Generalized Gibbs Ensemble Description of Real Space Entanglement Spectra of (2+1)-dimensional Chiral Topological Systems with $SU(2)$ Symmetry

TL;DR

This paper analyzes the low-lying entanglement spectra of (2+1)D chiral topological states with SU(2) symmetry using conformal field theory and generalized Gibbs ensembles, revealing the role of conservation laws including fractional ones.

Contribution

It introduces a GGE framework to explain ES splittings in chiral topological phases with SU(2) symmetry, incorporating fractional conservation laws at level two.

Findings

ES splittings are explained by a small set of conservation laws.

Fractional conservation laws are crucial at level two.

States including PEPS are confirmed to be chiral under this diagnostic.

Abstract

We provide a quantitative analysis of the splittings in low-lying numerical entanglement spectra (ES), at given momentum, of a number of quantum states that can be identified, based on "Li-Haldane state-counting", as ground states of (2+1)-dimensional chiral topological phases with global SU(2) symmetry. The ability to account for numerical ES splittings solely within the context of conformal field theory (CFT) is an additional diagnostic of the underlying topological theory, of finer sensitivity than "state-counting". We use the conformal boundary state description of the ES, which can be viewed as a quantum quench. In this language, the ES splittings arise from local conservation laws in the chiral CFT besides the energy, which we view as a Generalized Gibbs Ensemble (GGE). Global SU(2) symmetry imposes strong constraints on the number of such conservation laws, so that only a small number of parameters can be responsible for the splittings. We work out these conservation laws for chiral SU(2) Wess-Zumino-Witten CFTs at levels one and two, and for the latter we notably find that some of the conservation laws take the form of local integrals of operators of fractional dimension, as proposed by Cardy for quantum quenches. We analyze numerical ES from systems with SU(2) symmetry including chiral spin-liquid ground states of local 2D Hamiltonians and two chiral Projected Entangled Pair States (PEPS) tensor networks, which exhibit the "state-counting" of the SU(2)-level-one and -level-two theories. We find that the low-lying ES splittings can be well understood by the lowest of our conservation laws, and we demonstrate the importance of accounting for the fractional conservation laws at level two. Thus the states we consider, including the PEPS, appear chiral also under our more sensitive diagnostic.