Toric code-like models from the parameter space of $3D$ lattice gauge theories

TL;DR

This paper explores the parameter space of 3D lattice gauge theories derived from involutory Hopf algebras, revealing new models with partial confinement, disordered Hamiltonians, and stability under perturbations, extending the understanding of quantum double Hamiltonians.

Contribution

It introduces new models from the parameter space of 3D lattice gauge theories, including partially confining Hamiltonians and disordered quantum double models, expanding the landscape of topological phases.

Findings

Discovery of models with partial confinement of excitations.

Construction of disordered quantum double Hamiltonians.

Analysis of stability of topological order under magnetic field perturbations.

Abstract

A state sum construction on closed manifolds \'{a} la Kuperberg can be used to construct the partition functions of $3D$ lattice gauge theories based on involutory Hopf algebras, $\mathcal{A}$, of which the group algebras, $\mathbb{C}G$, are a particular case. Transfer matrices can be obtained by carrying out this construction on a manifold with boundary. Various Hamiltonians of physical interest can be obtained from these transfer matrices by playing around with the parameters the transfer matrix is a function of. The $2D$ quantum double Hamiltonians of Kitaev can be obtained from such transfer matrices for specific values of these parameters. A initial study of such models has been carried out in \cite{p1}. In this paper we study other regions of this parameter space to obtain some new and known models. The new model comprise of Hamiltonians which "partially" confine the excitations of the quantum double Hamiltonians which are usually deconfined. The state sum construction allows for parameters depending on the position in obtaining the transfer matrices and thus it is natural to expect disordered Hamiltonians from them. Thus one set of known models consist of the disordered quantum double Hamiltonians. Finally we obtain quantum double Hamiltonians perturbed by magnetic fields which have been considered earlier in the literature to study the stability of topological order to perturbations.