Emergence of Kac-Moody Symmetry in Critical Quantum Spin Chains

TL;DR

This paper demonstrates how Kac-Moody symmetry emerges in critical quantum spin chains with Lie-group symmetries, using numerical methods to construct lattice operators that reveal the algebraic structure at low energies.

Contribution

It introduces a method to construct lattice operators for Kac-Moody generators and verifies their algebraic relations numerically in specific spin chain models.

Findings

Lattice operators approximate Kac-Moody algebra at low energies

Kac-Moody towers can be organized from low-energy eigenstates

Level constants can be computed from lattice operators

Abstract

Given a critical quantum spin chain with a microscopic Lie-group symmetry, corresponding e.g. to $U(1)$ or $SU(2)$ spin isotropy, we numerically investigate the emergence of Kac-Moody symmetry at low energies and long distances. In that regime, one such critical quantum spin chain is described by a conformal field theory where the usual Virasoro algebra associated to conformal invariance is augmented with a Kac-Moody algebra associated to conserved currents. Specifically, we first propose a method to construct lattice operators corresponding to the Kac-Moody generators. We then numerically show that, when projected onto low energy states of the quantum spin chain, these operators indeed approximately fulfill the Kac-Moody algebra. The lattice version of the Kac-Moody generators allow us to compute the so-called level constant and to organize the low-energy eigenstates of the lattice Hamiltonian into Kac-Moody towers. We illustrate the proposal with the XXZ model and the Heisenberg model with a next-to-nearest-neighbor coupling.