Projective spin adaptation for the exact diagonalization of isotropic spin clusters

TL;DR

This paper introduces a simplified spin projection technique for exact diagonalization of isotropic spin clusters, improving efficiency by using a group-theoretical approach and an extended L"owdin's theorem for arbitrary local spins.

Contribution

It presents a new, simpler method for spin adaptation in ED based on a spin projector and an extended L"owdin's theorem applicable to any local spin quantum number.

Findings

The method simplifies the construction of spin-adapted bases.

It allows exact or approximate evaluation of the spin projector.

Examples demonstrate the effectiveness of the approach.

Abstract

Spin Hamiltonians, like the Heisenberg model, are used to describe magnetic properties of exchange-coupled molecules and solids. For finite clusters, physical quantities such as heat capacities, magnetic susceptibilities or neutron-scattering spectra, can be calculated based on energies and eigenstates obtained by exact diagonalization (ED). Utilizing spinrotational symmetry SU(2) to factor the Hamiltonian with respect to total spin S facilitates ED, but the conventional approach to spin-adapting the basis is more intricate than selecting states with a given magnetic quantum number M (the spin z-component), as it relies on irreducible tensor-operator techniques and spin-coupling coefficients. Here, we present a simpler technique based on applying a spin projector to uncoupled basis states. As an alternative to L\"owdin's projection operator, we consider a group-theoretical formulation of the projector, which can be evaluated either exactly or approximately using an integration grid. An important aspect is the choice of uncoupled basis states. We present an extension of L\"owdin's theorem for s = 1/2 to arbitrary local spin quantum numbers s, which allows for the direct selection of configurations that span a complete, linearly independent basis in an S sector upon spin projection. We illustrate the procedure with a few examples.