Variational density-matrix theory of liquid He4 at nonzero temperatures

We employ the minimum principle for the Helmholtz free energy to develop a self-consistent variational theory of liquid He4 at nonzero temperatures. Using the fact (demonstrated here) that the equilibrium density matrix of a boson system has non-negative matrix elements in coordinate representation, the trial density matrix in coordinate space is chosen to be an exponentiated sum of two-body functions. Adopting the separability assumption in conjunction with the hypernetted-chain approximation, we determine these functions optimally by a coupled set of Euler-Lagrange equations for the structure function S(k,T) and the energy (k,T) of the elementary excitations. In a phenomenological study we analyze the variation of S(k,T) and (k,T) with temperature, and discuss the relationship by employing experimental data on these functions at long wavelength and around the roton minimum.