Symmetry enrichment in three-dimensional topological phases

Ning, Shang-Qiang; Liu, Zheng-Xin; Ye, Peng

doi:10.1103/physrevb.94.245120

Cited by 25 publications

(21 citation statements)

References 87 publications

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“…Our present results combined together with Ref. [7] positively support the previous attempts based on continuum TQFTs [12,36,[38][39][40][41][42][43][44][45][46][47][48][49][50][51]. Various data derived from continuum TQFTs can be checked and compared through the discrete cocycle and lattice formulations [8-10, 29, 30, 37, 52-57].…”

Section: The Plan Of the Article And A Short Summarysupporting

confidence: 87%

Tunneling topological vacua via extended operators: (Spin-)TQFT spectra and boundary deconfinement in various dimensions

Wang

Ohmori

Putrov

et al. 2018

Progress of Theoretical and Experimental Physics

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Distinct quantum vacua of topologically ordered states can be tunneled into each other via extended operators. The possible applications include condensed matter and quantum cosmology. We present a straightforward approach to calculate the partition function on various manifolds and ground state degeneracy (GSD), mainly based on continuum/cochain Topological Quantum Field Theories (TQFT), in any dimension. This information can be related to the counting of extended operators of bosonic/fermionic TQFT. On the lattice scale, anyonic particles/strings live at the ends of line/surface operators. Certain systems in different dimensions are related to each other through dimensional reduction schemes, analogous to (de)categorification. Examples include spin TQFTs derived from gauging the interacting fermionic symmetry protected topological states (with fermion parity Z f 2 ) of symmetry group Z 4 × Z 2 and (Z 4 ) 2 in 3+1D, also Z 2 and (Z 2 ) 2 in 2+1D. Gauging the last three cases begets non-Abelian spin TQFT (fermionic topological order). We consider situations where a TQFT lives on (1) a closed spacetime or (2) a spacetime with boundary, such that the bulk and boundary are fully-gapped and short or long-range entangled (SRE/LRE). Anyonic excitations can be deconfined on the boundary. We introduce new exotic topological interfaces on which neither particle nor string excitations alone condensed, but only fuzzy-composite objects of extended operators can end (e.g. a stringlike composite object formed by a set of particles can end on a special 2+1D boundary of 3+1D bulk). We explore the relations between group extension constructions and partially breaking constructions (e.g. 0-form/higher-form/"composite" breaking) of topological boundaries, after gauging. We comment on the implications of entanglement entropy for some of such LRE systems. 1 We denote the spacetime dimensions as d + 1D 2 One can also consider a generalization of this relation by turning on a background flat connection A (G) for a global symmetry G. First, non-trivial holonomies along 1-cycles of M d will result in replacement H M d by the corresponding twisted Hilbert space H tw M d . Second, a non-trivial holonomy g ∈ G along the time S 1 will result in insertion of ρ(g) into the trace, where ρ is the representation of G on the Hilbert space:( 1.2) In condensed matter, this is related to the symmetry twist inserted on M d to probe the Symmetry Protected/Enriched Topological states (SPTs/SETs) [2,12]. In this work, instead we mainly focus on eqn. (1.1).

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Section: The Plan Of the Article And A Short Summarysupporting

confidence: 87%

Tunneling topological vacua via extended operators: (Spin-)TQFT spectra and boundary deconfinement in various dimensions

Wang

Ohmori

Putrov

et al. 2018

Progress of Theoretical and Experimental Physics

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show abstract

“…Because of the direct product structure in Eq. (1), the other global symmetries remain unaffected by the gauging procedure, so we end up with a Z N 0 gauge theory enriched by a symmetry group G/Z N 0 ¼ Q K i¼1 Z N i [9,[30][31][32][33][34][35][36][37].…”

Section: B Fspt Phases and Gauging Symmetrymentioning

confidence: 99%

Loop Braiding Statistics and Interacting Fermionic Symmetry-Protected Topological Phases in Three Dimensions

2018

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We study Abelian braiding statistics of loop excitations in three-dimensional gauge theories with fermionic particles and the closely related problem of classifying 3D fermionic symmetry-protected topological (FSPT) phases with unitary symmetries. It is known that the two problems are related by turning FSPT phases into gauge theories through gauging the global symmetry of the former. We show that there exist certain types of Abelian loop braiding statistics that are allowed only in the presence of fermionic particles, which correspond to 3D "intrinsic" FSPT phases, i.e., those that do not stem from bosonic SPT phases. While such intrinsic FSPT phases are ubiquitous in 2D systems and in 3D systems with antiunitary symmetries, their existence in 3D systems with unitary symmetries was not confirmed previously due to the fact that strong interaction is necessary to realize them. We show that the simplest unitary symmetry to support 3D intrinsic FSPT phases is Z 2 × Z 4 . To establish the results, we first derive a complete set of physical constraints on Abelian loop braiding statistics. Solving the constraints, we obtain all possible Abelian loop braiding statistics in 3D gauge theories, including those that correspond to intrinsic FSPT phases. Then, we construct exactly soluble state-sum models to realize the loop braiding statistics. These state-sum models generalize the well-known Crane-Yetter and Dijkgraaf-Witten models.

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“…The two conditions (14) and (15), together with the existence of (0,2,1,0) in the spectrum, strongly constrains the allowed values of (q e ,q m ) for M SO(3) and the allowed spectra of the U(1) spin liquids. For example, the (E f T M f ) θ spin liquid could satisfy Eq.…”

Section: A Triviality Of = πmentioning

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%

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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Symmetry enrichment in three-dimensional topological phases

Cited by 25 publications

References 87 publications

Tunneling topological vacua via extended operators: (Spin-)TQFT spectra and boundary deconfinement in various dimensions

Tunneling topological vacua via extended operators: (Spin-)TQFT spectra and boundary deconfinement in various dimensions

Loop Braiding Statistics and Interacting Fermionic Symmetry-Protected Topological Phases in Three Dimensions

Symmetry enriched U(1) quantum spin liquids

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