### Filling Constraints on Quantum Phases

**"Measuring space-group symmetry fractionalization in Z_2 spin liquids" **(with Mike Zaletel and Yuan Ming Lu) **arXiv:1501.01395**

We identify physical observables that distinguish Z_2 quantum spin liquids with different symmetry properties (SETs) and SU(2) spin rotation invariance. In the cylinder geometry we show that ground state quantum numbers for different topological sectors are robust invariants which can be used to identify the SET phase. More generally these invariants are related to 1D symmetry protected topological phases when viewing the cylinder geometry as a 1D spin chain. In particular we show that the Kagome spin liquid SET can be determined by measurements on one ground state, by wrapping the Kagome in a few different ways on the cylinder.

**"Constraints on topological order in Mott Insulators"**(with Mike Zaletel).

*Phys. Rev. Lett.*114, 077201 (2015). arxiv:14102894

Mott Insulators (quantum magnets with an odd number of spin per unit cell) must either develop order that doubles the unit cell or display exotic behavior. For example, according to the Hastings-Oshikawa-Lieb-Schultz-Mattis theorem, a fully gapped state of a 2D system that respects all symmetries must be topologically ordered i.e. contain anyon excitations. However, the theorem is silent on the nature of the topological order. We show that the double semion topological order is incompatible with time reversal and translation symmetry in Mott insulators. An application of our result is the Kagome lattice quantum antiferromagnet where recent numerical calculations of entanglement entropy indicate a ground state compatible with either toric code *or*double semion topological order. Our result rules out the latter possibility.