Measuring Modular Matrices by Shearing Lattices

Understand

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.