Hartree-Fock approximation for bosons with symmetry-adapted variational wave functions

TL;DR

This paper introduces a symmetry-adapted variational Hartree-Fock method for bosons, improving the accuracy of physical observables by optimizing wave functions that respect the system's symmetry, and demonstrates its effectiveness on various models.

Contribution

The authors develop a variation after projection scheme for bosonic Hartree-Fock wave functions, enhancing symmetry preservation and accuracy in modeling bosonic systems.

Findings

Accurately estimates ground state energies and correlations.

Effective for models with diverse symmetry groups.

Applicable to few-body and mesoscopic bosonic systems.

Abstract

The Hartree-Fock approximation for bosons employs variational wave functions that are a combination of permanents. These are bosonic counterpart of the fermionic Slater determinants, but with the significant distinction that the single-particle orbitals used to construct a permanent can be arbitrary and do not need to be orthogonal to each other. Typically, the variational wave function may break the symmetry of the Hamiltonian, resulting in qualitative and quantitative errors in physical observables. A straightforward method to restore symmetry is projection after variation, where we project the variational wave function onto the desired symmetry sector. However, a more effective strategy is variation after projection, which involves first creating a symmetry-adapted variational wave function and then optimizing its parameters. We have devised a scheme to realize this strategy and have tested it on various models with symmetry groups ranging from $\mathbb{Z}_2$, $\text{C}_L$, to $\text{D}_L$. In all the models and symmetry sectors studied, the variational wave function accurately estimates not only the energy of the lowest eigenstate but also the single-particle correlation function, as it approximate the target eigenstate very well on the wave function level. We have applied this method to study few-body bound states, superfluid fraction, and Yrast lines of some Bose-Hubbard models. This approach should be valuable for studying few-body or mesoscopic bosonic systems.