Some formal results for the valence bond basis

TL;DR

This paper provides formal mathematical results for the valence bond basis in SU(2) spin systems, including generating functions for correlation functions and extensions to triplet bonds, aiding both analytical and numerical studies.

Contribution

It introduces explicit formulas for higher-order correlation functions and extends the valence bond basis to include triplet bonds, enhancing computational methods.

Findings

Explicit formulas for second, fourth, and sixth order correlation functions.

A generating function for arbitrary order spin correlations.

Extension of the valence bond basis to include triplet bonds.

Abstract

In a system with an even number of SU(2) spins, there is an overcomplete set of states--consisting of all possible pairings of the spins into valence bonds--that spans the S=0 Hilbert subspace. Operator expectation values in this basis are related to the properties of the closed loops that are formed by the overlap of valence bond states. We construct a generating function for spin correlation functions of arbitrary order and show that all nonvanishing contributions arise from configurations that are topologically irreducible. We derive explicit formulas for the correlation functions at second, fourth, and sixth order. We then extend the valence bond basis to include triplet bonds and discuss how to compute properties that are related to operators acting outside the singlet sector. These results are relevant to analytical calculations and to numerical valence bond simulations using quantum Monte Carlo, variational wavefunctions, or exact diagonalization.