Analytical solutions of symmetric isotropic spin clusters

TL;DR

This paper introduces a versatile analytical method for solving symmetric isotropic spin clusters by leveraging basis adaptation to symmetry, enabling solutions for small magnetic systems with complex interactions.

Contribution

It presents a general approach to analytically solve isotropic spin clusters by adapting the basis to symmetry, facilitating solutions for systems with additional interactions.

Findings

Applicable to small rings and polyhedra

Handles complex interactions like biquadratic exchange

Simplifies solving spin clusters with symmetry adaptation

Abstract

Spin models like the Heisenberg Hamiltonian effectively describe the interactions of open-shell transition-metal ions on a lattice and can account for various properties of magnetic solids and molecules. Numerical methods are usually required to find exact or approximate eigenstates, but for small clusters with spatial symmetry, analytical solutions exist, and a few Heisenberg systems have been solved in closed form. This paper presents a simple, generally applicable approach to analytically solve isotropic spin clusters, based on adapting the basis to both total-spin and point-group symmetry to factor the Hamiltonian matrix into sufficiently small blocks. We demonstrate applications to small rings and polyhedra, some of which are straightforward to solve by successive spin-coupling for Heisenberg terms only; additional interactions, such as biquadratic exchange or multi-center terms necessitate symmetry adaptation.