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arXiv:1006.3924v1 [math-ph] 20 Jun 2010

On the nature of Bose-Einstein condensation

enhanced by localization

Thomas Jaeck1, Joseph V. Pul´ e2

School of Mathematical Sciences, University College Dublin

Belfield, Dublin 4, Ireland

Valentin A. Zagrebnov3

Universit´ e de la M´ editerran´ ee (Aix-Marseille II),

Centre de Physique Th´ eorique - UMR 6207, Luminy - Case 907

13288 Marseille, Cedex 09, France

Abstract

In a previous paper we established that for the perfect Bose gas and the mean-field Bose

gas with an external random or weak potential, whenever there is generalized Bose-

Einstein condensation in the eigenstates of the single particle Hamiltonian, there is

also generalized condensation in the kinetic energy states. In these cases Bose-Einstein

condensation is produced or enhanced by the external potential. In the present paper

we establish a criterion for the absence of condensation in single kinetic energy states

and prove that this criterion is satisfied for a class of random potentials and weak

potentials. This means that the condensate is spread over an infinite number of states

with low kinetic energy without any of them being macroscopically occupied.

Keywords: Generalized Bose-Einstein Condensation, Random Potentials, Integrated Den-

sity of States, Multiscale Analysis, Diagonal Particle Interactions.

PACS: 05.30.Jp, 03.75.Fi, 67.40.-w

AMS: 82B10, 82B23, 81V70

1PhD student at UCD and Universit´ e de la M´ editerran´ ee (Aix-Marseille II, France),

e-mail: Thomas.Jaeck@ucdconnect.ie, phone: +353 1 7162571

2e-mail: joe.pule@ucd.ie, phone: +353 1 7162568

3e-mail: Valentin.Zagrebnov@cpt.univ-mrs.fr, phone: +33 491 26 95 04

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1 Introduction

It can be easily seen from the explicit formula for the occupation numbers in the non-

interacting (perfect) Bose gas that, the condition for Bose-Einstein Condensation (BEC) to

occur, is that the density of states of the one particle Schr¨ odinger operator decreases fast

enough near the bottom of the spectrum. In the absence of any external potential, it is

known that this happens only in three dimension or higher. This is still true if one intro-

duces a mean-field interaction between particles. It has been known for some time that the

behavior of the density of states can be altered by the addition of suitable external potentials,

in particular weak potentials or random potentials. The subject of this paper is the study of

models of the Bose gas in the presence of such external potentials. The first case has been

extensively studied, see e.g. [1, 2], where sufficient conditions on the external potential were

derived for the occurrence of BEC. In the random case, it has been shown in [3] that the

so-called Lifshitz tails, which are a general feature of disordered systems, see for instance

[4], are able to produce BEC. In both cases, it is possible to obtain condensation even in

dimension 1 or 2.

While BEC has historically been associated with the macroscopic occupation of the ground

state only, it was pointed out in [5] that this phenomena is more thermodynamically stable

if it is interpreted as as the macroscopic accumulation of particles into an arbitrarily narrow

band of energy above the ground state, or generalized BEC. While it is clear that condensa-

tion in the ground state implies generalized BEC, there exist many situations in which the

converse is not true. For instance, it was shown in [2] that in the case of the weak potential,

the condensate can be in one state, in infinitely many states or even not in any state at all,

depending on the external potential. These situations correspond respectively to type I, II,

III generalized BEC in the classification established by the Van den Berg, Lewis and Pul´ e,

see e.g. [6]. In the random case, far less is known. The only case for which a rigorous proof

of the exact type of BEC has been established is the Luttinger-Sy model, see [7], where it

was shown that the ground-state only is macroscopically occupied. As far as we know, for

more complicated systems this is still an open question. The difficulty lies in the fact that

the characterization of the distribution of the condensate in individual states requires much

more detailed knowledge about the spectrum than the occurrence of generalized condensate.

Indeed, for the latter, it is enough to know the asymptotic behavior of the density of states,

while for the former, one needs in addition to know how fast the gap between two eigenvalues

vanishes in limit.

In the physics literature the phenomenon of BEC is generally understood to be the macro-

scopic occupation of the lowest kinetic energy (momentum) state, commonly referred as

zero-mode condensation. We refer the reader to [9] for a discussion of the motivation for

this type of condensation. This leads naturally to two questions in the case condensation is

produced or enhanced by the addition of external potentials.

The first one comes from the fact the condensates referred to here are to be found in the

eigenstates of the one-particle Schr¨ odinger operator and not the kinetic energy (momentum)

states. Therefore, it is not immediately clear, and in fact counter intuitive in the random case

because of the lack of translation invariance, that condensation occurs in the kinetic energy

states as well. This problem has been addressed in a previous paper, see [9], where we have

shown under fairly general assumptions on the external potential (random or weak) that the

amount of generalized BEC in the eigenstates in turn creates a generalized condensate in the

kinetic states, and moreover in the perfect gas the densities of condensed particles are iden-

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tical. These results were proved for the perfect Bose gas, and can be partially extended to

the mean-field Bose gas. Hence, the (generalized) condensation produced in these models by

the localization property of the one-particle Schr¨ odinger operator can be correctly described

as of “Bose-Einstein” type in the traditional sense. This opens up the possibility of formu-

lating a generalized version of the c-number Bogoliubov approximation ([9], [10]). In the

case of the weak external potential, perhaps this result is not so surprising since the model

is asymptotically translation invariant, but in the random case, it is less obvious since the

system is translation invariant only in the sense that translates of the potential are equally

probable and therefore for a given configuration the system is not translation invariant.

Having established generalized BEC in the kinetic states, the next question is about the

fine structure of that condensate. In our paper [9], we conjectured that the kinetic generalized

BEC is of type III, that is, no single kinetic state is macroscopically occupied, even though

the amount of generalized condensation is non-zero. Our motivation came from the fact

that the fast decrease of the density of states is usually associated with the corresponding

eigenstates becoming localized in the infinite volume limit. Hence, since the kinetic states

(plane waves) and the (localized) general eigenstates are “asymptotically orthogonal”, it

should follow that no condensation in any single-mode kinetic energy state could occur,

independently of whether the (localized) ground state is macroscopically occupied or not. In

[9], we were able to prove this conjecture in a simple example, the Luttinger-Sy model. Our

proof in that case used the absence of tunneling effect specific to that model, which we can

interpret as “perfect localization”.

In this paper we give a proof of the conjecture under a fairly weak localization hypothesis

and then we consider a family of continuous random models and a general class of weak ex-

ternal potential for which we are able to establish this localization property. Our results hold

for both the perfect and mean-field Bose gas, and for any dimension. Note that, in addition

to clarifying the nature of these condensates in low dimensions, we obtain an unexpected

conclusion. Indeed, we show that the presence of randomness or a weak potential, however

small, prevents condensation from occurring in any kinetic state, even if the corresponding

free Bose gas (without external potential) exhibits zero-mode condensation (isotropic system

in dimension 3, for example). This emphasizes the importance of the concept of generalized

BEC.

The structure of the paper is as follows: in Section 2 we give the general setting for which

our results are applicable and discuss generalized condensation in the kinetic energy states,

while in Section 3 we derive a criterion for the absence of condensation into any single kinetic

energy state. In Section 4 we establish that this criterion is satisfied for a class of random

potentials (Subsection 4.1) and for weak (scaled) potentials (Subsection 4.2).

2 Notation and models

Let {Λl:= (−l/2,l/2)d}l?1be a sequence of hypercubes of side l in Rd, centered at the origin

of coordinates with volumes Vl= ld. We consider a system of identical bosons, of mass m,

contained in Λl. For simplicity, we use a system of units such that ? = m = 1. First we

define the self-adjoint one-particle kinetic-energy operator of our system by:

h0

l:= −1

2∆D,(2.1)

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acting in the Hilbert space Hl:= L2(Λl), where ∆ is the usual Laplacian. The subscript D

stands for Dirichlet boundary conditions. We denote by {ψl

eigenfunctions and eigenvalues corresponding to h0

(counting multiplicity) as 0 < εl

following bound

k,εl

k}k?1the set of normalized

l. By convention, we order the eigenvalues

3... . Note that all kinetic states satisfy the

1? εl

2? εl

|ψl

k(x)| ? l−d/2. (2.2)

Next we define the Hamiltonian with an external potential

hl:= h0

l+ vl, (2.3)

also acting in Hl, where vl: Λl?→ [0,∞) is positive and bounded. Let {φl

be respectively the sets of normalized eigenfunctions and corresponding eigenvalues of hl.

Again, we order the eigenvalues (counting multiplicity) so that El

that the non-negativity of the potential implies that El

lower end of the spectrums of h0

This assumption will be proved for the models considered in this paper.

i}i?1and {El

i}i?1

1? El

2? El

3... . Note

1> 0. We shall also assume that the

land hlcoincide in the limit l → ∞, that is, liml→∞El

1= 0.

Now, we turn to the many-body problem. Let Fl:= Fl(Hl) be the symmetric Fock space

constructed over Hl. Then Hl:= dΓ(hl) denotes the second quantization of the one-particle

Schr¨ odinger operator hlin Fl. Note that the operator Hlacting in Flhas the form:

Hl=

?

i?1

El

ia∗(φl

i)a(φl

i),(2.4)

where a∗(ϕ),a(ϕ) are the creation and annihilation operators (satisfying the boson Canonical

Commutation Relations) for the one-particle state ϕ ∈ Hl. Then, the grand-canonical

Hamiltonian of the perfect Bose gas in an external potential is given by:

H0

l(µ) := Hl− µNl =

?

i?1

(El

i− µ) Nl(φl

i) (2.5)

where Nl(φ) := a∗(φ)a(φ) is the operator for the number of particles in the normalized state

φ, Nl :=

?

chemical potential.

iNl(φl

i) is the operator for the total number of particles in Λl and µ is the

As well as for the perfect gas, the results in this paper hold also when a mean-field

interaction is introduced, that is, for a Bose gas with the following many particle Hamiltonian

Hl(µ) := H0

l(µ) +

λ

2VlN2

l, (2.6)

where λ is a non-negative parameter.

We recall that the thermodynamic equilibrium Gibbs state ?−?lassociated with the Hamil-

tonian Hl(µ) is defined by:

?A?l:=TrFl{exp(−βHl(µl))A}

TrFlexp(−βHl(µl))

where the value of µ, µlis determined by fixing the mean density ρ > 0:

1

Vl?Nl?l= ρ.(2.7)

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When referring specifically to the perfect Bose gas state, we shall use the notation ?−?0

For simplicity, in the sequel we shall omit the explicit mention of the dependence on the

thermodynamic parameters (β,µ) unless it is necessary to refer to them.

l.

A normalized single particle state ϕ is macroscopically occupied if

lim

l→∞

1

Vl?Nl(ϕ)?l> 0

and in particular there is condensation in the ground state if

lim

l→∞

1

Vl?Nl(φl

1)?l> 0.

The concept of generalized condensation consists in considering the possible macroscopic

occupation of an arbitrary small band of energies at the bottom of the spectrum. To be

more precise, we say that there is generalized condensation in the states φl

iif

lim

δ↓0lim

l→∞

1

Vl

?

i:El

i?δ

?Nl(φl

i)?l> 0.

It is clear that the usual one-mode condensation implies generalized condensation, however

the converse is not true. Indeed, as was first established by the Dublin School in the eighties

[6], it is possible to classify generalized condensation into three types. Type I condensation,

when a finite number of states are macroscopically occupied (which includes the most com-

monly known notion of BEC as condensation in the ground state only), type II condensation,

when condensation occurs in a infinite number of states, and finally type III, when, although

the amount of generalized condensation is non-zero, no individual state is macroscopically

occupied. One can easily show that in the perfect Bose gas, under fairly general assumptions,

for both random and weak positive potentials, there is indeed generalized condensation in a

suitable range of density (or temperature).

In [9] we discussed the possibility of generalized condensation not in the states φl

kinetic energy states ψl

that for models which are diagonal in the occupation numbers of the eigenstates of the

Hamiltonian (2.3), the generalized BEC in the kinetic states is never less then that in the

eigenstates of the single particle Hamiltonian coincide. To be more precise, we proved that

ibut in the

k, and for both random and weak positive potentials we established

lim

δ↓0lim

l→∞

1

Vl

?

k:εl

k<δ

?Nl(ψl

k)?l? lim

δ↓0lim

l→∞

1

Vl

?

i:El

i<δ

?Nl(φl

i)?l.

We also showed that in the case of the perfect gas, the two quantities in the above inequality

are equal. Here, we shall give a “localization criterion” on the states φl

densation in the kinetic energy states ψl

macroscopically occupied.

iso that the con-

kis of type III, that is no kinetic energy state is

It is easy to see that both the perfect and mean-field gases satisfy the following commu-

tation relation

[Hl(µ),Nl(φl

j)] = 0, for all j. (2.8)

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