Microscopic description of 2d topological phases, duality and 3d state sums

TL;DR

This paper explores the microscopic and continuum descriptions of 2D topological phases, their dualities, and their relation to 3D state sum models, providing insights into their excitations and geometric interpretations.

Contribution

It introduces Turaev-Viro state sum models from continuum theories and demonstrates their equivalence to doubled topological phases on lattices via duality transformations.

Findings

Equivalence of models on honeycomb lattice with finite gauge group

Duality transformation between group algebra and spin network basis

Geometric interpretation of 3D topological excitations

Abstract

Doubled topological phases introduced by Kitaev, Levin and Wen supported on two dimensional lattices are Hamiltonian versions of three dimensional topological quantum field theories described by the Turaev-Viro state sum models. We introduce the latter with an emphasis on obtaining them from theories in the continuum. Equivalence of the previous models in the ground state are shown in case of the honeycomb lattice and the gauge group being a finite group by means of the well-known duality transformation between the group algebra and the spin network basis of lattice gauge theory. An analysis of the ribbon operators describing excitations in both types of models and the three dimensional geometrical interpretation are given.