Liquid helium on a surface: Ground state, excitations, condensate fraction, and impurity potential

We study the structure of films of ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ atoms adsorbed to a plane substrate. The ground state is described within a variational model developed earlier [E. Krotscheck, G.-X. Qian, and W. Kohn, Phys Rev. B 31, 4245 (1985)]. Euler-Lagrange equations are solved for the one-body and two-body correlations for three different models of the ${}_{}{}^4{}_{}{}^{}\mathrm{substrate}$ interaction. The one-body density shows structure indicating the formation of layers of one, two, three, . . . atoms above the substrate. The solution of the Euler-Lagrange equation also provides the dispersion relation and spatial shape of collective excitations in the system. The results of the ground-state optimization are used to calculate the condensate fraction (i.e., the number of particles with zero momentum parallel to the surface) and the binding energy of a ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ impurity.