scaling regimes in the small-field region. Our findings are summarized and discussed in section 6.

2 A reminder on the transition matrix formalism

This section provides a reminder of the transition matrix formalism which has been proposed in [8] to describe equilibration in small isolated quantum systems. Consider a system whose Hilbert space has dimension $N$. Let $\{|a\rangle\}$ ($a=1,\dots,N$) be a preferential basis, chosen once for all. In that basis the Hamiltonian ${\cal H}$ is some $N\times N$ matrix. We assume that its eigenvalues $E_{n}$ are non-degenerate. This condition will be satisfied in all systems considered in this work. The situation of degenerate energy eigenvalues will be considered briefly at the end of this section.

If the system is launched from one of the basis states, say $|a\rangle$, the probability for it to be in state $|b\rangle$ at a subsequent time $t$ reads

[TABLE]

where $\{|n\rangle\}$ ($n=1,\dots,N$) is a basis of normalized eigenstates, such that ${\cal H}|n\rangle=E_{n}|n\rangle$. The stationary state reached by the system is characterized by the time-averaged transition probabilities

[TABLE]

In the absence of spectral degeneracies, we have

[TABLE]

The transition matrix $Q$ is the central object of this formalism. Let us begin with a few general properties. The matrix $Q$ only depends on the eigenstates $|n\rangle$ of the Hamiltonian ${\cal H}$, and not on its eigenvalues $E_{n}$. Therefore, any two Hamiltonians with the same eigenstate basis share the same matrix $Q$, irrespective of their spectra. In full generality, the matrix $Q$ reads

[TABLE]

where the matrix $R$ is given by

[TABLE]

and $R^{T}$ is its transpose. The matrix $Q$ is therefore real symmetric and positive definite. The matrix $R$ is real, albeit not symmetric in general; it may therefore have complex spectrum. Finally, both matrices $Q$ and $R$ are doubly stochastic: their row and column sums equal unity:

[TABLE]

More generally, one can associate real matrices $Q$ and $R$ obeying the above properties with any pair of orthonormal bases $\{|a\rangle\}$ and $\{|n\rangle\}$, related to each other by an arbitrary unitary transformation $U$ such that

[TABLE]

Coming back to equilibration properties, the diagonal element

[TABLE]

i.e., the stationary return probability to the basis state $|a\rangle$, can be recast as

[TABLE]

where

[TABLE]

is the stationary density matrix issued from state $|a\rangle$. The right-hand side of (2.9) is called the purity of $\omega_{a}$. Its reciprocal, i.e.,

[TABLE]

provides a measure of the dimension of the subspace of Hilbert space where the system equilibrates if launched from state $|a\rangle$ [11, 12, 13, 14, 15, 16].

The equilibration properties of the system launched from a typical basis state are encoded in the trace of the matrix $Q$:

[TABLE]

where

[TABLE]

is the inverse participation ratio (IPR) [17, 18, 19, 20] of the eigenstate $|n\rangle$ of ${\cal H}$ in the preferential basis. The return probabilities $Q_{aa}$ and the IPR $I_{n}$ therefore somehow appear as dual to each other.

The ratio

[TABLE]

is such that

[TABLE]

Hence $D$ provides a measure of the dimension of the subspace which hosts the equilibration dynamics for a typical initial state. The trace $T$ and the effective dimension $D$ are always between the following extremal values.

Ideal equilibration.

If the eigenstates of ${\cal H}$ are uniformly spread over the preferential basis, we have

[TABLE]

for all $a$ and $n$. The bases $\{|a\rangle\}$ and $\{|n\rangle\}$ are said to be mutually unbiased [21, 22, 23]. Hence

[TABLE]

for all $a$ and $b$, and

[TABLE]

This ideal situation corresponds to a perfectly equilibrated stationary state, with no memory of the initial state at all.

No equilibration.

This situation is the exact opposite of the previous one. If ${\cal H}$ is diagonal in the preferential basis, its eigenstates $|n\rangle$ can be ordered so as to have

[TABLE]

for all $a$ and $n$, and

[TABLE]

for all $a$ and $b$, $\delta$ being the Kronecker symbol, and so

[TABLE]

In such a situation, the system keeps a full memory of its initial state, and so there is no equilibration at all.

Finally, let us mention that the above transition matrix formalism can be readily extended to situations where some of the energy eigenvalues are degenerate [8]. In such a situation, the expression (2.12) of the quantity $T$ becomes

[TABLE]

where the $|n,i\rangle$ form an orthonormal basis such that ${\cal H}|n,i\rangle=E_{n}|n,i\rangle$, for $i=1,\dots,\mu_{n}$, where $\mu_{n}$ is the multiplicity (degeneracy) of the $n$th energy eigenvalue $E_{n}$.

3 Random matrix theory approach

In this section we investigate the statistics of $T$ obtained by choosing the Hamiltonian matrix ${\cal H}$ at random, according to some probability distribution over $N\times N$ Hermitian matrices. We assume that the latter distribution is invariant under the action of a suitable chosen symmetry group, here U$(N)$ or O$(N)$ (see below). This assumption keeps in line with the random matrix theory approach to quantum chaos, which has been used extensively to investigate many properties of complex quantum systems [24, 25, 26, 27], including the energy spectra of closed systems and the scattering and transport properties of open systems. This line of thought implies in particular that the distribution of the matrix $U$, introduced in (2.7), encoding the eigenstates of ${\cal H}$ in the preferred basis, is given by the flat Haar measure on the appropriate symmetry group. Moreover, it will be sufficient for our purpose to consider the first two cumulants of $T$, i.e., the mean value $\langle T\rangle$ and the variance $\mathop{\rm var}\,T=\langle T^{2}\rangle-\langle T\rangle^{2}$.

The unitary case.

We consider first the situation where the Hamiltonian ${\cal H}$ is an $N\times N$ complex Hermitian matrix. The relevant symmetry group is then the unitary group U$(N)$. We have

[TABLE]

with

[TABLE]

where $\langle\dots\rangle_{\rm U}$ denotes an average with respect to the Haar measure on U$(N)$.

Moments of this kind have attracted much attention in the past, and several approaches have been proposed to evaluate them [28, 29, 30, 31, 32, 33, 34, 35, 36, 37]. Table 1 gives the values of the moments $A_{\rm U}$, $B_{\rm U}$, $C_{\rm U}$ and $D_{\rm U}$ for the unitary group U$(N)$ (see (3.3)) and $A_{\rm O}$, $B_{\rm O}$, $C_{\rm O}$ and $D_{\rm O}$ for the orthogonal group O$(N)$ (see (3.8)).

Using the results of table 1, we obtain

[TABLE]

The orthogonal case.

We now turn to the situation where the dynamics is invariant under time reversal, so that the Hamiltonian ${\cal H}$ is a real symmetric matrix. The relevant symmetry group is then the orthogonal group O$(N)$. We have

[TABLE]

with

[TABLE]

where $\langle\dots\rangle_{\rm O}$ now denotes an average with respect to the Haar measure on O$(N)$. Using again the results of table 1, we obtain

[TABLE]

For a very large dimension $N$, $T$ converges to the limits

[TABLE]

which only depend on the symmetry class of the Hamiltonian. These finite values testify extended eigenstates and good equilibration properties, as could be expected for a generic Hamiltonian. The limit values (3.11) are deterministic, in the sense that fluctuations of individual values of $T$ become negligible for large $N$, as testified by the $1/N^{2}$ falloff of the variances

[TABLE]

The limit values (3.11) coincide with the outcome of the following naive approach. In the unitary case, consider $T$ as the sum of $N^{2}$ independent quantities of the form $|u|^{4}$, modeling the entries $u$ as centered complex Gaussian variables such that $\langle|u|^{2}\rangle=1/N$. The identity $\langle|u|^{4}\rangle=2\langle|u|^{2}\rangle^{2}$ yields $\langle|u|^{4}\rangle=2/N^{2}$, and so $T_{\rm U}=N^{2}\langle|u|^{4}\rangle=2$. Similarly, in the orthogonal case, consider $T$ as the sum of $N^{2}$ independent quantities of the form $\omega^{4}$, modeling the entries $\omega$ as real Gaussian variables such that $\langle\omega^{2}\rangle=1/N$. The identity $\langle\omega^{4}\rangle=3\langle\omega^{2}\rangle^{2}$ yields $\langle\omega^{4}\rangle=3/N^{2}$, and so $T_{\rm O}=N^{2}\langle\omega^{4}\rangle=3$. In the same vein, the $1/N^{2}$ decay of the variances (3.12) also conforms with the above naive approach, viewing $T$ as the sum of $N^{2}$ independent random variables.

For finite values of $N$ (see figure 1), the results (3.4), (3.5) and (3.9), (3.10) have a simple rational dependence on $N$. The mean values $\langle T\rangle$ increase monotonically from unity and saturate to the limits (3.11). The variances vanish both for $N=1$ and $N\to\infty$, and therefore present a maximum. The maximum of $(\mathop{\rm var}\,T)_{\rm U}$, equal to 4/45, is reached for $N=2$, while the maximum of $(\mathop{\rm var}\,T)_{\rm O}$, equal to 4/25, is reached for $N=3$ and 4.

4 A spin in a tilted magnetic field

This section is devoted to the dynamics of a single quantum spin in the representation of SU$(2)$ of dimension

[TABLE]

The spin $S$ is either an integer (if $N$ is odd) or a half-integer (if $N$ is even). We choose the $z$-axis as preferred quantization axis. The preferred basis thus consists of the eigenstates $|a\rangle$ of $S_{z}$, where $a$ takes the $N$ values $a=-S,\dots,S$, with unit steps. The spin is subjected to a tilted external magnetic field ${\bf h}$ making an angle $\theta$ with the $z$-axis, with $0\leq\theta\leq\pi$. The corresponding Hamiltonian reads, in dimensionless units

[TABLE]

The spin $S$ and the tilt angle $\theta$ are the two parameters of the model.

4.1 The simplest case: $S={\textstyle{1\over 2}}$

Let us consider first the case of a spin $S={\textstyle{1\over 2}}$, so that $N=2$. The Hamiltonian

[TABLE]

has eigenvalues $E_{\pm}=\pm{\textstyle{1\over 2}}$. The matrix encoding the corresponding eigenstates is

[TABLE]

up to arbitrary phases. The matrices $R$ and $Q$ (see (2.3), (2.5)) therefore read

[TABLE]

We notice that the matrices $U$ and $R$ are respectively orthogonal and symmetric. We have finally

[TABLE]

4.2 The general case

In the general case ($S$ arbitrary), the eigenvalues of ${\cal H}$ are $E_{n}=n=-S,\dots,S$, again with unit steps. The matrix $U$ encoding the corresponding eigenstates in the preferred basis is nothing but the Wigner ‘small’ rotation matrix $d^{(S)}(\theta)$, describing the transformation of a spin $S$ under a rotation of angle $\theta$ around the $y$-axis. References [38, 39, 40, 41] give a full account of its properties. The rotation matrix $d^{(S)}(\theta)$ is a real orthogonal matrix, whose entries obey the symmetry property

[TABLE]

This identity allows us to recast the entries of the matrix $R$ as

[TABLE]

Expanding the product of rotation matrix elements over irreducible representations [38, 39, 40, 41], we obtain

[TABLE]

In this expression, the integer $j$ takes $N=2S+1$ values,

[TABLE]

are the Legendre polynomials, and

[TABLE]

are Clebsch-Gordan coefficients, up to signs.

The identity (4.10) has several consequences. First, it shows that $R$ is a real symmetric matrix. The properties that $U$ and $R$ are respectively orthogonal and symmetric, already observed for $S={\textstyle{1\over 2}}$, thus hold for an arbitrary spin $S$. Furthermore, (4.10) yields the spectral decomposition of the matrix $R$: $r_{j}=P_{j}(\cos\theta)$ are its eigenvalues, with the corresponding normalized eigenvectors $\varphi^{(j)}_{n}$ being independent of the angle $\theta$. Finally, the matrix $R$ being symmetric, (2.4) implies $Q=R^{2}$, and so the eigenvalues of $Q$ are $q_{j}^{\vphantom{2}}=r_{j}^{2}$. We thus obtain the explicit formula

[TABLE]

Here are a few immediate properties of the above expression. First, it is invariant when changing $\theta$ to $\pi-\theta$, as should be. The Legendre polynomials indeed obey the symmetry property $P_{j}(-x)=(-1)^{j}P_{j}(x)$, so that $T$ is an even function of $x=\cos\theta$. For $\theta=0$ or $\theta=\pi$, we have exactly $T=N$. The Hamiltonian indeed reads ${\cal H}=\pm S_{z}$, and so its eigenstates coincide with the preferred basis. This is one example of the situation referred to in section 2 as no equilibration. For $S={\textstyle{1\over 2}}$, i.e., $N=2$, the result (4.7) is recovered, as $P_{0}(x)=1$ and $P_{1}(x)=x$.

4.3 Behavior at large $N$ (semi-classical regime)

The goal of this section is to estimate the growth law of $T$ in the semi-classical regime of large values of $N$. The quantity $T$ is clearly an increasing function of $N$, at fixed angle $\theta$, as the expression (4.13) is a sum of positive terms.

Consider first the mean value $\langle T\rangle$, assuming that the magnetic field ${\bf h}$ acting on the spin is isotropically distributed. This amounts to $x=\cos\theta$ being uniformly distributed in the range $-1\leq x\leq 1$. The identity (see e.g. [42, Vol. II, Ch. 10])

[TABLE]

yields

[TABLE]

where $H_{N}$ are the harmonic numbers, and hence

[TABLE]

where $\gamma=0.577\,215\dots$ is Euler’s constant. The mean value $\langle T\rangle$ therefore grows logarithmically with $N$.

A similar logarithmic growth holds at any fixed value of the tilt angle $\theta$. This can be shown by means of the generating series [43, 44, 45]

[TABLE]

where ${\bf K}(k)$ is the complete elliptic integral of the first kind. We are interested in the singular behavior of the above expression as $z\to 1$. As the modulus $k$ diverges in that limit, it is advantageous to change the square modulus [42, Vol. II, Ch. 13] from $k^{2}$ to

[TABLE]

We have then ${\bf K}(k)=q^{\prime}\,{\bf K}(q)$, with $q^{\prime}=\sqrt{1-q^{2}}$. Furthermore, as $q\to 1$, i.e., $q^{\prime}\to 0$, the new elliptic integral diverges as ${\bf K}(q)\approx\ln\,(4/q^{\prime})$, so that

[TABLE]

and finally

[TABLE]

The quantity $T$ thus grows logarithmically in $N$ for any fixed value of the angle $\theta$, with an angle-dependent amplitude $1/(\pi\sin\theta)$. Hence the product $T\sin\theta$, plotted in figure 2 against the reduced angle $\theta/\pi$, grows to leading order as $(1/\pi)\ln N$, irrespective of $\theta$. Finally, the growth law (4.16) of the mean value $\langle T\rangle$ can be recovered by integrating (4.20) over $0\leq\theta\leq\pi$ with the measure ${\textstyle{1\over 2}}\sin\theta\,{\rm d}\theta$.

4.4 Scaling behavior at small angle

The quantity $T$ obeys a non-trivial scaling behavior in the regime where $N$ is large and the tilt angle $\theta$ is small, interpolating between the growth laws $T=T_{{\rm max}}=N$ for $\theta=0$ and $T\sim\ln N$ for any non-zero $\theta$ (see (4.20)).

This scaling behavior is inherited from the scaling formula obeyed by the Legendre polynomials for $j$ large and $\theta$ small, sometimes referred to as Hilb’s formula (see e.g. [42, Vol. II, Ch. 10]), i.e.,

[TABLE]

with

[TABLE]

and $J_{0}$ being the Bessel function. Inserting this into (4.13), we obtain the scaling law

[TABLE]

or, equivalently,

[TABLE]

with

[TABLE]

and

[TABLE]

The formula (4.23) is more adapted to small values of the scaling variable $u$. The function $F(u)$ decreases monotonically from $F(0)=1$ to zero. For small $u$, the expansion

[TABLE]

matches the exact small-angle expansion

[TABLE]

of the full expression (4.13). We have $F(u)=1/2$ for $u\approx 2.277$, so that $T$ drops from its maximal value $T=N$ at $\theta=0$ to $T=N/2$ for $\theta\approx 2.277/N$.

The formula (4.24) is more adapted to large values of $u$. The function $G(u)$ is monotonically increasing. It exhibits an infinite sequence of ‘pauses’ at the zeros $j_{n}$ of the Bessel function $J_{0}$, with heights $G(j_{n})=G_{n}$, around which it varies cubically, as $G(u)-G_{n}\sim(u-j_{n})^{3}$. The asymptotic behavior of the Bessel function

[TABLE]

implies that $G(u)$ grows logarithmically for large $u$, as

[TABLE]

where the finite part has been obtained by matching with (4.20). The zeros read approximately $j_{n}\approx n\pi$, so that the heights of the pauses also grow logarithmically, as

[TABLE]

There are 12 pauses with height less than 2, and 277 pauses with height less than 3. The first of these pauses predict the heights of the ripples which are visible on the data for finite values of $N$, shown in figure 2.

Figure 3 shows the above scaling functions. The left panel shows the function $F(u)$ entering (4.23). The right panel shows the function $G(u)=uF(u)$ entering (4.24). Blue symbols show the first three pauses.

5 A tight-binding particle on an electrified chain

In this section we investigate a tight-binding particle on a finite chain of $N$ sites subjected to a uniform electric field $F$. We assume $F>0$ for definiteness.

The preferential basis consists of the Wannier states $|a\rangle$ associated with the sites of the chain ($a=1,\dots,N$). In terms of the amplitudes $\psi_{a}=\langle a|\psi\rangle$, the eigenvalue equation ${\cal H}\psi=E\psi$ reads

[TABLE]

in reduced units, where the dimensionless electric field $F$ equals the potential drop across one unit cell, in units of the hopping rate.

We shall consider an open chain with Dirichlet boundary conditions ($\psi_{0}=\psi_{N+1}=0$). The choice of the constant offset in the electrostatic potential

[TABLE]

ensures that the latter vanishes in the middle of the sample. As a consequence, the energy spectrum is symmetric with respect to $E=0$. Indeed, if the wavefunction $\psi_{a}$ describes a state with energy $E$, then $(-1)^{a}\psi_{N+1-a}$ describes a state with energy $-E$. Both states have the same IPR. In particular, samples with odd sizes $N$ have a single non-degenerate eigenstate at $E=0$.

The problem of an electron in a uniform electric field is an old classic of Quantum Mechanics. The spectrum of a tight-binding particle on the infinite chain has been known for long [46, 47, 48]. It consists of an infinite sequence of localized states, the corresponding energies having a constant spacing ($\Delta E=F$ in our reduced units). Such a discrete spectrum has been dubbed a Wannier-Stark ladder [49, 50, 51]. The fate of these ladders on finite samples has been investigated by several authors in the 1970s [52, 53, 54, 55, 56, 57]. In order to investigate the behavior of $T$ in the various regimes of system size $N$ and applied electric field $F$, we have to thoroughly revisit these works.

5.1 General approach

A general approach consists in solving (5.1) on the half-line ($a\geq 0$), for an arbitrary value of the energy $E$, with boundary conditions $\psi_{0}=0$ and $\psi_{1}=1$. This approach is known under the name of ‘solving the Cauchy problem’. Equation (5.1) readily yields

[TABLE]

and so on. The amplitude $\psi_{a}=P_{a-1}(E)$ thus obtained is a polynomial with degree $a-1$ in $E$, dubbed a Lommel polynomial [53]. It also depends parametrically on $N$ and $F$.

On a finite chain of $N$ sites with Dirichlet boundary conditions, the energy levels $E_{n}$ are the $N$ zeros of the polynomial equation

[TABLE]

whereas the IPR $I$ of the eigenstate associated with energy $E$ reads

[TABLE]

in terms of the above amplitudes $\psi_{a}=P_{a-1}(E)$.

It is useful to begin with a detailed study of the simplest case, namely $N=2$. Equation (5.4) giving the two energy levels reads

[TABLE]

Inserting the expressions (5.3) into (5.5), we obtain

[TABLE]

Eliminating the energy $E$ between (5.6) and (5.7), we obtain

[TABLE]

with

[TABLE]

To sum up, the two energy levels $E_{n}$ are the zeros of (5.6), while the corresponding IPR $I_{n}$ are the zeros of (5.8). The quantity $T$ reads (see (2.12))

[TABLE]

In the present situation, both IPR are equal ($I_{1}=I_{2}$), and so the polynomial entering (5.8) is a perfect square. This peculiarity however plays no role in the exposition of the method.

The pattern for an arbitrary size $N$ is as follows. The energy levels $E_{n}$ are the zeros of the polynomial equation (5.4), while (5.5) allows one to express the IPR $I$ as a rational function of the energy $E$. As a consequence, the $I_{n}$ are the zeros of a polynomial equation of degree $N$, of the form

[TABLE]

where $\Pi_{N}(I)$ is the resultant of the algebraic elimination of $E$ between (5.4) and (5.5). This observation simplifies the calculation of $T$ for all system sizes $N$. This quantity is indeed the sum of the $I_{n}$ (see (2.12)), and so

[TABLE]

This procedure can be easily automated and implemented in a symbolic algebra software. We thus obtain the following expressions for $T$.

$\bullet$ $N=2:$

[TABLE]

$\bullet$ $N=3:$

[TABLE]

$\bullet$ $N=4:$

[TABLE]

$\bullet$ $N=5:$

[TABLE]

with

[TABLE]

The following observations can be made at this stage. The quantity $T$ is a rational even function of the applied field $F$, whose complexity grows rapidly with $N$. Its degree in $F$ reads $d_{N}=N^{2}/2$ if $N$ is even, and $d_{N}=(N^{2}-1)/2$ if $N$ is odd. We recall that the degree of a rational function is the larger of the degrees of its numerator and denominator. In the present situation, these degrees coincide.

In the small-field limit, we have

[TABLE]

The first term is the known value of $T$ for a free tight-binding particle [8], i.e.,

[TABLE]

which saturates to the finite limit

[TABLE]

for large system sizes. The coefficient $T^{(2)}$ will be calculated in section 5.2 for all $N$ by means of perturbation theory (see (5.32), (5.33)).

In the large-field limit, $T$ obeys a linear-plus-constant growth of the form

[TABLE]

asserting the poor equilibration properties characteristic of a localized regime, with

[TABLE]

As $N$ increases, more and more terms of the above asymptotic expansions stabilize. This localized regime will be further investigated in sections 5.4 and 5.6.

5.2 Perturbation theory

The goal of this section is to determine the coefficient $T^{(2)}$ of the expansion (5.18) by means of second-order perturbation theory. The derivation follows the lines of section 5.1 of Reference [8], devoted to the case of a weak diagonal random potential.

In the absence of an electric field, the energy eigenvalues of a free tight-binding particle [8] read

[TABLE]

($n=1,\dots,N$). The corresponding normalized eigenstates are

[TABLE]

Their IPR read

[TABLE]

For odd sizes $N$, the IPR of the central eigenstate at zero energy, with number $n={\textstyle{1\over 2}}(N+1)$, is enhanced by a factor $4/3$. The expression (5.19) can be recovered by summing (5.26) over $n$.

In the presence of an electric field, the un-normalized $n$th eigenstate reads

[TABLE]

with

[TABLE]

where $E_{n}$ and $|n\rangle$ are given by (5.24) and (5.25), while $W$ is the operator

[TABLE]

where the quantity in the parentheses is the ratio $V_{a}/F$ (see (5.2)). The property $\langle n|W|n\rangle=0$ has been used in the derivation of (5.27).

The expansion (5.27) yields the following expansions to order $F^{2}$:

[TABLE]

We thus obtain the following expression for the coefficient $T^{(2)}$:

[TABLE]

The final step of the derivation, i.e., the evaluation of the above sums, could not be done by analytical means. We have instead used the following trick. We know from the general approach of section 5.1 that $T^{(2)}$ is a rational number. A numerical evaluation of (5.31) with very high accuracy ($N=400$ can be reached in one minute of CPU time) reveals that the denominator of $T^{(2)}$ only grows linearly with the system size. It is indeed observed to be a divisor of $140(N+1)$ if $N$ is even, and of $105(N+1)$ if $N$ is odd. This serendipitous situation allows one to unambiguously reconstruct the numerator of $T^{(2)}$ by means of a polynomial fit. We are thus left with the exact closed-form expressions:

$\bullet$ $N$ even:

[TABLE]

$\bullet$ $N$ odd:

[TABLE]

Up to $N=5$, the above results agree with the expansions of the exact expressions (5.13)–(5.16). The coefficient $T^{(2)}$ turns out to be negative for $N=3$ and $N=5$, resulting in a non-monotonic dependence of $T$ on the applied field $F$.

For large system sizes, the expansion (5.18) becomes

[TABLE]

We shall come back to this result in section 5.5 (see (5.80)).

5.3 Wannier-Stark spectrum on the infinite chain

The problem of a tight-binding particle on the infinite electrified chain has been studied long ago [46, 47, 48, 49, 50, 51]. The energy spectrum is an infinite Wannier-Stark ladder of the form $E_{k}=kF$, forgetting about the constant offset $(N+1)/2$ in (5.1).

5.3.1 Spatial structure of Wannier-Stark states.

The central Wannier-Stark state with energy

[TABLE]

is described by the normalized wavefunction

[TABLE]

where $J_{a}$ is the Bessel function. We have $\psi_{-a}=(-1)^{a}\psi_{a}$, so that the probabilities $\psi_{a}^{2}$ are symmetric around the origin. All the other states are obtained from the above central one by translations along the chain. The state with energy

[TABLE]

is described by the wavefunction

[TABLE]

centered at site $k$.

There is a formal identity between the wavefunction (5.36) describing the central stationary Wannier-Stark state on the electrified chain and the time-dependent wavefunction [58, 59]

[TABLE]

describing the ballistic spreading of a quantum walker modelled as a free tight-binding particle launched from the origin at time $t=0$, with time $t$ playing the role of the inverse field $1/F$, in our reduced units.

The regime of most physical interest is that of a small electric field. The square wavefunction of the central Wannier-Stark state is shown in figure 4 for $F=0.02$. We summarize here for future use its behavior in various regions in the small-field regime. The following estimates stemming from the theory of Bessel functions [42, 60] can be recovered by semi-classical techniques based on the saddle-point method, as described in the physics literature [58, 59].

Allowed region ($|aF|<2$).

The wavefunction only takes appreciable values in an allowed region extending up to a distance $2/F$ on either side of the origin. The size of the allowed region,

[TABLE]

plays the role of a localization length. More precisely, setting

[TABLE]

we have

[TABLE]

The allowed region is limited by two transition regions, located symmetrically near $a=\pm L(F)/2=\pm 2/F$, corresponding to $\theta\to 0$ and $\theta\to\pi$.

Transition regions ($|aF|\approx 2$).

In the transition regions at the endpoints of the allowed region, the wavefunction exhibit sharp peaks. In the right transition region, setting

[TABLE]

we have

[TABLE]

where ${\mathop{\rm Ai}}(z)$ is the Airy function. The width of the transition regions,

[TABLE]

provides a second length scale which diverges in the small-field regime, albeit much less rapidly that $L(F)$ (see (5.40)).

Forbidden regions ($|aF|>2$).

The wavefunction falls off essentially exponentially fast in the forbidden regions. In the right forbidden region, setting

[TABLE]

we have

[TABLE]

5.3.2 IPR of Wannier-Stark states.

All Wannier-Stark states have the common IPR

[TABLE]

This quantity was already investigated [58] in the context of the time-dependent wavefunction (5.39). It has also attracted some attention in the mathematical literature (see [61, 62] and references therein).

In the large-field regime ($F\gg 1$), the central Wannier-Stark state is very tightly localized near the origin. The amplitudes $\psi_{a}$ indeed fall off as

[TABLE]

The expansion (5.22) of $I(F)$ can therefore be recovered easily. An all-order asymptotic expansion can be derived by means of the following expression of the Mellin transform [58]:

[TABLE]

Summing the contributions of the simple poles at $s=-2,-4,\dots$, we obtain

[TABLE]

The above result can be recovered by using the expression of $I(F)$ in terms of a hypergeometric function [61, 62]:

[TABLE]

In the small-field regime, we have

[TABLE]

The first term of the above expression exhibits a logarithmic correction, stemming from the double pole of $M(s)$ at $s=1$, with respect to the leading scaling $I(F)\sim 1/L(F)\sim F$. The presence of this logarithmic correction to scaling was already emphasized in the time-dependent problem [58], where it was put in perspective with the following phenomenon dubbed bifractality. The generalized IPR

[TABLE]

with $q>0$ arbitrary, exhibits an anomalous scaling behavior as $F\to 0^{+}$, of the form

[TABLE]

with

[TABLE]

The usual IPR $I(F)$ corresponds to $q=2$, i.e., precisely the break point between both branches of the above bifractal spectrum, where the generalized IPR is respectively dominated by the allowed region (for $q<2$) and by the transition regions (for $q>2$).

The subleading oscillatory contribution in (5.53), whose derivation requires more sophisticated techniques [62], results in a non-monotonic behavior of the IPR $I(F)$ as a function of the field $F$.

5.4 Main features of the IPR $I_{n}$ and of $T$

In this section we provide a picture of the salient features of the IPR $I_{n}$ of the various eigenstates, and of their sum $T$.

Figure 5 shows the IPR $I_{n}$ against the energy $E_{n}$ for all eigenstates of finite chains of various lengths ranging from $N=8$ to 100. The applied field reads $F=0.1$, so that the localization length is $L(F)=40$. The horizontal line shows the value $I(F)=0.035\,222\dots$ of the IPR of Wannier-Stark states. The IPR $I_{n}$ have been evaluated by means of a direct numerical diagonalization of the Hamiltonian matrix ${\cal H}$. The symmetry of the plot with respect to $E=0$ is manifest. As $N$ increases, the IPR of the few first (or last) eigenstates converge very fast to well-defined limiting values. In the small-field regime, these limits will be shown to scale as $F^{1/3}$ (see (5.68)). They are therefore much larger than the IPR $I(F)$ of Wannier-Stark states, which is proportional to $F$, with a logarithmic correction (see (5.53)). In the localized regime, i.e., for sample sizes $N$ larger than the localization length $L(F)$, there are approximately $N-L(F)$ bulk states in the central part of the energy spectrum:

[TABLE]

These eigenstates are exponentially close to the Wannier-Stark states whose allowed region is entirely contained in the sample (see e.g. [52, 53, 56]). Their IPR $I_{n}$ is therefore very close to $I(F)$. The remaining eigenstates are edge states, living near either boundary of the system, whose energies lie in the wings of the spectrum:

[TABLE]

Their IPR vary in a broad range between $I(F)$ and $F^{1/3}$. On shorter samples, whose length $N$ is smaller than the localization length $L(F)$, the distinction between bulk and edge states looses its meaning. All the IPR $I_{n}$ are larger than $I(F)$ and manifest a rather irregular dependence on the state number $n$. This is exemplified by the data for system sizes $N=8$ to 20 in figure 5.

The above picture substantiates the asymptotic linear-plus-constant growth of $T$ with the system size $N$, already advocated in (5.21):

[TABLE]

This estimate holds not only for large fields, but all over the localized regime, defined by the inequality $N>L(F)$, i.e., $NF>4$. The slope $I(F)$ is the IPR of Wannier-Stark states (see (5.48)). The offset $J(F)$ somehow encodes the contributions of all edge states. Its exact value is not known for arbitrary values of the field $F$. It can be noticed that the coefficients of its large-field expansion (5.23) do not exhibit any obvious regularity. The quantity $J(F)$ will be investigated in section 5.6 in the small-field regime (see (5.104)).

Figure 6 shows the ratio $T/N$ against $F$ for several system sizes $N$. The lowest black line shows the IPR $I(F)$, corresponding to the $N\to\infty$ limit of $T/N$, according to (5.59). The data for finite $N$ converge smoothly to that limit, for any fixed non-zero value of the electric field $F$.

The quantitative analysis of the IPR $I_{n}$ and of their sum $T$ in the small-field regime will require some special care. Two different characteristic lengths indeed diverge in the $F\to 0^{+}$ limit, namely $L(F)\sim 1/F$ (see (5.40)) and $\lambda(F)\sim 1/F^{1/3}$ (see (5.45)). Each of these diverging lengths is responsible for the occurrence of a non-trivial scaling regime (see sections 5.5 and 5.6).

5.5 Airy scaling regime $(N\sim\lambda(F)\sim 1/F^{1/3})$

This section is devoted to the regime where the system size $N$ and the size $\lambda(F)$ of the transition regions (see (5.45)) are large and comparable. We introduce the scaling variable

[TABLE]

proportional to the ratio $N/\lambda(F)$. Since the wavefunction of the Wannier-Stark states in the transition regions is an Airy function (see (5.44)), we dub this regime the Airy scaling regime.

5.5.1 Extremal edge states.

Let us first investigate the extremal edge states on a very large sample, with energies near the left edge of the spectrum ($E\approx-{\textstyle{1\over 2}}NF-2$). A direct approach goes as follows (see e.g. [54, 55]). Setting

[TABLE]

and replacing $N+1$ by $N$ in the offset of the electrostatic potential, the tight-binding equation (5.1) becomes

[TABLE]

Using a continuum description, this reads

[TABLE]

with

[TABLE]

and with the boundary conditions that $\chi$ vanishes for $a=0$ and $a\to+\infty$. Then, setting

[TABLE]

equation (5.63) boils down to Airy’s equation

[TABLE]

whose solution decaying as $z\to+\infty$ is $\chi(z)={\mathop{\rm Ai}}(z)$.

The extremal edge states are quantized by the Dirichlet boundary condition at $a=0$. The latter requires $c=c_{n}$, where $-c_{n}$ are the zeros of the Airy function ($n=1,2,\dots$). These zeros are real and negative, and so $c_{n}>0$. The extremal edge states on a very large sample therefore have energies

[TABLE]

The IPR of these states read

[TABLE]

with

[TABLE]

and

[TABLE]

The above results can be made more explicit for large $n$, which turns out to be the regime of most interest. The asymptotic behavior of the Airy function

[TABLE]

implies

[TABLE]

Then, for large $c$, averaging first the powers of the rapidly varying sine function, we obtain

[TABLE]

The finite part $c_{0}$ of the logarithm has been evaluated by more sophisticated techniques [63], yielding

[TABLE]

where $\gamma$ is again Euler’s constant. We thus obtain the estimate

[TABLE]

5.5.2 Scaling behavior of $T$.

The evaluation of $T$ for large but finite systems in the Airy scaling regime goes as follows. The IPR $I_{n}$ of the $n$th eigenstate scales as

[TABLE]

with

[TABLE]

The above estimates respectively match (5.26) and (5.68).

As a consequence, $T$ obeys a scaling law

[TABLE]

of the very same form as (1.1), corresponding to a particle in a weak disordered potential [8], albeit with a different scaling variable $x$, and a different function $\Phi(x)$. The first term is the value of $T$ for a free particle, given by (5.19), and so we have $\Phi(0)=0$. The following observation [8] also applies to the present situation. Had we chosen the limit value (5.20) for $T^{(0)}$ as the first term in (5.79), instead of the exact expression (5.19) for finite $N$, the scaling function $\Phi(x)$ would have acquired a finite offset depending on the parity of $N$, i.e., $\Phi(0)=-3/2$ (resp. $\Phi(0)=-1$) for $N$ even (resp. $N$ odd).

When the scaling variable $x$ is small, the perturbative result (5.34) implies

[TABLE]

The asymptotic behavior of the scaling function $\Phi(x)$ at large $x$ can be estimated as follows. The crossover between both estimates entering (5.78) remains steep, in the sense that it does not broaden out as the state number $n$ gets larger and larger. The sum $T$ can therefore be estimated as

[TABLE]

Using the expression (5.76) for the amplitudes $a_{n}$, we obtain after some algebra

[TABLE]

where $n_{c}(x)$ is the value of $n$ such that both arguments of the max function entering (5.81) coincide. To leading order in $\ln x\gg 1$, we have

[TABLE]

and so

[TABLE]

Figure 7 shows a log-log plot of the ratio $\Phi(x)/x^{4}$ against the scaling variable $x$. This ratio has been chosen in order to better reveal the structure of the scaling function $\Phi(x)$. Color lines show data pertaining to system sizes 100, 200 and 400. The straight line to the left shows the power-law result (5.80) stemming from perturbation theory. The line to the right shows the estimate (5.84). The data exhibit a very good convergence to the scaling function $\Phi(x)$, as well as a good agreement with both analytical predictions. Small oscillatory corrections are however visible at very high $x$.

5.6 Bessel scaling regime $(N\sim L(F)\sim 1/F)$

This section is devoted to the regime where the system size $N$ and the localization length $L(F)$ (see (5.40)) are large and comparable. We introduce the scaling variable

[TABLE]

proportional to the ratio $N/L(F)$, and dub this regime the Bessel scaling regime.

5.6.1 Generic edge states.

Let us first investigate the generic edge states on a very large sample in the localized regime ($y>4$), considering the left edge for definiteness. These edge states are very well described by the restriction of the Wannier-stark states (5.38), i.e.,

[TABLE]

to a part of their allowed region. The allowed region of the above wavefunction is $k-2/F<a<k+2/F$. Left edge states therefore correspond to the range $-2/F<k<2/F$, i.e., $-{\textstyle{1\over 2}}y-2<E<-{\textstyle{1\over 2}}y+2$, in agreement with (5.58).

The above edge states are quantized by the Dirichlet boundary condition $\psi_{0}=0$. Setting

[TABLE]

the expression (5.42) of the wavefunction in the allowed region yields the quantization condition

[TABLE]

As the angle $\alpha$ increases from [math] to $\pi$, the edge state number $n$ ranges from 1 to $n_{{\rm edge}}$, where

[TABLE]

is our estimate for the number of edge states in each wing of the spectrum in the small-field regime. The quantization condition (5.88) yields the density of states

[TABLE]

normalized as

[TABLE]

The IPR of the above edge states read

[TABLE]

with

[TABLE]

and $\psi_{a,n}$ is given by (5.86), together with the quantization condition (5.88). The sums $S_{m,n}$ can be estimated by using the expression (5.42) of the wavefunction in the allowed region, averaging first the powers of the sine function, and transforming the sums over $a$ into integrals over $\theta$ in the range $0<\theta<\alpha_{n}$, according to (5.41). We thus obtain

[TABLE]

and so

[TABLE]

The cutoff $\varepsilon$ is independent of the state number $n$. It can therefore be determined by matching the Airy and Bessel regimes. For $n\ll n_{{\rm edge}}$, i.e., $nF\ll 1$, (5.88) reads

[TABLE]

Identifying the expressions (5.68), (5.76) in the Airy regime and (5.96), (5.97) in the Bessel regime yields

[TABLE]

We are thus left with the estimate

[TABLE]

5.6.2 Scaling behavior of $T$.

The evaluation of $T$ for finite systems in the Bessel scaling regime goes as follows. The cases $y>4$ and $y<4$ have to be dealt with separately.

$\bullet$ $y>4$.

Consider first the simple situation of samples in the localized regime, such that $N>L(F)$, i.e., $y>4$. As already discussed in section 5.4, there are $N-2n_{{\rm edge}}=N-L(F)=(y-4)/F$ bulk states, whose IPR essentially coincide with $I(F)$, and $2n_{{\rm edge}}=L(F)=4/F$ edge states, whose IPR vary according to (5.99). We thus obtain

[TABLE]

where the contribution of all edge states reads

[TABLE]

with

[TABLE]

The result (5.100) has the expected linear-plus-constant form (5.21), (5.59). It yields the following estimate for the offset $J(F)$ in the small-field regime:

[TABLE]

Finally, (5.100) can be recast as

[TABLE]

with

[TABLE]

$\bullet$ $y<4$.

Consider now shorter samples, such that $N<L(F)$, i.e., $y<4$. As discussed in section 5.4, there is no clear-cut distinction between bulk and edge states in this situation. As a consequence, the formula (5.99) for the IPR of edge states is not sufficient to derive an estimate of $T$ for $y<4$. We nevertheless hypothesize that $T$ still obeys a logarithmic scaling law in $F$ of the form (5.105), even though the amplitudes $U(y)$ and $V(y)$ cannot be predicted analytically. Figure 8 presents numerical results for these amplitudes. For each value of $y$, $T$ has been calculated for 15 roughly equidistant system sizes $N$ on a logarithmic scale between $N=10$ and $N=5000$. These values of $T$ obey accurately the logarithmic dependence on $F$, i.e., on $N=y/F$, hypothesized in the scaling form (5.105). The resulting estimates for $U(y)$ and $V(y)$ are shown as symbols in both panels of figure 8. Black lines show fits of the form $U(y)=(u_{1}\ln y+u_{2})\sqrt{y}+(u_{3}\ln y+u_{4})y$ and $V(y)=3/2+(v_{1}\ln y+v_{2})\sqrt{y}+(v_{3}\ln y+v_{4})y$. The form of these fits is freely inspired by the expected crossover at small $y$ with the estimate (5.84) for large $x$ in the Airy regime, i.e.,

[TABLE]

In any case, the proposed fits describe the data accurately. Finally, the blue straight lines show the analytical results (5.106), (5.107) for $y>4$. The data suggest that the functions $U(y)$ and $V(y)$ are continuous at $y=4$, as well as their first derivatives.

6 Discussion

A novel way of investigating equilibration – or lack of equilibration – in small isolated quantum systems has been put forward recently [8]. The key quantity of that approach is the trace $T$ of the matrix $Q$ of asymptotic transition probabilities in some preferential basis chosen once for all. The quantity $T$ is also the sum of the inverse participation ratios of all eigenstates of the Hamiltonian in the same basis. This transition matrix formalism therefore requires the knowledge of all energy eigenvectors, so that its scope is limited to small systems. This is however consistent with the purpose of the approach, which is precisely to quantify the degree of equilibration of small quantum systems. The quantity $T$ provides a measure of the degree of equilibration of the system if launched from a typical basis state of the preferential basis. It may vary between $T_{{\rm min}}=1$ and $T_{{\rm max}}=N$, the dimension of the Hilbert space; the larger $T$, the poorer the equilibration.

The models investigated in [8] and in the present work embrace the most significant physical systems with a finite-dimensional Hilbert space. The outcomes of both papers allow one to get a full understanding of the behavior of the quantity $T$. Quantum systems whose dynamics is governed by a generic Hamiltonian exhibit good typical equilibration properties, so that $T$ saturates to a non-trivial limit for large $N$. This limit is however always found to be larger than $T_{{\rm min}}=1$. It equals $T^{(0)}=3/2$ for a free tight-binding particle on a large open interval [8], and $T_{\rm U}=2$ or $T_{\rm O}=3$ for a random Hamiltonian whose distribution is invariant under the groups U$(N)$ or O$(N)$ (section 3).

An interesting situation is met when varying a system parameter through some scaling regime induces a continuous crossover from good to bad equilibration properties. Three systems have been shown to exhibit such a crossover phenomenon, where $T$ is found to obey non-trivial scaling behavior. Table 2 summarizes the following comparative discussion of these three models. For a spin $S$ in a tilted magnetic field (section 4), making an angle $\theta$ with the preferred quantization axis, we have $T=T_{{\rm max}}=N$ for $\theta=0$, while $T\sim\ln N$ for any fixed non-zero tilt angle $\theta$. Both regimes are connected by the scaling laws (4.23), (4.24), with scaling variable $u=N\theta$. For a tight-binding particle on a finite open chain in a weak random potential [8], $T$ obeys the scaling law recalled in the Introduction (see (1.1)), with scaling variable $x=Nw^{2/3}$, which describes the crossover between the finite value $T=T^{(0)}=3/2$ of a free particle in the ballistic regime and the linear growth $T\approx N{\overline{Q}}$ in the localized regime, testifying poor equilibration, with ${\overline{Q}}\approx Aw^{4/3}$ for a weak disorder. For a tight-binding particle on a finite electrified chain (section 5), $T$ obeys a similar but richer behavior in the situation where the applied field $F$ is small, so that the localization length $L(F)=4/F$ is large. The quantity $T$ indeed exhibits two successive scaling laws describing the crossover between the ballistic regime and the insulating one, namely a first scaling law of the form (5.79) with scaling variable $x=NF^{1/3}$ in the so-called Airy scaling regime, followed by a second scaling law of the form (5.105) with scaling variable $y=NF$ in the so-called Bessel scaling regime.

Numerical results on the XXZ spin chain [9] demonstrate that a transition from good to bad equilibration properties, in the sense of a change in the growth law of $T$, can be observed in a clean quantum many-body system which does not exhibit many-body localization. There, the transition is found to coincide with a thermodynamical phase transition, with $T\sim L$ in the gapless phase of the model ($|\Delta|<1$), and $T\sim\exp(aL)$ in the gapped phase ($|\Delta|>1$). It would be worth sorting out in a broader setting the connection between changes in the scaling behavior of $T$ and various types of phase transitions, involving or not the appearance of many-body localized phases.

It is a pleasure to thank P Krapivsky, K Mallick and G Misguich for fruitful discussions, and especially V Pasquier for his interest in this work and for a careful reading of the manuscript.