Note on a Non Linear Perturbation of the Ideal Bose Gas

TL;DR

This paper demonstrates that introducing a U(1) symmetry breaking field in an ideal Bose gas is equivalent to adding a nonlinear function of the zero mode number operator, leading to unconventional Bose-Einstein condensation at negative chemical potential.

Contribution

It establishes the equivalence between a symmetry-breaking perturbation and a nonlinear modification of the Hamiltonian in the thermodynamic limit, extending to a class of diagonal models.

Findings

Models exhibit non-conventional BEC at negative chemical potential.

Limit pressures of the models coincide in the thermodynamic limit.

Equivalence extends to a broader class of diagonal models.

Abstract

In this work we show that the introduction of a U(1) symmetry breaking field in the energy operator of the boson-free gas, is equivalent, in the thermodynamic limit, to the inclusion, in the Hamiltonian of the ideal gas, of a non-linear function of the number operator associated with the zero mode. In other words, the limit pressures coincide. Moreover, both models undergo non conventional Bose-Einstein condensation (BEC) for strictly negative values of the chemical potential $\mu.$ Finally, the proof of equivalence of limit pressures is extended to a class of full-diagonal models.

Equations

$$[{\hat{a}}_{\mathbf{p}},{\hat{a}}_{\mathbf{q}}^{†}]={\delta }_{\mathbf{pq}}I,$$

$${\Xi }_l(\beta ,\mu ):={\mathrm{Tr}}_{{\mathcal{F}}_B}\exp \left(-\beta {\hat{H}}_l(\mu )\right),p_l(\beta ,\mu ):=\frac{1}{\beta V_l}\ln {\Xi }_l(\beta ,\mu );$$

$$Z_{N,l}(\beta ,\varrho ):={\mathrm{Tr}}_{{\mathcal{H}}_B^{(N)}}{e}^{-\beta {\widehat{H_l}}^{(N)}},f_l(\beta ,{\rho }_l):=-\frac{1}{\beta V_l}\ln Z_{N,l}(\beta ,\varrho ),$$

$$⟨\cdot {⟩}_{{\hat{H}}_l(\mu )}={\Xi }^{-1}(\beta ,\mu ){\mathrm{Tr}}_{{\mathcal{F}}_B}\cdot \exp \left(-\beta {\hat{H}}_l(\mu )\right),$$

$$⟨\cdot {⟩}_{{\hat{H}}_l^{(N)}}=Z_{N,l}(\beta ,\varrho ){\mathrm{Tr}}_{{\mathcal{H}}_B^{(N)}}\cdot \exp \left(-\beta {\hat{H}}_l^{(N)}\right),$$

$$f(\beta ,\varrho ):=\underset{N_l,V_l\to \infty }{\lim }{f}_l(\beta ,{\varrho }_l),\text{mbox}assumingthat\underset{N_l,V_l\to \infty }{\lim }{\varrho }_l=\varrho =\text{mbox}constant,$$

$$p(\beta ,\mu ):=\underset{V_l\to \infty }{\lim }{p}_l(\beta ,\mu ),\text{mbox}with$$

$$\underset{V_l\to \infty }{\lim }{⟨\frac{\hat{N}}{V_l}⟩}_{{\hat{H}}_l(\mu )}=\underset{V_l\to \infty }{\lim }{\rho }_l=\rho (\mu )=\text{mbox}constant.$$

$${\hat{H}}_l^I\ge -\frac{C_2}{V_l}\hat{N}+\frac{C_1}{V_l}{\hat{N}}^2$$

$$\underset{V_l\to \infty }{\lim }{⟨\frac{{\hat{n}}_0}{V_l}⟩}_{{\hat{H}}_l(\mu )}={\rho }_0>0.$$

$$\underset{\delta \to 0^{+}}{\lim }\underset{V_l\to \infty }{\lim }\frac{1}{V_l}\sum _{\mathbf{p}\in {\Lambda }^{\ast },{\lambda }_l(\mathbf{p})<\delta }⟨{\hat{n}}_{\mathbf{p}}{⟩}_{{\hat{H}}_l(\mu )}>0.$$

$$\left|{⟨{\hat{a}}_0^{†}⟩}_{{\hat{H}}_l(\mu )}\right|=\left|{⟨{\hat{a}}_0⟩}_{{\hat{H}}_l(\mu )}\right|=0,$$

$$\left|{⟨{\hat{a}}_0^{†}⟩}_{{\hat{H}}_{l,\nu ,\phi }(\mu )}\right|=\left|{⟨{\hat{a}}_0⟩}_{{\hat{H}}_{l,\nu ,\phi }(\mu )}\right|=\sqrt{V}{\eta }_{l,\nu ,\phi }\ne 0,$$

$$\displaystyle\lim_{\nu\to 0}\displaystyle\lim_{V\to\infty}\eta^{2}_{l}=\left\{\begin{array}[]{cl}\rho_{\mathbf{0}}\neq 0&\mbox{if }\rho>\rho_{c}\\ 0&\mbox{if }\rho\leq\rho_{c}\end{array}\right.$$

$$\underset{\nu \to 0}{\lim }\underset{V_l\to \infty }{\lim }{\eta }_{l,\nu ,\phi }=\underset{V\to \infty }{\lim }\underset{\nu \to 0}{\lim }{\eta }_{l,\nu ,\phi }=0.$$

$$\underset{\nu \to 0}{\lim }\underset{V_l\to \infty }{\lim }{\eta }_{l,\nu ,\phi }\ne \underset{V\to \infty }{\lim }\underset{\nu \to 0}{\lim }{\eta }_{l,\nu ,\phi }$$

$$\prec \cdot \succ :=\underset{h\to 0}{\lim }\underset{V\to \infty }{\lim }{⟨-⟩}_{{\hat{H}}_{l,\nu ,\phi }(\mu )}$$

$${\hat{H}}_{l,\nu ,\phi }^0=\sum _{\mathbf{p}}{\lambda }_l(\mathbf{p}){\hat{a}}_{\mathbf{p}}^{†}{\hat{a}}_{\mathbf{p}}-\sqrt{V}\nu \left({\hat{a}}_0{e}^{-i\phi }+{\hat{a}}_0^{†}{e}^{i\phi }\right),$$

$$\underset{\nu \to 0^{+}}{\lim }\underset{V\to \infty }{\lim }{⟨{\hat{\rho }}_{0,l}⟩}_{{\hat{H}}_{l,\nu ,\phi }(\mu )}={\rho }_0,\underset{\nu \to 0^{+}}{\lim }\underset{V\to \infty }{\lim }{⟨{\hat{\eta }}_l⟩}_{{\hat{H}}_{l,\nu ,\phi }(\mu )}=\sqrt{{\rho }_0}{e}^{i\phi }$$

$$\underset{\nu \to 0^{+}}{\lim }\underset{V\to \infty }{\lim }{⟨{\hat{\rho }}_{0,l}⟩}_{{\hat{H}}_{l,\nu ,\phi }(\mu )}=\underset{h\to 0^{+}}{\lim }\underset{V\to \infty }{\lim }|{⟨{\hat{\eta }}_l⟩}_{{\hat{H}}_{l,\nu ,\phi }(\mu )}{|}^2={\rho }_0.$$

$${\hat{a}}_0=-\frac{\nu }{\mu }{e}^{i\phi }\sqrt{V}+{\hat{b}}_0,{\hat{a}}_0^{†}=-\frac{\nu }{\mu }{e}^{-i\phi }\sqrt{V}+{\hat{b}}_0^{†}.$$

$${\hat{H}}_{l,\nu }^0(\mu )=-\mu {\hat{b}}_0^{†}{\hat{b}}_0+\sum _{\mathbf{p}\in {{\Lambda }_l}^{\ast }\backslash \{0\}}\left(\frac{{\mathbf{p}}^2}{2}-\mu \right){\hat{a}}_{\mathbf{p}}^{†}{\hat{a}}_{\mathbf{p}}+\frac{{\nu }^{2}V}{\mu }.$$

$$\mu ={\mu }_{\ast }=-\frac{\nu }{\sqrt{{\rho }_0}},$$

$${⟨{\hat{b}}_0^{†}⟩}_{{\hat{H}}_{l,\nu }^0({\mu }_{\ast })}={⟨{\hat{b}}_0⟩}_{{\hat{H}}_{l,\nu }^0({\mu }_{\ast })}=0.$$

$${⟨{\hat{a}}_0^{†}⟩}_{{\hat{H}}_{l,\nu }^0({\mu }_{\ast })}=e^{-i\phi }\sqrt{V{\rho }_0},{⟨{\hat{a}}_0⟩}_{{\hat{H}}_{l,\nu }^0({\mu }_{\ast })}=e^{i\phi }\sqrt{V{\rho }_0}$$

$${⟨{\hat{b}}_0^{†}{\hat{b}}_0⟩}_{{\hat{H}}_{l,\nu }^0({\mu }_{\ast })}={\left(\exp \beta \left(-{\mu }_{\ast }\right)-1\right)}^{-1},{⟨{\hat{n}}_{\mathbf{p}}⟩}_{{\hat{H}}_{l,\nu }^0({\mu }_{\ast })}={\left(\exp \beta \left(\frac{{\mathbf{p}}^2}{2}-{\mu }_{\ast }\right)-1\right)}^{-1}.$$

$${\rho }_{c,l}(\beta ,{\mu }_{\ast })=\frac{1}{V}\sum _{\mathbf{p}\in {{\Lambda }_l}^{\ast }\backslash \{0\}}{\left(\exp \beta \left(\frac{{\mathbf{p}}^2}{2}-{\mu }_{\ast }\right)-1\right)}^{-1}.$$

$${\rho }_c(\beta ,{\mu }_{\ast })=\frac{1}{(2\pi {)}^d}\int {\left(\exp \beta \left(\frac{{\mathbf{p}}^2}{2}-{\mu }_{\ast }\right)-1\right)}^{-1}{d}^3\mathbf{p}.$$

$${⟨\frac{{\hat{b}}_0^{†}{\hat{b}}_0}{V}⟩}_{{\hat{H}}_{l,\nu }^0({\mu }_{\ast })}={⟨\left(\frac{{\hat{a}}_0^{†}}{\sqrt{V}}-\sqrt{{\rho }_0}{e}^{-i\phi }\right)\left(\frac{{\hat{a}}_0}{\sqrt{V}}-\sqrt{{\rho }_0}{e}^{i\phi }\right)⟩}_{{\hat{H}}_{l,\nu }^0({\mu }_{\ast })}$$

$$=\underset{V\to \infty }{\lim }\frac{1}{V}{\left(\exp \beta \left(-{\mu }_{\ast }\right)-1\right)}^{-1}=0.$$

Full text

1. Introduction

Until 2013 it was believed that, from the point of view of a physical experiment, to confine a homogeneous system of Bose atoms, and to make it pass, subsequently, to the thermodynamic limit, would be an impossible task to perform. Thus [3],

In the magnetic traps, not only is the number of particles quite small, compared to the usual case, but the “boundary,” formed by a quadratic potential well, extends literally throughout the whole system. In order to take the thermodynamic limit in such a system it is necessary to weaken the potential so that, as the number of particles increases, the average density remains constant. This is well-defined mathematically, but is of course physically unrealizable. On the other hand, taking the box size to infinity in the homogeneous case is also unrealized experimentally.

Moreover, trapped gases were, generally speaking, spatially inhomogeneous. In this framework, to overcome the difficulties in defining pressure and volume for a gas confined in an inhomogeneous trap, it has been necessary to define macroscopic parameters that behave like them.

However, in 2013, BEC in a quasi uniform three dimensional potential of an optical trap box (cilindrical optical box) BEC was observed [4, 5].

The authors of ref.[4] point out that:

We have observed the Bose-Einstein condensation of an atomic gas in the (quasi)uniform three-dimensional potential of an optical box trap. Condensation is seen in the bimodal momentum distribution and the anisotropic time-of-flight expansion of the condensate. The critical temperature agrees with the theoretical prediction for a uniform Bose gas. The momentum distribution of a noncondensed quantum-degenerate gas is also clearly distinct from the conventional case of a harmonically trapped sample and close to the expected distribution in a uniform system. We confirm the coherence of our condensate in a matter-wave interference experiment. Our experiments open many new possibilities for fundamental studies of many-body physics.

Later, in the same article, they indicate that,

The thermodynamics of our gas are therefore very close to the textbook case of a uniform system and very different from the case of a harmonically trapped sample.

In this sense, this seems to be a suitable experimental scenario to test the consistency of the Bogoliubov’s theory -based on the concept of quasiaverages  [12]-, about the spontaneous rupture of the $U(1)$ symmetry and simultaneous emergence of Bose Einstein condensation in the case of an ideal Bose gas. In other words, experiments could be carried out, in this framework, to study the thermodynamic behavior of an Ideal Bose gas system when it is disturbed by a nonvanishing external field, that breaks the symmetry $U(1)$. The question is: what is, in this case, the nature of such a condensation?

2. Basic Notions

2.1. Grand canonical and canonical ensembles

Let $\hat{H}_{l}$ be a selfadjoint operator on the Hilbert space ${\mathcal{F}}_{\mathrm{B}}$ (Fock space), representing the energy operator of a Bose particle system. Let $\beta=T^{-1}$ and $\mu\in\mathbb{R}$ be the inverse temperature and the so-called chemical potential, respectively.

$\hat{H}_{l}(\mu)$ is defined as $\hat{H}_{l}(\mu):=\hat{H}_{l}-\mu\hat{N},$ where $\hat{N}$ is the total number operator given by $\hat{N}=\displaystyle\sum_{{\mathbf{p}}}\hat{n}_{\mathbf{p}},$ being $\hat{n}_{\mathbf{p}}=\hat{a}^{{\dagger}}_{\mathbf{p}}\hat{a}_{\mathbf{p}}$ the number operator associated to the ${\mathbf{p}}-$ mode.

The operators $\hat{a}^{{\dagger}}_{\mathbf{p}},\hat{a}_{\mathbf{p}}$ defined on the Fock space, well-known as creations and annihilation operators, respectively, satisfies the commutation rules:

[TABLE]

being $\delta_{\mathbf{p}\mathbf{q}}$ the kronecker delta and $I$ the identity operator.

$\hat{H}_{l}$ can be decomposed as the followig sum: $\hat{H}_{l}=\hat{H}^{0}_{l}+\hat{H}^{I}_{l},$ where $\hat{H}^{0}_{l}=\displaystyle\sum_{\mathbf{p}}\lambda_{l}({\mathbf{p}})\hat{a}^{{\dagger}}_{\mathbf{p}}\hat{a}_{\mathbf{p}}$ and $\hat{H}^{I}_{l}=\displaystyle\sum_{{\mathbf{p}},{\mathbf{q}},{\mathbf{r}},{\mathbf{s}}}U_{{\mathbf{p}},{\mathbf{q}},{\mathbf{r}},{\mathbf{s}}}\hat{a}^{{\dagger}}_{\mathbf{p}}\hat{a}^{{\dagger}}_{\mathbf{q}}\hat{a}_{\mathbf{r}}\hat{a}_{\mathbf{s}}$ are the seccond quantizations of the laplacian operator and of the interaction $U,$ respectively, both defined on the region of confinement of particles $\Lambda\subset\mathbb{R}^{d},$ with $d\in\mathbb{N}.$

We shall assume in this work, periodic boundary conditions. In this case all the subscripts $\mathbf{p}$ belong to the set $\Lambda^{*}$ (dual of $\Lambda$) defined as $\Lambda^{*}_{l}=\{\mathbf{p}=(p_{1},\dots,p_{d})\in{\mathbb{R}}^{d}:p_{\alpha}={2\pi n_{\alpha}}/{l},n_{\alpha}\in{\mathbb{Z}},\alpha=1,2,...,d\},$ and $\lambda_{l}(\mathbf{p})={\mathbf{p}^{2}}/{2}.$

With these definitions, at finite volume, it is posible to introduce the grand canonical partition function $\Xi_{l}(\beta,\mu)$ and the pressure $p_{l}(\beta,\mu):$

[TABLE]

the canonical partition function $Z_{N,l}(\beta,\varrho)$ and the free energy $f_{l}(\beta,\varrho_{l}),$ where $\varrho_{l}=\frac{N}{V_{l}}:$

[TABLE]

and, finally, the Gibbs states in the grand canonical ensemble and in the canonical ensemble:

[TABLE]

[TABLE]

respectively.

The limit free energy $f(\beta,\varrho)$ and the limit pressure $p(\beta,\mu)$ are defined as:

[TABLE]

and

[TABLE]

[TABLE]

On the other hand, stable systems are defined as those for which there exists $\mu_{*}\in\mathbb{R}$ such that only for $\mu\in(-\infty,\mu_{*}],$ $p(\beta,\mu)<\infty,$ while superstable sytems satisfies $p(\beta,\mu)<\infty.$ for all values of $\mu.$ Finally, if the following inequality (in the sense of operators)

[TABLE]

holds, the system is superstable.

2.2. Types of BEC

• •

Condensation of type I corresponds to a macroscopic occupation of a finite nuber of states. Thus, a macroscopic occupation of the the ground state, or traditional Bose-Einstein condensation, is given by the fulfilment of the condition

[TABLE]

For the latter, in the condensed-uncondensed phase transition the appropiate order parameter is $\rho_{\mathbf{0}}=\rho-\rho_{c},$ being $\rho_{c}$ a critical density.

• •

Condensation of type II holds when there exists an infinite number of states macoscopically occupied.

• •

Condensation of the type III holds when there are not macroscopically occupied states but the following condition holds:

[TABLE]

The third type of Bose condensation, denominated generalized BEC (GBEC), was introduced by M. Girardeau in 1960 [1].

GBEC is more robust that the other kinds of condensation in the sense that it is independent on the shape of the confining region. Indeed, in the case of the free Bose gas, it always occurs for particle density values larger than a critical one.

These kinds of critical phenomena are in agreement with the standard phase transitions theory that identifies critical points with the emergence of singularities in the thermodynamic functions in the thermodynamic limit.

However, there is a fourth type of condensation independent on temperature and, for this reason, called non conventional. It is in the study of this phenomenon that we are interested in this work.

3. BEC and spontaneous symmetry breaking (SSB)

The standard strategy devoted to associate symmetry breaking with certain phase transition consists in introducing a small term on the original energy operator, preserving its self-adjointness but eliminating the symmetry corresponding to some conservation law.

Thus, in the case of Bose systems, it is posible to break the global $U(1)$ symmetry by adding the extra term $-\sqrt{V}\nu\left(\hat{a}_{\mathbf{0}}e^{-i\varphi}+\hat{a}^{{\dagger}}_{\mathbf{0}}e^{i\varphi}\right)$ to the original energy operator $\hat{H}_{l}(\mu)=\hat{H}_{l}-\mu\hat{N},$ which satisfies $[\hat{H}_{l}(\mu),\hat{N}],$ being $\hat{N}$ the total number operator, obtaining the new Hamiltonian $\hat{H}_{l,\nu,\varphi}(\mu)=\hat{H}_{l}(\mu)-\sqrt{V}\nu\left(\hat{a}_{\mathbf{0}}e^{-i\varphi}+\hat{a}^{{\dagger}}_{\mathbf{0}}e^{i\varphi}\right)$ for which $[\hat{H}_{l,\nu,\varphi}(\mu),\hat{N}]\neq 0,$ being $\nu\in\mathbb{R}^{+},\;\varphi\in[0,2\pi).$ In this case, the typical selection rules (degeneracy of the thermal averages),

[TABLE]

being $\left<-\right>_{\hat{H}_{l}(\mu)}$ the thermal average associated to $\hat{H}_{l}(\mu),$ at finite volume $V=l^{d},\;d\in\mathbb{N},\;d\geq 3,$ do not hold anymore, i.e.:

[TABLE]

where $\left<-\right>_{\hat{H}_{l,\nu,\varphi}(\mu)}$ is the thermal average corresponding to the perturbed operator $\hat{H}_{l,\nu,\varphi}(\mu).$

For a Bose system undergoing BEC, in the thermodynamic limit, we have

[TABLE]

being $\rho_{c}$ a critical density of particles.

From a mathematical point of view, for $\rho\leq\rho_{c},$ in the uncondensed phase, it is possible to make a limit exchange, obtaining:

[TABLE]

However, for $\rho>\rho_{c},$

[TABLE]

In this context the limit thermal averages defined as

[TABLE]

have been denominated Bogoliubov’s quasiaverages or anomalous averages. In fact, this notion was introduced for the first time by N.N.Bogoliubov [12].

Thus, the degeneracy of regular averages, produced by the presence of additive conservation laws (or equivalently, by the invariance of the Hamiltonian with respect to certain groups of transformations) is reflected by the dependence of quasi averages on the extra infinitesimal term. In this sense Bogoliubov claimed that the latter are more “physical” than the regular averages [12]. However this procedure, in some cases, has been applied without having necessarily a clear physical meaning.

We are assuming that other types of degeneracy do not exist and, thus, the introduction of the term […] is sufficient for the removal of the degeneracy. (N. N. Bogoliubov [12])

Let $\hat{\rho}_{\mathbf{0},l}=V^{-1}{\hat{a}^{{\dagger}}_{\mathbf{0}}\hat{a}_{\mathbf{0}}},$ $\hat{\eta}_{l}={V}^{-\frac{1}{2}}{\hat{a}_{\mathbf{0}}}.$ In the case of the free Bose gas, for

[TABLE]

the following limits

[TABLE]

hold [12]. In other words:

[TABLE]

In what follows, in order to simplify the notation, we will omit the angle $\varphi$ in the subscripts of any mathematical expression (thermal averages, Hamiltonians, etc.).

Following step by step the strategy developed by Bogoliubov, let

[TABLE]

Substituting this operators in the original energy operator, we obtain:

[TABLE]

In that follows, it will be assumed that,

[TABLE]

where $\rho_{0}$ is a strictly positive real constant.

Clearly,

[TABLE]

This implies that:

[TABLE]

Besides,

[TABLE]

Then, we define

[TABLE]

Passing to the thermodynamic limit, we get:

[TABLE]

On the other hand,

[TABLE]

[TABLE]

This leads to,

[TABLE]

[TABLE]

[TABLE]

The role of the coupled external source, which is very unique, should not be exaggerated from a physical point of view. It is rather the superposition or transition from the ground state to a coherent state that best reflects, in that sense, the spontaneous break of symmetry. Such superposition disappears when the thermodynamic limit is reached. There, both states (with the same energy) the fundamental and the coherent - traslation of the first one- become orthogonal between them. Mathenatically speaking, unlike the finite systems, in the thermodynamic limit we have infinite inequivalent representations of the Bose commutation rules or, in other words, infinite representations of the broken symmetry. In this sense, the Bogoliubov’s approach consists in explicitely fixing one of them.

4. Non linear perturbation of the Ideal Bose Gas

Our purpose, in this work, is to replace in eq.(1) the external source $-\sqrt{V}\nu\left(\hat{a}_{\mathbf{0}}e^{-i\varphi}+\hat{a}^{{\dagger}}_{\mathbf{0}}e^{i\varphi}\right)$ by $-2\nu\sqrt{V}\sqrt{\hat{n}_{\mathbf{0}}+1}.$ In this case, the latter exxpression is expandable in powers series of the operator $\hat{n}_{\mathbf{0}}$ (spectral theorem). This substitution is motivated by the fact that:

[TABLE]

where $\{\varphi_{n}\}$ is a set of orthonormal eigenfunctions of $\hat{n}_{\bf{0}}.$

Thus, in this section we consider a model of a Bose gas whose energy operator correponds to the sum of the Hamiltonian of the free Bose gas with a nonlinear perturbation represented by the square root of the number operator associated to the zero mode.

[TABLE]

The Hamiltonian given by eq.(2) represents a stable model defined in the domain $\mathcal{D}=\{(\beta,\mu):\beta>0,\;\mu<0\}.$

On the other hand, let $\hat{H}^{0}_{l,\nu}(\mu)$ is defined as:

[TABLE]

Note that $[\hat{H}^{\rm{app}}_{l,\nu}(\mu),\hat{N}]=0,$ i.e. the energy operator given by eq.(2) preserves the $U(1)$ symmetry. However, $[\hat{H}^{0}_{l,\nu}(\mu),\hat{N}]\neq 0,$ i.e., the continuous gauge symmetry associated with the $U(1)$ group is broken by the external field $-\nu\sqrt{V}(e^{i\varphi}\hat{a}^{\dagger}_{\mathbf{0}}+e^{-i\varphi}\hat{a}_{\mathbf{0}}).$

In the next section a strong connection between the critical behaviour of both models, in the thermodynamic limit, will be stablished.

The main purpose of this work is to determine explicit expressions for the limit pressures of the model given by eq.(2) in the framework of the so called Laplace principle (see Apendix) and the Large Deviations Method based in two theorems proved by S. R. S. Varadhan  [2]. Moreover, it shall be proven the existence of a phase characrterized by the emergence of non conventional Bose-Einstein condensation, i.e., the existence of an independent on temperature condensate.

4.1. Limit pressure and nonconventional condensation

Theorem 4.1**.**

For $(\beta,\mu)\in\mathcal{D},\;\nu>0,$ in the thermodynamic limit,

[TABLE]

where $p^{\rm{app}}(\beta,\mu,\nu),p^{\rm{id}^{\prime}}(\beta,\mu)$ are the the limit pressures of the system whose Hamiltonian is given by the energy operator of eq.(2) and the energy operator given by eq.(1), but excluding the mode $\mathbf{0},$ respectively.

Proof.

Let,

[TABLE]

Note that the function $h(x)=ax+b\sqrt{x+c},\;a,\;b\in\mathbb{R},\;x\in[0,\infty),\;c>0$ is either an infinitely increasing or an infinitely decreasing mapping on $[0,\infty)$ except the case $a<0,\;b>0.$

Let $\{g_{l}\}$ be a sequence of functions defined on $[0,\infty)$ given as,

[TABLE]

whose first and second derivatives are,

[TABLE]

[TABLE]

respectively.

From these facts, it follows that $g_{l}(x)$ is a concave function attaining its global maximum at

[TABLE]

for a large enough value of $V$ such that $x^{*}_{l}\geq 0$ and

[TABLE]

being $\mu<0,$ $\lambda(\mathbf{0})=0.$

Use will be made of the so-called large deviations method, based on the Laplace principle, for obtaining a closed analytical expression for $p^{\rm{app}}(\beta,\mu,\nu).$ Since $\hat{H}^{\rm{app}}_{l,\nu}(\mu)$ is a diagonal operator with respect to the number operators, the finite pressure can be written as,

[TABLE]

[TABLE]

[TABLE]

Noting, that

[TABLE]

[TABLE]

[TABLE]

It is not hard to see that $\{p^{\rm{app},\mathbf{0}}_{l}(\beta,\mu,\nu)\}$ is a sequence of Darboux sums, then, in the thermodynamic limit the Laplace principle leads to the following expression,

[TABLE]

∎

Theorem 4.2**.**

For $(\beta,\mu)\in\mathcal{D},\;\nu>0,$ in the thermodybanic limit, the Bose Gas with Hamiltonian given by eq.(2) undergoes non conventional condensation if and only if, the ideal gas whose energy operator is given by eq.(1) also displays independent on temperature condensation. Moreover,

[TABLE]

and the amount of condensate satisfies:

[TABLE]

Proof.

Note that:

[TABLE]

From this it follows that:

[TABLE]

Thus, in the thermodynnamic limit, for fixed values of $\beta$ y $\mu<0,$

[TABLE]

On the other hand, using the Griffiths Lemma  [13]

[TABLE]

[TABLE]

In this case,

[TABLE]

[TABLE]

From eqs.(8) and (9) we get:

[TABLE]

[TABLE]

being, as before:

[TABLE]

[TABLE]

Since,

[TABLE]

for the fixed parameters $(\beta,\mu)\in\mathcal{D},\;\nu>0,$ and from the expresions in eqs.(10) and (11), we have that both systems, simultaneously, undergo non conventional condensation. Moreover, the amount of condensate is given as:

[TABLE]

∎

The Bogoliubov’s approach considers a chemical potential $\mu_{*}$ such that $\mu_{*}=-\frac{\nu}{\sqrt{\rho_{\mathbf{0}}}},$ being $\rho_{\mathbf{0}}$ a real and strictly positive constant

On the other hand, unlike the system given by the Hamiltonian in eq.(1), the system whose energy operator is represented by eq.(2) preserves the U(1) symmetry.

If $\rho^{{\rm id}}_{0,l}\left(\beta,\mu_{l},\nu\right)=\rho_{0}={\rm constant}\neq 0,\;\rho_{0}>0,\mu_{l}<0,$ we have that:

[TABLE]

Thus, for values of $\mu_{l}$ in a small neighborhood of zero,

[TABLE]

By solving the second order equation in $\mu_{l},$ we obtain:

[TABLE]

Finally, taking the thermodynamic limit:

[TABLE]

For the free Bose gas, this result means that in the domain $\mathcal{D},$ in spite of that the chemical potential $\mu_{l}$ depends on the inverse temperature $\beta$ at finite volume, in the thermodynamic limit $\mu$ dependes only on the fixed parameters $\rho_{0},\nu.$

4.2. Full diagonal models

Let $\hat{H}^{\rm{FD}}_{l}(\mu)$ be the energy operator defined as:

[TABLE]

$\hat{H}^{\rm{FD}}_{l}(\mu)$ belongs to a class of energy operators so-called full diagonal Bose Hamiltonians. Clearly, $\hat{H}^{\rm{FD}}_{l}(\mu)$ satisfies the commutation rule $[\hat{H}^{\rm{FD}}_{l}(\mu),\hat{N}]=0.$ For example, $\hat{H}^{0}_{l}$ is a full diagonal mode. If $a>0,$ and $v(\mathbf{p}-\mathbf{p}^{\prime})\geq 0$ these are superstable systems, i.e., their limit pressures exist for all real value of $\mu.$

Let $\hat{H}^{\rm{FD}}_{l,\nu}(\mu),\;\hat{H}^{\rm{FD,app}}_{l,\nu}(\mu)$ be the following operators:

[TABLE]

[TABLE]

In this case, $[\hat{H}^{\rm{FD,app}}_{l,\nu}(\mu),\hat{N}]=0,$ and $[\hat{H}^{\rm{FD}}_{l,\nu}(\mu),\hat{N}]\neq 0.$

Theorem 4.3**.**

[TABLE]

Proof.

For this kind of models in ref. [11] it has been proved that:

[TABLE]

[TABLE]

In our case ${\rm sgn}\;\nu=+.$

Let define $\delta_{p_{l}},$ y $\delta_{H}$ as

[TABLE]

[TABLE]

respectively.

Note that $\hat{H}^{\rm{FD,app}}(\mu)$ preserves the $U(1)$ symmetry. This fact and the left hand side Bogoliubov’s inequality (see the Apendix) lead to:

[TABLE]

Moreover, in the thermodynamic limit we get,

[TABLE]

From the right hand Bogoliubov’s inequality and the Jensen inequality (see Apendix) we obtain:

[TABLE]

[TABLE]

Finally, taking the limit $V\to\infty$ and using the expresions in eq.(16) and the inequalities (17) and (18), we obtain: $0\leq\displaystyle\lim_{V\to\infty}\delta p_{l}\leq 0.$ Hence

[TABLE]

∎

A well-known example of a full diagonal Hamiltonian is asociated to the mean field model, whose energy operato,r with an additional term broken the $U(1)$ symmetry, is given by the expression:

[TABLE]

where $a>0,$ $V$ is the volume of the region enclosing the particle system and $\nu\in\mathbb{R}.$

In this case, the operator $\hat{H}^{\rm{MF,app}}_{l,\nu}$ has the following form:

[TABLE]

As before, $\nu>0.$

4.3. Conclusions

• a.

For fixed parameters $\mu<0,$ $\nu>0,$ the pressures and the density of particles in the condensates of the systems whose operators are given by eqs.(1) and (2), in the thermodynamic limit, coincide. Thus,

[TABLE]

[TABLE]

i.e., both models are equivalent in a thermodynamic sense and they undergo, simultaneously, non conventional BEC in $\mathcal{D}.$

• b.

The full diagonal models, with coupled external sources given in eqs. (13) and (14), are thermodynamically equivalent.

• c.

Despite what has been said in a) and b), the external source $-2\nu\sqrt{V}\sqrt{\hat{n}_{\mathbf{0}}+1}$ does not remove the degeneracy of the regular averages.

5. Apendix

5.1. Laplace principle

Proposition 5.1**.**

Let $G:I\rightarrow R$ be a continuous function defined on the interval $I,$ and bounded above by the constant $M$ for all $x\in I$. It is assumed that there exists $\alpha>0$ such that for $|x|$ large enough,

[TABLE]

Then,

[TABLE]

5.2. Bogoliubov’s Inequalities

Let $\hat{H}_{a,l}$ and $\hat{H}_{b,l}$ be selfadjoint operators defined on $\mathcal{D}\subset{\mathcal{F}}_{B}$. $p_{a,l}(\beta,\mu),$ $p_{b,l}(\beta,\mu)$ represent the grand canonical pressures and the free canonical energies corresponding to the operators $\hat{H}_{a,l}$, $\hat{H}_{b,l}$. In this case the following well known Bogolubov inequalities,

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hold, where $\langle-\rangle_{\hat{H}_{a,l}(\mu)},$ $\langle-\rangle_{\hat{H}_{b,l}(\mu)}$ are the Gibbs states in the grand canonical ensemble associated to the Hamiltonians $\hat{H}_{a,l},\hat{H}_{b,l},$ respectively.

5.3. Jensen’s Inequality

Let $\hat{H}_{l},$ be a self-adjoint operator, diagonal with respect to the number operators.Since the spectrum of $\hat{H}_{l},$ coincides with the set of non negative integers, this model can be classically understood by using non negative random variables defined on a suitable probability space $\Omega_{l}.$

Let $\Omega_{l}$ be the countable set of sequences $\omega=\{\omega(p)\in{\mathbb{N}}:p\in\Lambda^{*}_{l}\}\subset{\mathbb{N}}\cup\{0\}$ satisfying

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The basic random variables are the occupation numbers $\{n_{p}:j=1,2,....\}$. They are defined as the functions $n_{p}:\Omega_{l}\rightarrow{\mathbb{N}}$ given as $n_{p}(\omega)=\omega(p)$ for any $\omega\in\Omega_{l}$. The total number of particles in the configuration $\omega$ is denoted as $N(\omega).$ Then the total number, excluded the zero mode is denoted as $N^{\prime}(\omega).$

In this framework, the Gibbs state can be written by replacing $\hat{H}_{l},$ by a function $H_{l}:\Omega_{l}\rightarrow{\mathbb{R}},$ representing the proyection of the energy operator on the occupation-number basis of the Bose Fock space.

Let ${\mathbb{P}}$ be a probability defined for any $\omega\in\Omega_{l}$ as

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For arbitrary $S\subset\Omega$ this implies that

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In this case, $\left<\hat{X}\right>_{\hat{H}_{l}(\mu)}\equiv{\mathbb{E}}[X],$ being $X:\Omega_{l}\rightarrow\mathbb{R}$ the function corresponding to the proyection of the operator $\hat{X}$ on the occupation-number basis.

Thus, the expectation of $X$ respect to ${\mathbb{P}}$ is defined as:

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If $X:\mathbb{R}\rightarrow\mathbb{R}$ is a concave function, the following Jensen’s inequality:

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holds.

5.4. Griffiths Lemma

Lemma 5.2**.**

(Griffiths [13])* Let $\{g_{n}:I\to\mathbb{R},\;I\equiv(a,b)\subset\mathbb{R}\}_{n\in\mathbb{N}}$ be a sequence of convex functions on $I$ with a pointwise limit $g(x),$ which, of course is convex. Let $G^{+}_{n}(x)$ $[\mbox{resp.}\;G^{-}_{n}(x)]$ be the right (resp. left) derivatives of $g_{n}(x),$ and similarly for $G^{+}(x),\;G^{-}(x).$ Then, for all $x\in I,$*

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In particular, if all the $g_{n}$ and $g$ are differentiable at some point $x\in I,$ then

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Proof.

Fix $x\in I$ and $x\pm y\in I,$

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Fix $y$ and take the limit $n\to\infty.$ Then,

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and similarly for $\displaystyle\lim_{n\to\infty}\inf G^{-}_{n}(x).$ Now let $y\downarrow 0.$

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