Nonanalytic Elementary Excitation Spectrum at Long Wavelengths in a Bose Gas and LiquidHe4

A rigorous microscopic analysis of a weakly interacting Bose gas at zero temperature shows that the elementary excitation spectrum $\omega \mathrm{}\left(k\right)\mathrm{}$ is not an analytic function of the wave vector $k$ in the long-wavelength limit but has the form $k[c_0+c_2{k}^2+c_{L4\mathrm{}}{k}^4\ln (\frac{1}{k})+\cdots ]$. It is pointed out that the structure factor $S\left(k\right)\mathrm{}$ is not an analytic function of $k$ and that the specific heat $C_v\mathrm{}\left(T\right)\mathrm{}$ at low temperatures $T$ has the form $C_v=T^3[C_0+C_2{T}^2+C_{L4\mathrm{}}{T}^4\ln (\frac{1}{T})+\cdots ]$. The extension of these results to liquid ${\mathrm{He}}^4$ is discussed.