Boson topological insulators: A window into highly entangled quantum phases

TL;DR

This work investigates how global symmetries realize themselves in highly entangled quantum phases, using short-range entangled SPT insights to place no-go constraints and to reinterpret 2d and 3d topological orders. It develops a unified view where 2d $Z_2$ topological orders with symmetry may only occur as surfaces of 3d SPTs, and shows a parallel interpretation of 3d $U(1)$ spin liquids as SPT phases of emergent monopoles, via explicit coupled-layer constructions. The authors classify various symmetry-enriched topological orders, demonstrate anomaly cancellation between bulk theta terms and boundary topological orders, and prove that certain gapless 2d proposals (like the Algebraic Vortex Liquid) cannot exist in strictly 2d systems under the stated symmetries. Overall, the results clarify how surface states of 3d SPTs constrain possible 2d phases, provide concrete realizations of beyond-cohomology SPTs, and illuminate the interplay between bulk topological field theories and boundary excitations in highly entangled quantum matter.

Abstract

We study several aspects of the realization of global symmetries in highly entangled phases of quantum matter. Examples include gapped topological ordered phases, gapless quantum spin liquids and non-fermi liquid phases. An insightful window into such phases is provided by recent developments in the theory of short ranged entangled Symmetry Protected Topological (SPT) phases . First they generate useful no-go constraints on how global symmetry may be implemented in a highly entangled phase. Possible symmetry implementation in gapped topological phases and some proposed gapless spin/bose liquids are examined in this light. We show that some previously proposed spin liquid states for 2d quantum magnets do not in fact have consistent symmetry implementation unless they occur as the surface of a 3d SPT phase. A second SPT-based insight into highly entangled states is the development of a view point of such states as SPT phases of one of the emergent excitations. We describe this in the specific context of time reversal symmetric 3d U(1) quantum spin liquids with an emergent photon. Different such spin liquids are shown to be equivalent to different SPT insulating phases of the emergent monopole excitation of such phases. The highly entangled states also in turn enrich our understanding of SPT phases. We use the insights obtained from our results to provide an explicit construction of bosonic SPT phases in 3d in a system of coupled layers. This includes construction of a time reversal symmetric SPT state that is not currently part of the cohomology classification of such states.

Figures (1)

• Figure 1: Coupled-layer construction of SPT states. The particle composite in the ellipses are condensed, and only the four surface particles in the dotted ellipses survived as deconfined topological quasi-particles.