Emergence of Kac-Moody symmetry in critical quantum spin chains

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 with conformal invariance is augmented with a Kac-Moody algebra associated with 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 allows 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.

Automated summary

This paper investigates the emergence of Kac-Moody symmetry in a critical quantum spin chain with microscopic Lie group symmetry. The authors propose a method to construct lattice operators corresponding to Kac-Moody generators and numerically show their effectiveness in low-energy states. The lattice version of Kac-Moody generators allows for the computation of the level constant and organization of low-energy eigenstates.