Doubled Lattice Chern-Simons-Yang-Mills Theories with Discrete Gauge Group

TL;DR

This paper constructs and analyzes doubled lattice Chern-Simons-Yang-Mills theories with discrete gauge groups, revealing topological features, ground-state degeneracies, and exact spectra for various groups, with implications for quantum computation.

Contribution

It introduces a lattice formulation of doubled gauge theories with discrete groups, exploring their topological properties and exact spectra, and discusses potential for non-Abelian topological states.

Findings

Ground-state degeneracy depends on the gauge group and topology.

Exact spectra of the electric Hamiltonian are determined for several discrete groups.

The role of center symmetry in charge confinement is highlighted.

Abstract

We construct doubled lattice Chern-Simons-Yang-Mills theories with discrete gauge group $G$ in the Hamiltonian formulation. Here, these theories are considered on a square spatial lattice and the fundamental degrees of freedom are defined on pairs of links from the direct lattice and its dual, respectively. This provides a natural lattice construction for topologically-massive gauge theories, which are invariant under parity and time-reversal symmetry. After defining the building blocks of the doubled theories, paying special attention to the realization of gauge transformations on quantum states, we examine the dynamics in the group space of a single cross, which is spanned by a single link and its dual. The dynamics is governed by the single-cross electric Hamiltonian and admits a simple quantum mechanical analogy to the problem of a charged particle moving on a discrete space affected by an abstract electromagnetic potential. Such a particle might accumulate a phase shift equivalent to an Aharonov-Bohm phase, which is manifested in the doubled theory in terms of a nontrivial ground-state degeneracy on a single cross. We discuss several examples of these doubled theories with different gauge groups including the cyclic group $\mathbb{Z}(k)\subset U(1)$, the symmetric group $S_3\subset O(2)$, the binary dihedral (or quaternion) group $\bar{D}_2\subset SU(2)$, and the finite group $\Delta(27)\subset SU(3)$. In each case the spectrum of the single-cross electric Hamiltonian is determined exactly. We examine the nature of the low-lying excited states in the full Hilbert space, and emphasize the role of the center symmetry for the confinement of charges. Whether the investigated doubled models admit a non-Abelian topological state which allows for fault-tolerant quantum computation will be addressed in a future publication.