arXiv:1308.5589v2 Announce Type: replace Abstract: We study the grand canonical Gibbs states of a finite Hubbard model coupled to massless phonons with the dispersion relation $|k|^s$, $1\leq s<d$, at finite temperature. We keep the decoupling setting of the zero-temperature analysis, in which the electron-phonon form factor is common to all sites. An explicit unitary transformation factorizes the Gibbs state into the electron Gibbs state and the free phonon Gibbs state up to a phase, and the phonons exhibit Bose-Einstein condensation in the thermodynamic limit with the critical density of the free Bose gas. Assuming the convergence of this phase as the infrared cutoff is removed, the condensed state is the mixture of mutually disjoint component states with broken gauge symmetry, described by their expectations on the Weyl algebra. Physically, phonons are considered not to undergo Bose-Einstein condensation arxiv:1207.4621, and the study of a mechanism that suppresses this mathematical and formal condensation is left for future research.
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