Identifying topological order by entanglement entropy

Abstract

Topological phases are unique states of matter that incorporate long-range quantum entanglement and host exotic excitations with fractional quantum statistics. Here we report a practical method to identify topological phases in arbitrary realistic models by accurately calculating the topological entanglement entropy using the density matrix renormalization group (DMRG). We argue that the DMRG algorithm systematically selects a minimally entangled state from the quasi-degenerate ground states in a topological phase. This tendency explains both the success of our method and the absence of ground-state degeneracy in previous DMRG studies of topological phases. We demonstrate the effectiveness of our procedure by obtaining the topological entanglement entropy for several microscopic models, with an accuracy of the order of 10−3, when the circumference of the cylinder is around ten times the correlation length. As an example, we definitively show that the ground state of the quantum S = 1/2 antiferromagnet on the kagome lattice is a topological spin liquid, and strongly constrain the conditions for identification of this phase of matter. Topological entanglement entropy provides a robust measure for detecting the long-range entanglement that characterizes quantum ground states displaying topological order. A new method for calculating this entropy isolates minimally entangled states from the ground states of a topological phase—offering a reliable test for identifying topological spin liquids.