Measuring Modular Matrices by Shearing Lattices

TL;DR

This paper introduces a numerical method to measure modular matrices in topologically ordered phases by calculating the non-Abelian Berry phase during lattice deformations, demonstrated on a gauged p+ip superconductor.

Contribution

It presents a novel approach to directly compute modular matrices via adiabatic lattice deformations, linking numerical techniques with topological quantum field theory.

Findings

Successfully measured modular matrices for a gauged p+ip superconductor

Results align with topological quantum field theory predictions

Method applicable to other topologically ordered systems

Abstract

A topologically ordered phase on a torus possesses degenerate ground states that transform nontrivially under the modular transformations of the torus, generated by Dehn twists. Representation of modular transformations on the ground states (modular matrices) characterizes the topological order. We show that the modular matrices can be numerically measured as the non-Abelian Berry phase of adiabatic deformations of the lattice model placed on a torus. We apply this method to the example of a gauged $p_x+i p_y$ superconductor, and show that the result is consistent with the topological quantum field theory descriptions.