Lattice Laughlin states on the torus from conformal field theory

TL;DR

This paper develops analytical expressions for lattice Laughlin states on the torus using conformal field theory, enabling the study of fractional quantum Hall physics with periodic boundary conditions and demonstrating their relation to continuum Laughlin states.

Contribution

It introduces a method to construct and analyze lattice Laughlin states on the torus, extending conformal field theory techniques to periodic boundary conditions.

Findings

States become orthonormal for large lattices

The S-matrix matches that of continuum Laughlin states

Lattice states approximate continuum Laughlin states with suitable spacing

Abstract

Conformal field theory has turned out to be a powerful tool to derive two-dimensional lattice models displaying fractional quantum Hall physics. So far most of the work has been for lattices with open boundary conditions in at least one of the two directions, but it is desirable to also be able to handle the case of periodic boundary conditions. Here, we take steps in this direction by deriving analytical expressions for a family of conformal field theory states on the torus that is closely related to the family of bosonic and fermionic Laughlin states. We compute how the states transform when a particle is moved around the torus and when the states are translated or rotated, and we provide numerical evidence in particular cases that the states become orthonormal up to a common factor for large lattices. We use these results to find the S-matrix of the states, which turns out to be the same as for the continuum Laughlin states. Finally, we show that when the states are defined on a square lattice with suitable lattice spacing they practically coincide with the Laughlin states restricted to a lattice.