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arXiv:2606.30277

Existence and absence of Bose-Einstein condensation in the interacting random Kac-Luttinger model

C. Boccato, J. Kerner, M. Pechmann, W. Spitzer

math-phmath.PRmath.SPquant-phSubmitted Jun 29, 2026• Announced Jun 30, 2026• 0 citations
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Abstract

In this paper, we study interacting bosons at zero temperature in a random and higher-dimensional continuum model introduced by Kac and Luttinger. For weak interactions we prove that there is condensation in the lowest eigenstate of the one-particle Hamiltonian (type-I BEC). For strong interactions, however, we show that condensation in a localized state cannot occur. We also prove generalized condensation, where a family of eigenstates of the one-particle Hamiltonian is macroscopically occupied as a whole. Combining these results yields a scenario where there is generalized condensation into a family of eigenstates of the one-particle Hamiltonian, but none of them is macroscopically occupied itself (type-III BEC). This proves a transition in the type of condensation. To the best of our knowledge, this is the first rigorous result in this direction for a random continuum model in higher dimensions.

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In this paper, we study interacting bosons at zero temperature in a random and higher-dimensional continuum model introduced by Kac and Luttinger. For weak interactions we prove that there is condensation in the lowest eigenstate of the one-particle Hamiltonian (type-I BEC). For strong interactions, however, we show that condensation in a localized state cannot occur. We also prove generalized condensation, where a family of eigenstates of the one-particle Hamiltonian is macroscopically occupied as a whole. Combining these results yields a scenario where there is generalized condensation into a family of eigenstates of the one-particle Hamiltonian, but none of them is macroscopically occupied itself (type-III BEC). This proves a transition in the type of condensation. To the best of our knowledge, this is the first rigorous result in this direction for a random continuum model in higher dimensions.
We present a rigorous study of the Bose-Einstein condensation in the Luttinger-Sy model. We prove the existence of the condensation in this one-dimensional model of the perfect boson gas placed in the Poisson random potential of singular point impurities. To tackle the off-diagonal long-range order we calculate explicitly the corresponding space-averaged one-body reduced density matrix. We show that mathematical mechanism of the Bose-Einstein condensation in this random model is similar to condensation in a one-dimensional nonrandom hierarchical model of scaled intervals. For the Luttinger-Sy model we prove the Kac-Luttinger conjecture, i.e., that this model manifests a type I BEC localized in a single "largest" interval of logarithmic size.
Bose–Einstein condensation can be understood as a macroscopic occupation of a single-particle state. But an alternative definition, which requires only a macroscopic occupation of an arbitrarily small energy band of single-particle states, is also known, referred to as generalized Bose–Einstein condensation, and considered to be thermodynamically more stable. Depending on the number of macroscopically occupied single-particle states in the condensate, three types are then distin-guished. If the number of macroscopically occupied single-particle states is finite but at least one, then the condensation is said to be of type-I. A type-II condensate occurs if infinitely many single-particle states are macroscopically occupied. If a generalized Bose–Einstein condensation without any macroscopically occupied single-particle states is present, then one speaks of a type-III con-densation. In noninteracting Bose gases, it is generally easier to prove the occurrence of generalized Bose–Einstein condensation than to determine its type. While a main step for the former task is to show that the critical density is finite, quite detailed knowledge of the energy gaps of the cor-responding single-particle Hamiltonian at the lower edge of the spectrum is needed for the latter one. In this dissertation, we study the occurrence of a Bose–Einstein condensate in Bose gases that are placed in metrically transitive random potentials. We mainly consider the Poisson random potential. The thermodynamic limit and Dirichlet boundary conditions are used throughout this work. We explore noninteracting Bose gases in arbitrary dimension at positive temperatures and a one-dimensional interacting Bose gas at zero temperature. We provide fairly general sufficient conditions for the random potential that imply the almost sure occurrence of generalized Bose–Einstein condensation in noninteracting Bose gases. These conditions are fulfilled by a Poisson random potential, depending on its single-impurity potential. They are also met by the one-dimensional Luttinger–Sy model, which is a central system in the research area of Bose gases in random potentials, and a generalization thereof. Moreover, we state sufficient energy gap conditions such that one can conclude a type-I condensation in Poisson random potentials. We consider different probabilistic notions here, such as convergence in the rth mean, r P N, in probability, and with probability almost one. In particular, we prove that in the Luttinger–Sy model with infinitely strong random potential a type-I condensation occurs in the rth mean, r P N, if the particle density is greater than the critical one. For the Luttinger–Sy model with a finitely strong random potential, we show the same statement, but with probability almost one. Finally, we discuss the Luttinger–Sy model with a repulsive contact interaction between the particles at zero temperature. We provide sufficient requirements such that a generalized Bose–Einstein condensate almost surely occurs. We also show that the type of the condensate is altered by the interparticle interaction. In particular, we demonstrate a transition from a type-I to a type-III condensate when the strength of the contact interaction is increased. Furthermore, we prove that the interparticle interaction reduces the local particle density.
In this paper we investigate the effect of repulsive pair interactions on Bose–Einstein condensation in a well-established random one-dimensional system known as the Luttinger–Sy model at positive temperature. We study separately hard core interactions as well as a class of more general repulsive interactions, also allowing for a scaling of certain interaction parameters in the thermodynamic limit. As a main result, we prove in both cases that for sufficiently strong interactions all eigenstates of the non-interacting one-particle Luttinger–Sy Hamiltonian are P \mathbb {P} -almost surely not macroscopically occupied. We also outline an extension of the results to sufficiently localized one-particle states.
We investigate the phase transition exhibited by the Bose gas in systems which are not translation-invariant. Though it has been known since the sixties that Bose-Einstein condensation (BEC) cannot occur in translation invariant systems for dimension 1 or 2, one can nevertheless enhance this phase transition in low-dimensional Bose gases by the addition of suitable external potentials (thus losing translation invariance in the process). However, the resulting condensate is then found to be in localised states, while BEC is usually understood to be the macroscopic occupation of extended kinetic eigenstates. It is therefore not clear whether the phase transition obtained by means of localisation is of the same nature as the one related to the usual concept of BEC. In this thesis, we consider two classes of localised systems. The first one is a family of random systems, where the Bose gas is contained in a disordered medium, which is modelled by a random external potential. Our second model consists of weak (scaled) external potentials. We first recall necessary conditions on these external potentials to enhance condensation in the localised states. We then show under very general assumptions that in these models, BEC in the usual sense occurs also, in a generalised sense. This means that the particles condense on kinetic eigenstates with arbitrary small energy. For the non-interacting Bose gas, we can moreover show that the densities of both condensates are actually equal. Next, we investigate BEC on a finer scale, asking whether one can obtain condensation in a single kinetic eigenstate. We show that in spite of the existence of a phase transition, and the occurence of generalised BEC, no condensation exists in any single kinetic eigenstate. In particular, the so-called "ground-state condensation" does not occur in these localised systems. Finally, we establish a possible generalisation of the Bogoliubov c-number approximation to take into account the very specific properties of BEC in the presence of localisation, and discuss how to interpret the result of the corresponding variational problem.
Bose--Einstein condensation for particles with repulsive short-range pair interactions in a Poisson random external potential in Rd\mathbb R^dRd
We study Bose gases in ddd dimensions, d≥2d \ge 2d≥2, with short-range repulsive pair interactions, at positive temperature, in the canonical ensemble and in the thermodynamic limit. We assume the presence of hard Poissonian obstacles and focus on the non-percolation regime. For sufficiently strong interparticle interactions, we show that almost surely there cannot be Bose--Einstein condensation into a sufficiently localized, normalized one-particle state. The results apply to the eigenstates of the underlying one-particle Hamiltonian.