Symmetry-projected cluster mean-field theory applied to spin systems

We introduce $S_z$ spin-projection based on cluster mean-field theory and apply it to the ground state of strongly-correlated spin systems. In cluster mean-field, the ground state wavefunction is written as a factorized tensor product of optimized cluster states. In previous work, we have focused on unrestricted cluster mean-field, where each cluster is $S_z$ symmetry adapted. We here remove this restriction by introducing a generalized cluster mean-field (GcMF) theory, where each cluster is allowed to access all $S_z$ sectors, breaking $S_z$ symmetry. In addition, a projection scheme is used to restore global $S_z$, which gives rise to $S_z$ spin-projected generalized cluster mean-field (S$_z$GcMF). Both of these extensions contribute to accounting for inter-cluster correlations. We benchmark these methods on the 1D, quasi-2D, and 2D $J_1-J_2$ and $XXZ$ Heisenberg models. Our results indicate that the new methods (GcMF and S$_z$GcMF) provide a qualitative and semi-quantitative description of the Heisenberg lattices in the regimes considered, suggesting them as useful references for further inter-cluster correlations, which are discussed in this work.