Quantum spin Hamiltonians for the SU(2)_k WZW model

TL;DR

This paper introduces a method using null vectors in conformal field theories to construct quantum spin chain Hamiltonians with ground states related to SU(2)_k WZW model correlators, enabling analytical and numerical analysis of their properties.

Contribution

It presents a general approach to derive spin chain Hamiltonians from conformal field theories, including explicit models and analytical tools for correlation functions.

Findings

Constructed Hamiltonians with ground states as SU(2)_k WZW correlators

Analyzed k=2 model showing logarithmic entropy growth and algebraic decay of correlations

Derived linear equations for spin correlators enabling analytical and numerical solutions

Abstract

We propose to use null vectors in conformal field theories to derive model Hamiltonians of quantum spin chains and corresponding ground state wave function(s). The approach is quite general, and we illustrate it by constructing a family of Hamiltonians whose ground states are the chiral correlators of the SU(2)_k WZW model for integer values of the level k. The simplest example corresponds to k=1 and is essentially a nonuniform generalization of the Haldane-Shastry model with long-range exchange couplings. At level k=2, we analyze the model for N spin 1 fields. We find that the Renyi entropy and the two-point spin correlator show, respectively, logarithmic growth and algebraic decay. Furthermore, we use the null vectors to derive a set of algebraic, linear equations relating spin correlators within each model. At level k=1, these equations allow us to compute the two-point spin correlators analytically for the finite chain uniform Haldane-Shastry model and to obtain numerical results for the nonuniform case and for higher-point spin correlators in a very simple way and without resorting to Monte Carlo techniques.