Symmetry fractionalization and anomaly detection in three-dimensional topological phases

We study a class of anomalies associated with time-reversal and spatial reflection symmetry in (2+1)D bosonic topological phases of matter. In these systems, the topological quantum numbers of the quasiparticles, such as the fusion rules and braiding statistics, possess a Z2 symmetry which can be associated with either timereversal (denoted Z T 2 ) or spatial reflections. Under this symmetry, correlation functions of all Wilson loop operators in the low energy topological quantum field theory (TQFT) are invariant. However, the theories that we study possess a severe anomaly associated with the failure to consistently localize the symmetry action to the quasiparticles, precluding even defining a consistent notion of symmetry fractionalization in such systems. We present simple sufficient conditions which determine when Z T 2 symmetry localization anomalies exist in general. We present an infinite series of TQFTs with such anomalies, some examples of which include USp(4)2 Chern-Simons (CS) theory and SO(4) 4 CS theory. The theories that we find with these Z T 2 anomalies can all be obtained by gauging the unitary Z2 subgroup of a different TQFT with a Z T 4 symmetry. We further show that the anomaly can be resolved in several distinct ways: (1) the true symmetry of the theory is Z T 4 , or (2) the theory can be considered to be a theory of fermions, with T 2 = (−1) N f corresponding to fermion parity. Finally, we demonstrate that theories with the Z T 2 localization anomaly can be compatible with Z T 2 if they are "pseudo-realized" at the surface of a (3+1)D symmetry-enriched topological phase. The "pseudo-realization" refers to the fact that the bulk (3+1)D system is described by a dynamical Z2 gauge theory and thus only a subset of the quasi-particles are truly confined to the surface.

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“…[ρ] (G, A) could be related to the symmetry fractionalization of string excitations in the (3+1)D system [45,[49][50][51].…”

Section: Introductionmentioning

confidence: 99%

Time-reversal and spatial-reflection symmetry localization anomalies in (2+1)-dimensional topological phases of matter

Barkeshli

Cheng

2018

Phys. Rev. B

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“…This shows that eZmZ is still anomalous with T × Z 2 symmetry, and we also see T plays a role in protecting this state, even though it appears to act on e and m trivially. 13 Again, it will be useful for us to understand what this state becomes when the symmetry is enhanced to the full (O(2) × T )/Z 2 symmetry of M bT . Because only the Z 2 acts nontrivially in this state, when the symmetry is enhanced, U(1) and T should still act trivially.…”

Section: B Superfluid Coexisting With Another Anomalous Topological mentioning

confidence: 99%

“…The simplest way to construct these 4 + 1-d surface states is the following layer construction, which has been widely used to construct topological states [8,13,50,75].…”

Section: Appendix C: Anomalous Spin Liquids As Surface States Of Somementioning

confidence: 99%

“…Though much of the early work on spin liquids dealt with SET phases, it is only in the last few years that there has been tremendous and systematic progress in understanding their full structure and classification in two-dimensional systems [1][2][3][4][5][6][7][8][9][10][11]. Some limited progress has been made for three-dimensional SET phases as well [12][13][14]. A different broad class of spin liquid phases have gapless excitations.…”

Section: Introductionmentioning

confidence: 99%

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Symmetry enriched U(1) quantum spin liquids

2018

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We classify and characterize three-dimensional U(1) quantum spin liquids [deconfined U(1) gauge theories] with global symmetries. These spin liquids have an emergent gapless photon and emergent electric/magnetic excitations (which we assume are gapped). We first discuss in great detail the case with time-reversal and SO(3) spin rotational symmetries. We find there are 15 distinct such quantum spin liquids based on the properties of bulk excitations. We show how to interpret them as gauged symmetry-protected topological states (SPTs). Some of these states possess fractional response to an external SO(3) gauge field, due to which we dub them "fractional topological paramagnets." We identify 11 other anomalous states that can be grouped into three anomaly classes. The classification is further refined by weakly coupling these quantum spin liquids to bosonic symmetry protected topological (SPT) phases with the same symmetry. This refinement does not modify the bulk excitation structure but modifies universal surface properties. Taking this refinement into account, we find there are 168 distinct such U(1) quantum spin liquids. After this warm-up, we provide a general framework to classify symmetry enriched U(1) quantum spin liquids for a large class of symmetries. As a more complex example, we discuss U(1) quantum spin liquids with time-reversal and Z 2 symmetries in detail. Based on the properties of the bulk excitations, we find there are 38 distinct such spin liquids that are anomaly-free. There are also 37 anomalous U(1) quantum spin liquids with this symmetry. Finally, we briefly discuss the classification of U(1) quantum spin liquids enriched by some other symmetries.

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“…Recently there has been great interest in symmetry enriched long-range entangled phases: phases which cannot be smoothly connected to a trivial state even in the absence of any symmetry, and which acquire symmetry protected distinctions among themselves. In the last few years tremendous and systematic progress has been made in understanding twodimensional symmetry enriched long-range entangled states [34,[44][45][46][47][48][49][50][51][52][53][54], and the studies of three-dimensional symmetry enriched long-range entangled states just began [55][56][57]. Very recently, in Ref.…”

Section: Introductionmentioning

confidence: 99%

Bulk characterization of topological crystalline insulators: Stability under interactions and relations to symmetry enriched U (1) quantum spin liquids

Zou

2018

Phys. Rev. B

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Topological crystalline insulators (TCIs) are nontrivial quantum phases of matter protected by crystalline (and other) symmetries. They are originally predicted by band theories, so an important question is their stability under interactions. In this paper, by directly studying the physical bulk properties of several band-theory-based nontrivial TCIs that are conceptually interesting and/or experimentally feasible, we show they are stable under interactions. These TCIs include (1) a weak topological insulator, (2) a TCI with a mirror symmetry and its time-reversal symmetric generalizations, (3) a doubled topological insulator with a mirror symmetry, and (4) two TCIs with symmetry-enforced-gapless surfaces. We describe two complementary methods that allow us to determine the properties of the magnetic monopoles obtained by coupling these TCIs to a U (1) gauge field. These methods involve studying different types of surface states of these TCIs. Applying these methods to our examples, we find all of them have nontrivial monopoles, which proves their stability under interactions. Furthermore, we discuss two levels of relations between these TCIs and symmetry enriched U (1) quantum spin liquids (QSLs). First, these TCIs are directly related to U (1) QSLs with crystalline symmetries. Second, there is an interesting correspondence between U (1) QSLs with crystalline symmetries and U (1) QSLs with internal symmetries. In particular, the TCIs with symmetry-enforced-gapless surfaces are related to the "fractional topological paramagnets" introduced in Zou et al. [arXiv:1710.00743].

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Symmetry fractionalization and anomaly detection in three-dimensional topological phases

Cited by 23 publications

References 54 publications

Time-reversal and spatial-reflection symmetry localization anomalies in (2+1)-dimensional topological phases of matter

Time-reversal and spatial-reflection symmetry localization anomalies in (2+1)-dimensional topological phases of matter

Symmetry enriched U(1) quantum spin liquids

Bulk characterization of topological crystalline insulators: Stability under interactions and relations to symmetry enriched U (1) quantum spin liquids

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