Quantized topological terms in weak-coupling gauge theories with a global symmetry and their connection to symmetry-enriched topological phases

Hung, Ling-Yan; Wen, Xiao-Gang

doi:10.1103/physrevb.87.165107

Cited by 95 publications

(118 citation statements)

References 102 publications

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“…One can now gauge the Z 2 symmetry to obtain a topologically ordered state that is distinct from the regular Z 2 gauge theory 38,39,[43][44][45] . In fact, the particle excitations in this theory are semions and anti-semions, and this is termed the double-semion theory.…”

Section: Methodsmentioning

confidence: 99%

“…Recently, 3D SPT phases have been discussed from number of different theoretical perspectives, including twisted vortex condensates 18 and the statistical magneto-electric effect 20 . A useful perspective on SPT phases appears on gauging the symmetry discussed by Levin and Gu 5 in 2D, and recently extended in other studies 20,21,[38][39][40] to 3D phases. Finally, we note a special feature of 3D SPT phases is that their surface states could be gapped and fully symmetric if they develop topological order just at the surface 9,[20][21][22] .…”

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confidence: 99%

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Symmetry-protected topological phases from decorated domain walls

2014

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Symmetry-protected topological phases generalize the notion of topological insulators to strongly interacting systems of bosons or fermions. A sophisticated group cohomology approach has been used to classify bosonic symmetry-protected topological phases, which however does not transparently predict their properties. Here we provide a physical picture that leads to an intuitive understanding of a large class of symmetry-protected topological phases in d ¼ 1,2,3 dimensions. Such a picture allows us to construct explicit models for the symmetry-protected topological phases, write down ground state wave function and discover topological properties of symmetry defects both in the bulk and on the edge of the system. We consider symmetries that include a Z 2 subgroup, which allows us to define domain walls. While the usual disordered phase is obtained by proliferating domain walls, we show that symmetry-protected topological phases are realized when these domain walls are decorated, that is, are themselves symmetry-protected topological phases in one lower dimension. This construction works both for unitary Z 2 and anti-unitary time reversal symmetry.

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Section: Methodsmentioning

confidence: 99%

mentioning

confidence: 99%

Symmetry-protected topological phases from decorated domain walls

2014

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“…In general, coupling topological phases with various symmetries [symmetry-protected topological (SPT) phases or symmetry-enriched topological (SET) phases] to the corresponding gauge field results in nontrivial responses (like nontrivial statistics, Hall conductance, symmetry fractionalization, etc.) and provides an important tool in distinguishing these phases [4][5][6][7][8][9][10].…”

Section: Introductionmentioning

confidence: 99%

Towards Gauging Time-Reversal Symmetry: A Tensor Network Approach

Xie

Vishwanath

2015

Phys. Rev. X

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It is well known that unitary symmetries can be "gauged," i.e., defined to act in a local way, which leads to a corresponding gauge field. Gauging, for example, the charge-conservation symmetry leads to electromagnetic gauge fields. It is an open question whether an analogous process is possible for time reversal which is an antiunitary symmetry. Here, we discuss a route to gauging time-reversal symmetry that applies to gapped quantum ground states that admit a tensor network representation. The tensor network representation of quantum states provides a notion of locality for the wave function coefficient and hence a notion of locality for the action of complex conjugation in antiunitary symmetries. Based on that, we show how time reversal can be applied locally and also describe time-reversal symmetry twists that act as gauge fluxes through nontrivial loops in the system. As with unitary symmetries, gauging time reversal provides useful access to the physical properties of the system. We show how topological invariants of certain timereversal symmetric topological phases in D ¼ 1, 2 are readily extracted using these ideas.

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“…SPT phases are short-range entangled [1] with a global symmetry and have been studied intensively in stronglycorrelated bosonic systems . Much progress has also been made in two-dimensional (2D) SETs [36][37][38][39][40][41][42][43][44][45][46][47], which are partially driven by tremendous efforts in quantum spin liquids (QSL) [36,48] that respect a certain global symmetry (e.g., spatial reflection, time-reversal, Ising Z 2 , U(1) and SU(2) spin rotations, etc.). In contrast to SPTs, SETs are long-range entangled [1] and support emergent excitations, such as anyons in 2D systems.…”

Section: Introductionmentioning

confidence: 99%

Symmetry enrichment in three-dimensional topological phases

Ning

Liu

2016

Phys. Rev. B

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While two-dimensional symmetry-enriched topological phases (SETs) have been studied intensively and systematically, three-dimensional ones are still open issues. We propose an algorithmic approach of imposing global symmetry Gs on gauge theories (denoted by GT) with gauge group Gg. The resulting symmetric gauge theories are dubbed "symmetry-enriched gauge theories" (SEG), which may be served as low-energy effective theories of three-dimensional symmetric topological quantum spin liquids. We focus on SEGs with gauge group Gg = ZN 1 × ZN 2 × · · · and on-site unitary symmetry group Gs = ZK 1 × ZK 2 × · · · or Gs = U(1) × ZK 1 × · · · . Each SEG(Gg, Gs) is described in the path integral formalism associated with certain symmetry assignment. From the path-integral expression, we propose how to physically diagnose the ground state properties (i.e., SET orders) of SEGs in experiments of charge-loop braidings (patterns of symmetry fractionalization) and the mixed multi-loop braidings among deconfined loop excitations and confined symmetry fluxes. From these symmetry-enriched properties, one can obtain the map from SEGs to SETs. By giving full dynamics to background gauge fields, SEGs may be eventually promoted to a set of new gauge theories (denoted by GT * ). Based on their gauge groups, GT * s may be further regrouped into different classes each of which is labeled by a gauge group G * g . Finally, a web of gauge theories involving GT, SEG, SET and GT * is achieved. We demonstrate the above symmetry-enrichment physics and the web of gauge theories through many concrete examples.

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Quantized topological terms in weak-coupling gauge theories with a global symmetry and their connection to symmetry-enriched topological phases

Cited by 95 publications

References 102 publications

Symmetry-protected topological phases from decorated domain walls

Symmetry-protected topological phases from decorated domain walls

Towards Gauging Time-Reversal Symmetry: A Tensor Network Approach

Symmetry enrichment in three-dimensional topological phases

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