SCF framework, HF stability and RPA correlation for Jordan-Wigner-transformed spin Hamiltonians on arbitrary coupling topologies

TL;DR

This paper develops a self-consistent field framework for Jordan-Wigner-transformed spin Hamiltonians, enabling stable Hartree-Fock solutions and improved correlation energy estimates via RPA on arbitrary topologies.

Contribution

It introduces a novel SCF scheme that optimizes the single-particle density matrix and derives an analytic Hessian, enhancing stability analysis and correlation energy calculations for spin systems.

Findings

RPA improves mean-field accuracy for XXZ and J1-J2 models.

The SCF approach is effective on arbitrary coupling topologies.

Analytic Hessian aids in diagnosing Hartree-Fock stability.

Abstract

Mapping spins to fermions via the Jordan-Wigner (JW) transformation can render mean-field (Hartree-Fock, HF) descriptions effective for strongly correlated spin systems. As established in recent work, the application of such approaches is not limited by the nonlocal structure of JW strings or by site ordering, because string operators can be absorbed into Thouless rotations of a Slater determinant, and the variational optimization of a unitary Lie-Algebraic similarity transformation removes any ordering dependence. Leveraging these ideas, we develop a self-consistent field (SCF) scheme that expresses the mean-field energy as a functional of the single-particle density matrix, providing an alternative to gradient-based optimization of Thouless parameters. We derive the analytic orbital Hessian to diagnose HF stability and compute ground-state correlation energy through the random-phase approximation (RPA). Benchmark results for the XXZ and J1-J2 model on one- and two-dimensional lattices demonstrate that RPA significantly improves mean-field accuracy.