$\mathbb{Z}_2$ topologically ordered phases on a simple hyperbolic lattice

TL;DR

This paper explores $$ topologically ordered phases on a hyperbolic lattice, revealing how ground state degeneracy and entanglement entropy depend on the lattice's branching structure, linked to many anyon superselection sectors.

Contribution

It introduces a model of $$ topological phases on a hyperbolic lattice and analyzes how topological properties vary with lattice geometry, a novel approach in topological matter.

Findings

Ground state degeneracy depends on Cayley tree branches and generation.

Topological entanglement entropy varies with lattice structure.

Large number of anyon superselection sectors observed.

Abstract

In this work, we consider 2D $\mathbb{Z}_2$ topologically ordered phases ($\mathbb{Z}_2$ toric code and the modified surface code) on a simple hyperbolic lattice. Introducing a 2D lattice consisting of the product of a 1D Cayley tree and a 1D conventional lattice, we investigate two topological quantities of the $\mathbb{Z}_2$ topologically ordered phases on this lattice: the ground state degeneracy on a closed surface and the topological entanglement entropy. We find that both quantities depend on the number of branches and the generation of the Cayley tree. We attribute these results to a huge number of superselection sectors of anyons.