Symmetry breaking and restoration on a fermionic quantum ring

Background: The Hartree-Fock mean-field approximation is standard in combination with energy density functionals (EDF) that account for some dynamical correlations. Breaking and restoring the symmetries of the system allow for the inclusion of additional static correlations. However, exact solutions to evaluate the effectiveness of these methods are rare. Purpose: To benchmark the Hartree-Fock method with broken and restored rotational symmetry in a system of identical interacting fermions on a one-dimensional quantum ring using model interactions. Method: The ground-state wave function is found using the Hartree-Fock method both with rotational invariance and with the symmetry broken at the mean-field level. Rotational symmetry is then restored with an angular momentum projection method. The ground-state energies are compared to variational Monte Carlo predictions. This is done for a range of different interactions between the particles. Results: Breaking the rotational symmetry in the Hartree-Fock mean-field brings little improvement to the ground-state energy in weakly repulsive systems or attractive systems confined on small rings. Larger improvements are found in strongly repulsive systems and attractive systems on larger rings in which the particles form a self-bound system. Symmetry restoration brought only small improvements in most cases but was able to account for most of the remaining correlation energy (after symmetry breaking) in repulsive systems. Conclusions: The effectiveness of incorporating correlations through rotational symmetry breaking followed by angular momentum projection is demonstrated for one-dimensional quantum rings using model interactions, encouraging generalisations to other symmetries, extensions to higher dimensions, as well as applications in the EDF framework.