The role of thermal fluctuations in a Bose system in lowering the roton minimum

A study is made of the influence of thermal fluctuations on the stability of the homogeneous state of a Bose system with condensate in lowering the roton minimum. In contrast to the case of quantum fluctuations, which at $T=0$ are responsible for a premature loss of stability with respect to arbitrarily small perturbations, this occurring moreover first for long-wavelength (phonon) and not roton perturbations [7, 8], thermal fluctuations delay the appearance of local instability. Another difference from the case $T=0$ with a real roton minimum is the fact that to describe the complete region of stability at $T>0$ the single-loop approximation is inadequate. An additional metastable homogeneous phase at $T=0$ with “complex roton minimum” is found.