Exact solution of Z_2 Chern-Simons model on a triangular lattice

TL;DR

This paper provides an exact Hamiltonian solution for the Z_2 Chern-Simons model on a triangular lattice, revealing gapless bulk spectra, edge states, and implications for protected qubits.

Contribution

It constructs an exact Hamiltonian for the Z_2 Chern-Simons model on a triangular lattice and analyzes its spectrum and edge states, including the effects of magnetic terms.

Findings

Bulk spectrum is gapless without magnetic perturbation.

Adding a magnetic term opens a gap in the bulk spectrum.

Edges host non-gauge invariant gapless degrees of freedom.

Abstract

We construct the Hamiltonian description of the Chern-Simons theory with Z_n gauge group on a triangular lattice. We show that the Z_2 model can be mapped onto free Majorana fermions and compute the excitation spectrum. In the bulk the spectrum turns out to be gapless but acquires a gap if a magnetic term is added to the Hamiltonian. On a lattice edge one gets additional non-gauge invariant (matter) gapless degrees of freedom whose number grows linearly with the edge length. Therefore, a small hole in the lattice plays the role of a charged particle characterized by a non-trivial projective representation of the gauge group, while a long edge provides a decoherence mechanism for the fluxes. We discuss briefly the implications for the implementations of protected qubits.