Translation-symmetry protected topological orders on lattice

TL;DR

This paper classifies and characterizes translation-symmetry protected topological orders in quantum spin systems, introducing new indices and predicting gapless excitations at phase transitions, with potential generalizations to higher dimensions.

Contribution

It systematically identifies 16 new translation-symmetry protected topological orders in 2D and 256 in 3D, using Z2 indices and physical measurements.

Findings

Identified 16 subclasses of Z2A topological orders in 2D.

Calculated topological degeneracies and crystal momenta for these phases.

Predicted gapless fermionic excitations at phase transitions.

Abstract

In this paper we systematically study a simple class of translation-symmetry protected topological orders in quantum spin systems using slave-particle approach. The spin systems on square lattice are translation invariant, but may break any other symmetries. We consider topologically ordered ground states that do not spontaneously break any symmetry. Those states can be described by Z2A or Z2B projective symmetry group. We find that the Z2A translation symmetric topological orders can still be divided into 16 sub-classes corresponding to 16 new translation-symmetry protected topological orders. We introduced four $Z_2$ topological indices $\zeta_{\v{k}}=0,1$ at $\v {k}=(0,0)$, $(0,\pi)$, $(\pi, 0)$, $(\pi ,\pi)$ to characterize those 16 new topological orders. We calculated the topological degeneracies and crystal momenta for those 16 topological phases on even-by-even, even-by-odd, odd-by-even, and odd-by-odd lattices, which allows us to physically measure such topological orders. We predict the appearance of gapless fermionic excitations at the quantum phase transitions between those symmetry protected topological orders. Our result can be generalized to any dimensions. We find 256 translation-symmetry protected Z2A topological orders for a system on 3D lattice.