Constraints on Topological Order in Mott Insulators

We point out certain symmetry induced constraints on topological order in Mott insulators (quantum magnets with an odd number of spin $\frac{1}{2}$ moments per unit cell). We show, for example, that the double-semion topological order is incompatible with time reversal and translation symmetry in Mott insulators. This sharpens the Hastings-Oshikawa-Lieb-Schultz-Mattis theorem for 2D quantum magnets, which guarantees that a fully symmetric gapped Mott insulator must be topologically ordered, but is silent about which topological order is permitted. Our result applies to the kagome lattice quantum antiferromagnet, where recent numerical calculations of the entanglement entropy indicate a ground state compatible with either toric code or double-semion topological order. Our result rules out the latter possibility.

Figure 1

(a) The adiabatic processes ${\mathcal{F}}_{x/y}^b$. (b) The four minimally entangled basis states are represented as the node of a graph. The process ${\mathcal{F}}_x^c$, with $c=1,b,s,s^{\prime }$, permutes these basis state, illustrated with labeled edges. The action of time reversal $\mathcal{T}$ is also a permutation of the MESs; the only permutation consistent with fusion acts as a mirror reflection across the diagonal, since it must exchange $\mathcal{T}:s\leftrightarrow s^{\prime }$