Quasiparticle statistics and braiding from ground-state entanglement

Abstract

Topologically ordered phases are gapped states, defined by the properties of excitations when taken around one another. Here we demonstrate a method to extract the statistics and braiding of excitations, given just the set of ground-state wave functions on a torus. This is achieved by studying the Topological Entanglement Entropy (TEE) on partitioning the torus into two cylinders. In this setting, general considerations dictate that the TEE generally differs from that in trivial partitions and depends on the chosen ground state. Central to our scheme is the identification of ground states with minimum entanglement entropy, which reflect the quasi-particle excitations of the topological phase. The transformation of these states allows for a determination of the modular S and U matrices which encode quasi-particle properties. We demonstrate our method by extracting the modular S matrix of a chiral spin liquid phase using a Monte Carlo scheme to calculate TEE, and prove that the quasi-particles obey semionic statistics. This method offers a route to a nearly complete determination of the topological order in certain cases.

Exact matrix product states for quantum Hall wave functions

“…The m choices on the bond provide the m "minimal entanglement states." 54 The "thin torus" wave functions are also a limiting case of our construction. 55,56 As L → 0, we can truncate the MPS by keeping only the states of the CFT with the lowest energy within each family (the "highest weight states"), which generates a χ = 1 MPS.…”

Competing Abelian and non-Abelian topological orders inν=1/3+1/3quantum Hall bilayers

“…of each anyon a; the "shift" S [105], or equivalently the bulk Hall viscosity [47]; the topological spins θ a = e 2πiha and the chiral central charge c − of the edge theory [47,99,101]. Below we provide a brief summary of these measurements in the context of FQH systems, and refer to Refs.…”

“…However, these phases do have different topological orders, and we can therefore apply a number of recent developments [47,[99][100][101] which demonstrate how the topological order of a system can be extracted from its entanglement properties.…”

“…There is a special basis for the ground state manifold, the minimally entangled basis, in which each basis state |a can be identified with an anyon type a [99,102,103].…”

Topological central charge from Berry curvature: Gravitational anomalies in trial wave functions for topological phases

“…Similarly, we assume the left-moving central chargec is equal to c, otherwise the non-chiral theory would be anomalous. Ψ e is a conformal block given by 20) where R * is the Ricci scalar for the metric g * = e −Φ g. K is the local counterterm needed to make the partition function invariant under coordinate transformations, analogous to the U(1) counterterm Eq. (4.10).…”