Momentum polarization: An entanglement measure of topological spin and chiral central charge

Topologically ordered states are quantum states of matter with topological ground-state degeneracy and quasiparticles carrying fractional quantum numbers and fractional statistics. The topological spin theta(a) = 2 pi h(a) is an important property of a topological quasiparticle, which is the Berry phase obtained in the adiabatic self-rotation of the quasiparticle by 2 pi. For chiral topological states with robust chiral edge states, another fundamental topological property is the edge state chiral central charge c. In this paper we propose an approach to compute the topological spin and chiral central charge in lattice models by defining a quantity that we call the momentum polarization. Momentum polarization is defined on the cylinder geometry as a universal subleading term in the average value of a partial translation operator. We show that the momentum polarization is a quantum entanglement property which can be computed from the reduced density matrix, and our analytic derivation based on edge conformal field theory shows that the momentum polarization measures the combination h(a) - c/24 of topological spin and central charge. Results are obtained for two example systems, the non-Abelian phase of the honeycomb lattice Kitaev model and the nu = 1/2 Laughlin state of a fractional Chern insulator described by a variational Monte Carlo wave function, which verify the analytic formula with high accuracy and further suggest that this result remains robust even when the edge states cannot be described by a conformal field theory. Our result provides an efficient approach to extract characteristic quantities of topological states of matter from finite size numerics.