Global constraints on $Z_2$ fluxes in two different anisotropic limits of a hypernonagon Kitaev model

TL;DR

This paper investigates global constraints on $Z_2$ fluxes in a 3D hypernonagon Kitaev model, revealing surface and volume constraints that govern flux excitations in two anisotropic limits of the chiral spin liquid ground state.

Contribution

It identifies and analyzes the global surface and volume constraints on $Z_2$ fluxes in a 3D Kitaev model with a hypernonagon lattice, extending understanding of flux excitations in such systems.

Findings

Global surface and volume constraints on fluxes are identified.

Flux excitations occur in pairs of closed loops satisfying these constraints.

Constraints influence the nature of flux excitations in the anisotropic limits.

Abstract

The Kitaev model is an exactly-soluble quantum spin model, whose ground state provides a canonical example of a quantum spin liquid. Spin excitations from the ground state are fractionalized into emergent matter fermions and $Z_2$ fluxes. The $Z_2$ flux excitation is pointlike in two dimensions, while it comprises a closed loop in three dimensions because of the local constraint for each closed volume. In addition, the fluxes obey global constraints involving (semi)macroscopic number of fluxes. We here investigate such global constraints in the Kitaev model on a three-dimensional lattice composed of nine-site elementary loops, dubbed the hypernonagon lattice, whose ground state is a chiral spin liquid. We consider two different anisotropic limits of the hypernonagon Kitaev model where the low-energy effective models are described solely by the $Z_2$ fluxes. We show that there are two kinds of global constraints in the model defined on a three-dimensional torus, namely, surface and volume constraints: the surface constraint is imposed on the even-odd parity of the total number of fluxes threading a two-dimensional slice of the system, while the volume constraint is for the even-odd parity of the number of the fluxes through specific plaquettes whose total number is proportional to the system volume. In the two anisotropic limits, therefore, the elementary excitation of $Z_2$ fluxes occurs in a pair of closed loops so as to satisfy both two global constraints as well as the local constraints.