Boson topological insulators: A window into highly entangled quantum phases

TL;DR

This paper explores the realization of global symmetries in highly entangled quantum phases, revealing how symmetry constraints and SPT phases influence the properties and classifications of topological quantum matter.

Contribution

It introduces new insights into symmetry implementation in topological phases, especially linking 2D spin liquids to 3D SPT surfaces and constructing novel 3D bosonic SPT states beyond existing classifications.

Findings

Certain 2D spin liquids are surface states of 3D SPT phases.

Different 3D U(1) quantum spin liquids correspond to distinct SPT phases of emergent monopoles.

Constructed a 3D bosonic SPT phase outside the cohomology classification.

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.