A lower bound is derived on the free energy (per unit volume) of a
homogeneous Bose gas at density Ο and temperature T. In the dilute
regime, i.e., when a3Οβͺ1, where a denotes the scattering length of
the pair-interaction potential, our bound differs to leading order from the
expression for non-interacting particles by the term 4Οa(2Ο2β[ΟβΟcβ]+2β). Here, Οcβ(T) denotes the critical density for
Bose-Einstein condensation (for the non-interacting gas), and []+β denotes
the positive part. Our bound is uniform in the temperature up to temperatures
of the order of the critical temperature, i.e., TβΌΟ2/3 or smaller.
One of the key ingredients in the proof is the use of coherent states to extend
the method introduced in [arXiv:math-ph/0601051] for estimating correlations to
temperatures below the critical one.Comment: LaTeX2e, 53 page