High- and low-frequency behaviour of response functions in a Bose liquid: One-loop approximation

The filed-theoretic analysis of Wong and Gould for a Bose fluid is used to study response functions at arbitrary temperatures. Several rigorous identities are derived for density and momentum current response functions in the high- and low-frequency limits. These involve the condensate fraction n₀ and the superfluid density ?_s. By explicit calculations using the one-loop diagrammatic approximation for the proper irreducible functions, these rigorous results are verified at finite temperatures. In particular, the zero-frequency Gavoret-Nozières relation, the high-frequency Wong-Gould f-sum rule, and the impulse approximation at high momentum transfers are all satisfied in this simple microscopic calculation. On the other hand, the superfluid f-sum rule is found not to be satisfied.