We present a variational ansatz for the ground state of a strongly correlated Bose system. This ansatz goes beyond the Jastrow-Feenberg functional form and explicitly enforces coordination shells in the structure of the wave function. We apply this ansatz to liquid helium-4 with a simple three-variable parametrization of the pair functions. The optimized wave function is found to give an excellent description of the mid-range correlations in the fluid. We also demonstrate the possibility to use this ansatz to study inhomogeneous systems. The phase separation and free surface emerge naturally in this wave function, even though it is constructed of short-range two-body functions and does not contain one-body terms. Because no explicit description of the surface is necessary, this provides a powerful description tool for cluster states.
We study the zero-temperature phase diagram of two-dimensional helium-4 using neural quantum states. Our variational description allows us to address liquid and solid phases using the same functional form as well as exploring possible melting scenarios-for instance, via an intermediate hexatic phase. Notably, this is achieved by performing fixed pressure variational Monte Carlo calculations. Within the isobaric ensemble framework, we are able to clearly identify the first-order liquid-solid phase transition. We additionally compute the Rényi-2 entanglement entropy across the liquid-solid phase transition and find a sharp decrease upon freezing. Calculations for larger systems follow the metastable liquid and solid branches in the transient region, while signatures of a hexatic order coexisting with a small condensate fraction are observed only for smaller systems as a finite-size effect.
Variational calculations of the ground-state energy of hard-sphere, helium-4, Coulomb, and nuclear Yukawa fluids are carried out. The long-range part
v(r>d) and a constant part $\ensuremath{\lambda}$ at
r<d of the bare interaction are assumed to contribute to the average field. The correlation function
f(r) is obtained by minimizing the two-body cluster contribution due to the remainder interaction, and the distance
d is determined by minimizing the energy as calculated with the hierarchy of hypernetted-chain-type equations. The convergence of two- and three-particle distribution functions given by these equations is discussed. The method is shown to be exact in the low-density limit, and the comparison of the present results with the available exact (standard) results in the high-density limit (region) verifies its accuracy over the entire density range. It is shown that the variationally obtained value of
d in the high-density Coulomb fluid can be understood with conventional meanfield theory. A solidification criterion is obtained by studying the energies in liquid and solid phases with the present
f(r). This criterion explains the solidification of the Coulomb, hard-sphere, and helium systems, and the inability of the nuclear Yukawa system to solidify. The variational method in the form presented here could be very useful for studying complicated quantum systems.
The properties of a macroscopic assembly of weakly repulsive bosons at zero temperature are well described by Gross-Pitaevskii mean-field theory. According to this formalism the system exhibits a quantum transition from superfluid to cluster supersolid as a function of pressure. We develop a thermodynamically rigorous treatment of the different phases of the system by adopting a variational formulation of the condensate wave function---represented as a sum of Gaussians---that is amenable to exact manipulations. Not only is this description quantitatively accurate, but it is also capable to predict the order (and sometimes even the location) of the transition. We consider a number of crystal structures in two and three dimensions and determine the phase diagram. Depending on the lattice, the transition from fluid to solid can be first order or continuous, a lower coordination entailing a milder transition. In two dimensions, crystallization would occur at the same pressure on three distinct lattices (square, honeycomb, and stripes), all providing metastable phases with respect to the triangular crystal. A similar scenario holds in three dimensions, where the simple-cubic and diamond crystals also share a common melting point; however, the stable crystal at low pressure is typically fcc. Upon compression and depending on the shape of the potential, the fcc crystal may transform into hcp. We conclude by sketching a theory of the solid-fluid interface and of quantum nucleation of the solid from the fluid.
Estimates are given for the zero-momentum condensate fraction of ground-state
4He described by a Jastrow wave function at the solidification density of the fluid (2%) and at the melting density of the solid (\ensuremath{\sim}1%). Indications are given that these estimates may be sensitively dependent on the form of the pair function utilized.
We have computed the one-body density matrix ${\ensuremath{\rho}}_{1}$ in solid
4He at
T=0K using the shadow wave-function (SWF) variational technique. The accuracy of the SWF has been tested with an exact projector method. We find that off-diagonal long-range order is present in ${\ensuremath{\rho}}_{1}$ for a perfect hcp and bcc solid
4He for a range of densities above the melting one, at least up to 54 bars. This is first microscopic indication that Bose Einstein condensation (BEC) is present in perfect solid
4He. At melting, the condensate fraction in the hcp solid is $5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ and it decreases by increasing the density. The key process giving rise to BEC is the formation of vacancy interstitial pairs. We also present values for Leggett's upper bound on the superfluid fraction deduced from the exact local density.