Bose-Einstein Condensation and Spontaneous Symmetry Breaking
Elliott H. Lieb, Robert Seiringer, Jakob Yngvason
备注：
Contribution to the proceedings of the 21st Max Born Symposium, Wroclaw, Poland, June 26--28, 2006. (References added. To be published in Reports on Mathematical Physics.)
After recalling briefly the connection between spontaneous symmetry breaking and off-diagonal long range order for models of magnets a general proof of spontaneous breaking of gauge symmetry as a consequence of Bose-Einstein Condensation is presented. The proof is based on a rigorous validation of Bogoliubov's $c$-number substitution for the ${\bf k}={\bf 0}$ mode operator $a_{\bf 0}$.
We apply the Bogoliubov inequality to the Bose-Hubbard model to rule out the possibility of Bose-Einstein condensation. The result holds in one and two dimensions, for any filling at any nonzero temperature. This result can be considered as complementary to analogous, classical result known for interacting bosons in continuum.
The paper presents a review of results concerning the non-conventional dynamical condensation versus conventional Bose-Einstein condensation, including the case of generalised van den Berg-Lewis-Pulé condensate. The review is based on detailed discussion of two models: a simple toy model and the Bogoliubov Weakly Imperfect Bose-Gas model, which was invented for explanation of superflfluidity of liquid 4He, but which is also instructive for analysis of non-conventional condensation regarding some recent interpretations of experimental data.
We consider systems of bosons trapped in a box, in the Gross-Pitaevskii regime. We show that low-energy states exhibit complete Bose-Einstein condensation with an optimal bound on the number of orthogonal excitations. This extends recent results obtained in \cite{BBCS1}, removing the assumption of small interaction potential.
We consider N bosons in a box with volume one, interacting through a two-body potential with scattering length of the order $N^{-1+κ}$, for $κ>0$. Assuming that $κ\in (0;1/43)$, we show that low-energy states of the system exhibit complete Bose-Einstein condensation by providing explicit bounds on the expectation and on higher moments of the number of excitations.
The equilibrium states for a model with two kinds of Bose condensation
本文研究了由Bru和Zagrebnov定义的玻色气体模型的平衡吉布斯态，该模型具有两个玻色凝聚型的相变。这两种相变对应于两种不同的机制，通过这两种机制可以发生这些冷凝。第一（非常规）玻色凝聚介导的零模相互作用项的哈密顿量。第二个是由于饱和的过渡非常类似于传统的玻色-爱因斯坦（BE）凝聚在理想玻色气体。由于非零模式中的排斥相互作用，该模型表现出广义的III型，即，非广泛BE缩合。我们的主要结果是，在理想玻色气体中，常规凝聚伴随着正则和巨正则系综的强等价性的损失，而非常规的，由于相互作用，不打破系综的等价性。有趣的是，（广义）凝聚的类型，I、II或III（用van den Berg、Lewis和Pule的术语），对系综的等价性没有影响。通过计算对应平衡吉布斯态的正则对易关系（CCR）的循环表示的生成泛函，证明了这些结果。
A well-known conjecture in mathematical physics asserts that the interacting Bose gas exhibits Bose-Einstein condensation (BEC) in the thermodynamic limit. We consider the Bose gas on certain hyperbolic spaces. In this setting, one obtains a short proof of BEC in the infinite-volume limit from the existence of a volume-independent spectral gap of the Laplacian.
Non-relativistic interacting bosons at zero temperature, in two and three dimensions, are expected to exhibit a fascinating critical phase, famously known as condensate phase. Even though a proof of Bose-Einstein condensation in the thermodynamic limit is still beyond reach of the current available methods, in the past decades the mathematical physics community has gained an enhanced comprehension of other aspects of the macroscopic behavior of dilute Bose gases at zero temperature. In these notes we review part of these advances, by focusing on the strict relation among the occurrence of Bose…
We study in detail relevant spectral properties of the adjacency matrix of inhomogeneous amenable networks, and in particular those arising by negligible additive perturbations of periodic lattices. The obtained results are deeply connected to the systematic investigation of the Bose--Einstein condensation for the so called Pure Hopping model describing the thermodynamics of Bardeen--Cooper pairs of Bosons in arrays of Josephson junctions.
We consider a Bose gas consisting of $N$ particles in $\mathbb{R}^3$, trapped by an external field and interacting through a two-body potential with scattering length of order $N^{-1}$. We prove that low energy states exhibit complete Bose-Einstein condensation with optimal rate, generalizing previous work in \cite{BBCS1, BBCS4}, restricted to translation invariant systems. This extends recent results in \cite{NNRT}, removing the smallness assumption on the size of the scattering length.
We consider systems of N bosons trapped on the two-dimensional unit torus, in the Gross-Pitaevskii regime, where the scattering length of the repulsive interaction is exponentially small in the number of particles. We show that low-energy states exhibit complete Bose-Einstein condensation, with almost optimal bounds on the number of orthogonal excitations.
We study Bose gases in $d$ dimensions, $d \ge 2$, 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.
