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

Abstract

We introduce Sz spin-projection based on cluster mean-field theory and apply it to the ground state of strongly correlated spin systems. In cluster mean-fields, 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 Sz symmetry adapted. We here remove this restriction by introducing a generalized cluster mean-field (GcMF) theory, where each cluster is allowed to access all Sz sectors, breaking Sz symmetry. In addition, a projection scheme is used to restore global Sz, which gives rise to the Sz spin-projected generalized cluster mean-field (SzGcMF). Both of these extensions contribute to accounting for inter-cluster correlations. We benchmark these methods on the 1D, quasi-2D, and 2D J1 − J2 and XXZ Heisenberg models. Our results indicate that the new methods (GcMF and SzGcMF) 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.

Linear Combinations of Cluster Mean-Field States Applied to Spin Systems

“…In this study, we maintain the constraint that each cluster represents an S z = 0 eigenstate (where we use the same symbol for the operator and the eigenvalues), consistent with the principles of both restricted (RcMF) and unrestricted cMF (UcMF). Despite our previous findings, which highlighted the nearly exact nature of generalized cMF (GcMF) around Δ = −1 for the XXZ model, our current emphasis lies in broadening our approach by incorporating additional cluster coverings rather than expanding the Hilbert space for each cluster.…”

“…When those correlations are strong, however, the HF treatment is inadequate. Cluster mean-field (cMF − ) theory generalizes this basic idea but provides a more nuanced and flexible framework. It also uses a wave function which is correct for non-interacting constituents, but where in HF these constituents are the individual electrons, cMF uses multielectronic fragments.…”

“…In recent publications, , we have successfully applied cluster-based methodologies to address strongly correlated spin systems, demonstrating their effectiveness in dealing with the challenging J 1 – J 2 and XXZ Heisenberg models. These investigations are extensions of our previous work, where we introduced cMF theory for Fermionic systems .…”

“…This is motivated as we want to explore both XXZ and J 1 – J 2 Heisenberg lattices, mainly because the intermediate spin-liquid-like region J 2 / J 1 ≈ 0.5 of the square J 1 – J 2 Heisenberg model is very difficult for conventional methods to describe, and the XXZ model does not have S 2 as a symmetry and comes with a diverse phase spectrum. For a more comprehensive understanding of the interplay between cMF and other advanced methodologies, as well as insights into the advantages of cMF relative to these approaches, readers are directed to refs – and the associated references therein.…”

“…The computation of matrix elements in LC-cMF follows a similar procedure to what we have described in our prior publications. , Calculating matrix elements involving states with distinct tilings does not significantly increase the computational cost. The overall efficiency of this approach is contingent on the number of coverings included and the optimization strategy for the coefficients (whether by selecting specific singlet and triplet states or by concurrently optimizing the coefficients for each covering).…”