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

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.