The Bogoliubov free energy functional is analysed. The functional serves as a model of a translation-invariant Bose gas at positive temperature. We prove the existence of minimizers in the case of repulsive interactions given by a sufficiently regular two-body potential. Furthermore, we prove existence of a phase transition in this model and provide its phase diagram.

We analyse the canonical Bogoliubov free energy functional in three dimensions at low temperatures in the dilute limit. We prove existence of a first-order phase transition and, in the limit V → 8πa, we determine the critical temperature to be Tc = T fc (1 + 1.49ρ 1/3 a) to leading order. Here, T fc is the critical temperature of the free Bose gas, ρ is the density of the gas and a is the scattering length of the pairinteraction potential V . We also prove asymptotic expansions for the free energy. In particular, we recover the Lee-Huang-Yang formula in the limit V → 8πa.Contents 1 arXiv:1511.05953v3 [math-ph]

Spontaneous symmetry breaking (ssb) is mathematically tied to some limit, but must physically occur, approximately, before the limit. Approximate ssb has been independently understood for Schrödinger operators with double well potential in the classical limit (Jona-Lasinio et al, 1981;Simon, 1985) and for quantum spin systems in the thermodynamic limit (Anderson, 1952;Tasaki, 2019). We relate these to each other in the context of the Curie-Weiss model, establishing a remarkable relationship between this model (for finite N ) and a discretized Schrödinger operator with double well potential.

Unlike bosons, fermions always have a non-trivial entanglement. Intuitively, Slater determinantal states should be the least entangled states. To make this intuition precise we investigate entropy and entanglement of fermionic states and prove some extremal and near extremal properties of reduced density matrices of Slater determinantal states.Mathematics subject classification numbers: 81V99, 82B10, 94A17