Topological order in spin nematics from the quantum melting of a disclination lattice

TL;DR

This paper explores topological phases in spin nematics, revealing stable quantum liquids of disclinations with fractionalized excitations, protected edge modes, and distinct topological orders, extending understanding of quantum spin liquids.

Contribution

It introduces a continuum gauge theory framework for Z$_2$ spin nematics, demonstrating stable topological quantum liquids with fractionalized quasiparticles and analyzing their edge states and microscopic models.

Findings

Existence of stable quantum liquids of disclinations with Z$_2$ topological order.

Presence of protected gapless edge modes in these topological phases.

Distinct thermodynamic properties despite similar topological orders in related states.

Abstract

The topological defects of Spin($n+1$) nematics in two spatial dimensions, known as disclinations, are characterized by the $\pi_1(\mathbb{R}P^n) = \textrm{Z}_2$ homotopy group for $n\ge2$. We argue that incompressible quantum liquids of disclinations can exist as stable low-temperature phases and host composite quasiparticles which combine a fractional amount of fundamental Z$_2$ charge with a unit of topological charge. The four-fold topological ground state degeneracy on a torus admits a fermionic or semionic quasiparticle exchange statistics. The topological non-triviality of these states is visible in the existence of protected gapless edge modes. While the fermionic nematic and gapped Z$_2$ spin liquids have equivalent topological orders, they are still thermodynamically distinct due to having different edge modes, in analogy to the topologically non-trivial and trivial states of quantum spin-Hall systems. The analysis proceeds by recasting the Z$_2$ gauge theory of spin nematics as a continuum limit theory with a larger gauge structure. Nematic fractionalization parallels that of a quantum Hall liquid, but the large gauge symmetry restores the time-reversal symmetry and restricts the quasiparticle fusion rules and statistics. The conclusions from field theory analysis are complemented with the construction of plausible host microscopic models.