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arXiv:1609.00985 · Physical review. B./Physical review. B

Symmetry enrichment in three-dimensional topological phases

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

Published Dec 15, 2016• 33 citations
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Abstract

While two-dimensional symmetry-enriched topological phases $(\text{SET}\mathrm{s})$ have been studied intensively and systematically, three-dimensional ones are still open issues. We propose an algorithmic approach of imposing global symmetry ${G}_{s}$ on gauge theories (denoted by $\text{GT})$ with gauge group ${G}_{g}$. The resulting symmetric gauge theories are dubbed ``symmetry-enriched gauge theories'' $(\text{SEG})$, which may be served as low-energy effective theories of three-dimensional symmetric topological quantum spin liquids. We focus on $\text{SEG}\mathrm{s}$ with gauge group ${G}_{g}={\mathbb{Z}}_{{N}_{1}}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{N}_{2}}\ifmmode\times\else\texttimes\fi{}\ensuremath{\cdots}$ and onsite unitary symmetry group ${G}_{s}={\mathbb{Z}}_{{K}_{1}}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{K}_{2}}\ifmmode\times\else\texttimes\fi{}\ensuremath{\cdots}$ or ${G}_{s}=\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{K}_{1}}\ifmmode\times\else\texttimes\fi{}\ensuremath{\cdots}$. Each $\text{SEG}({G}_{g},{G}_{s})$ 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., $\text{SET}$ orders) of $\text{SEG}\mathrm{s}$ in experiments of charge-loop braidings (patterns of symmetry fractionalization) and the mixed multiloop braidings among deconfined loop excitations and confined symmetry fluxes. From these symmetry-enriched properties, one can obtain the map from $\mathsf{\text{SEG}}\mathrm{s}$ to $\mathsf{\text{SET}}\mathrm{s}$. By giving full dynamics to background gauge fields, $\mathsf{\text{SEG}}\mathrm{s}$ may be eventually promoted to a set of new gauge theories (denoted by ${\mathsf{\text{GT}}}^{*})$. Based on their gauge groups, ${\mathsf{\text{GT}}}^{*}\mathrm{s}$ 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 $\mathsf{\text{GT}},\mathsf{\text{SEG}},\mathsf{\text{SET,}}$ and ${\mathsf{\text{GT}}}^{*}$ is achieved. We demonstrate the above symmetry-enrichment physics and the web of gauge theories through many concrete examples.

DOI
10.1103/physrevb.94.245120

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While two-dimensional symmetry-enriched topological phases (SETs)(\text{SET}\mathrm{s})(SETs) have been studied intensively and systematically, three-dimensional ones are still open issues. We propose an algorithmic approach of imposing global symmetry Gs{G}_{s}Gs​ on gauge theories (denoted by GT)\text{GT})GT) with gauge group Gg{G}_{g}Gg​. The resulting symmetric gauge theories are dubbed ``symmetry-enriched gauge theories'' (SEG)(\text{SEG})(SEG), which may be served as low-energy effective theories of three-dimensional symmetric topological quantum spin liquids. We focus on SEGs\text{SEG}\mathrm{s}SEGs with gauge group Gg=ZN1\ifmmode×\else\texttimes\fiZN2\ifmmode×\else\texttimes\fi\ensuremath⋯{G}_{g}={\mathbb{Z}}_{{N}_{1}}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{N}_{2}}\ifmmode\times\else\texttimes\fi{}\ensuremath{\cdots}Gg​=ZN1​​\ifmmode×\else\texttimes\fiZN2​​\ifmmode×\else\texttimes\fi\ensuremath⋯ and onsite unitary symmetry group Gs=ZK1\ifmmode×\else\texttimes\fiZK2\ifmmode×\else\texttimes\fi\ensuremath⋯{G}_{s}={\mathbb{Z}}_{{K}_{1}}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{K}_{2}}\ifmmode\times\else\texttimes\fi{}\ensuremath{\cdots}Gs​=ZK1​​\ifmmode×\else\texttimes\fiZK2​​\ifmmode×\else\texttimes\fi\ensuremath⋯ or Gs=U(1)\ifmmode×\else\texttimes\fiZK1\ifmmode×\else\texttimes\fi\ensuremath⋯{G}_{s}=\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{K}_{1}}\ifmmode\times\else\texttimes\fi{}\ensuremath{\cdots}Gs​=U(1)\ifmmode×\else\texttimes\fiZK1​​\ifmmode×\else\texttimes\fi\ensuremath⋯. Each SEG(Gg,Gs)\text{SEG}({G}_{g},{G}_{s})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\text{SET}SET orders) of SEGs\text{SEG}\mathrm{s}SEGs in experiments of charge-loop braidings (patterns of symmetry fractionalization) and the mixed multiloop braidings among deconfined loop excitations and confined symmetry fluxes. From these symmetry-enriched properties, one can obtain the map from SEGs\mathsf{\text{SEG}}\mathrm{s}SEGs to SETs\mathsf{\text{SET}}\mathrm{s}SETs. By giving full dynamics to background gauge fields, SEGs\mathsf{\text{SEG}}\mathrm{s}SEGs may be eventually promoted to a set of new gauge theories (denoted by GT∗){\mathsf{\text{GT}}}^{*})GT∗). Based on their gauge groups, GT∗s{\mathsf{\text{GT}}}^{*}\mathrm{s}GT∗s may be further regrouped into different classes, each of which is labeled by a gauge group Gg∗{G}_{g}^{*}Gg∗​. Finally, a web of gauge theories involving GT,SEG,SET,\mathsf{\text{GT}},\mathsf{\text{SEG}},\mathsf{\text{SET,}}GT,SEG,SET, and GT∗{\mathsf{\text{GT}}}^{*}GT∗ is achieved. We demonstrate the above symmetry-enrichment physics and the web of gauge theories through many concrete examples.
Topological spin liquids can be described by topological gauge theories with global symmetry. Due to the presence of both nontrivial bulk deconfined gauge fluxes and global symmetry, topological spin liquids are examples of the so-called ``symmetry enriched topological phases'' (SETs). In this paper, we find that, in some twisted versions of topological gauge theories (with discrete Abelian gauge group Gg{G}_{g}Gg​), implementing a global symmetry (denoted by Gs{G}_{s}Gs​) is anomalous although symmetry charge carried by topological pointlike excitations is normally fractionalized and classified by the second cohomology group. To demonstrate the anomaly, we fully gauge the global symmetry, rendering a new gauge theory that is not gauge invariant. Therefore, the SET order of the ground state is anomalous, which cannot exist in the three-dimensional system alone. Such an anomalous state construction generalizes the ``2D surface topological order'' to three dimensions. A concrete example with ${G}_{g}={\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{4}$ and Gs=Z2{G}_{s}={\mathbb{Z}}_{2}Gs​=Z2​ is calculated.
Symmetry plays a crucial role in enriching topological phases of matter. Phases with intrinsic topological order that are symmetric are called symmetry-enriched topological phases (SET). In this paper, we focus on SETs in three spatial dimensions, where the intrinsic topological orders are described by Abelian gauge theory and the symmetry groups are also Abelian. As a series work of our previous research [Phys. Rev. B 94, 245120 (2016); (arXiv:1609.00985)], we study these topological phases described by twisted gauge theories with global symmetry and consider all possible topologically inequivalent charge matrices. Within each equivalence class, there is a unique pattern of symmetry fractionalization on both point-like and string-like topological excitations. In this way, we classify Abelian topological order enriched by Abelian symmetry within our field-theoretic approach. To illustrate, we concretely calculate many representative examples of SETs and discuss future directions.
A large class of symmetry-protected topological phases (SPT) in boson/spin systems have been recently predicted by the group cohomology theory. In this work, we consider bosonic SPT states at least with charge symmetry [U(1) or ZN{}_{N}N​] or spin-Sz{S}^{z}Sz rotation symmetry [U(1) or ZN{}_{N}N​] in two (2D) and three dimensions (3D) and the surface of 3D. If both are U(1), we apply external electromagnetic field/``spin gauge field'' to study the charge/spin response. For the SPT examples we consider {i.e., Uc{}_{c}c​(1)$\ensuremath{\rtimes}$Z2T{}_{2}^{T}2T​, Us{}_{s}s​(1)$\ifmmode\times\else\texttimes\fi{}$Z2T{}_{2}^{T}2T​, Uc{}_{c}c​(1)$\ifmmode\times\else\texttimes\fi{}$[Us{}_{s}s​(1)$\ensuremath{\rtimes}$Z2{}_{2}2​]; subscripts ccc and sss are short for charge and spin; Z2T{}_{2}^{T}2T​ and Z2{}_{2}2​ are time-reversal symmetry and $\ensuremath{\pi}$ rotation about Sy{S}^{y}Sy, respectively}, many variants of Witten effect in the 3D SPT bulk and various versions of anomalous surface quantum Hall effect are defined and systematically investigated. If charge or spin symmetry reduces to ZN{}_{N}N​ by considering charge-NNN or spin-NNN condensate, instead of the linear response approach, we gauge the charge/spin symmetry, leading to a dynamical gauge theory with some remaining global symmetry. The 3D dynamical gauge theory describes a symmetry-enriched topological phase (SET), i.e., a topologically ordered state with global symmetry which admits nontrivial ground-state degeneracy depending on spatial manifold topology. For the SPT examples we consider, the corresponding SET states are described by dynamical topological gauge theory with topological BF\mathit{BF}BF term and axionic $\ensuremath{\Theta}$ term in 3D bulk. In addition, the surface of SET is described by the chiral boson theory with quantum anomaly.
Symmetry fractionalization on topological excitations is one of the most remarkable quantum phenomena in topological orders with symmetry, i.e., symmetry-enriched topological phases. While much progress has been theoretically and experimentally made in two dimensions (2D), the understanding on symmetry fractionalization in 3D is far from complete. A long-standing challenge is to understand symmetry fractionalization on looplike topological excitations, which are spatially extended objects. In this paper, we construct a powerful topological-field-theoretical framework approach for 3D topological orders, which leads to a systematic characterization and classification of symmetry fractionalization. For systems with Abelian gauge groups (Gg{G}_{g}Gg​) and Abelian symmetry groups (Gs{G}_{s}Gs​), we successfully establish equivalence classes that lead to a finite number of patterns of symmetry fractionalization, although there are notoriously infinite number of Lagrangian-descriptions of the system. Based on this, we compute topologically distinct types of fractional symmetry charges carried by particles. Then, for each type, we compute topologically distinct statistical phases of braiding processes among loop excitations and external symmetry fluxes. As a result, we are able to unambiguously list all physical observables for each pattern of symmetry fractionalization. We present detailed calculations on many concrete examples. As an example, we find that the symmetry fractionalization in an untwisted ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ topological order with Z2{\mathbb{Z}}_{2}Z2​ symmetry is classified by ${({\mathbb{Z}}_{2})}^{6}\ensuremath{\bigoplus}{({\mathbb{Z}}_{2})}^{2}\ensuremath{\bigoplus}{({\mathbb{Z}}_{2})}^{2}\ensuremath{\bigoplus}{({\mathbb{Z}}_{2})}^{2}$. If the topological order is twisted, the classification reduces to (Z2)6{({\mathbb{Z}}_{2})}^{6}(Z2​)6 in which particle excitations always carry integer charge. Inspired by the field-theoretical analysis, we find that our classification of symmetry fractionalization with twist $\ensuremath{\omega}$ can be formally organized into an algebraic formalism: ${⨁}_{{\ensuremath{\nu}}_{i}}{\mathcal{H}}^{4}({G}_{g}{\ensuremath{\leftthreetimes}}_{{\ensuremath{\nu}}_{i}}{G}_{s},U(1))/{\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\omega}}({\ensuremath{\nu}}_{i}),$ where anomaly-free symmetry fractionalization on particles ${\ensuremath{\nu}}_{i}\ensuremath{\in}{\mathcal{H}}_{non}^{2}({G}_{s},{G}_{g})$. Despite the lack of detailed dependence of ${\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\omega}}({\ensuremath{\nu}}_{i})$ on $\ensuremath{\omega}$ and ${\ensuremath{\nu}}_{i}$, the present field-theoretical approach allows us to efficiently calculate and understand ${\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\omega}}({\ensuremath{\nu}}_{i})$. Several future directions are presented at the end of this paper.
We study (2+1)-dimensional phases with topological order, such as fractional quantum Hall states and gapped spin liquids, in the presence of global symmetries. Phases that share the same topological order can then differ depending on the action of symmetry, leading to symmetry-enriched topological (SET) phases. Here, we present a KKK-matrix Chern-Simons approach to identify distinct phases with Abelian topological order, in the presence of unitary or antiunitary global symmetries. A key step is the identification of a smooth edge sewing condition that is used to check if two putative phases are indeed distinct. We illustrate this method by classifying Z2{Z}_{2}Z2​ topological order (Z2{Z}_{2}Z2​ spin liquids) in the presence of an internal Z2{Z}_{2}Z2​ global symmetry for which we find six distinct phases. These include two phases with an unconventional action of symmetry that permutes anyons leading to symmetry-protected Majorana edge modes. Other routes to realizing protected edge states in SET phases are identified. Symmetry-enriched Laughlin states and double-semion theories are also discussed. Somewhat surprisingly, we observe that (i) gauging the global symmetry of distinct SET phases leads to topological orders with the same total quantum dimension, and (ii) a pair of distinct SET phases can yield the same topological order on gauging the symmetry.
The hallmark of a two-dimensional (2d) topologically ordered phase is the existence of deconfined ``anyon'' excitations that have exotic braiding and exchange statistics, different from those of ordinary bosons or fermions. As opposed to conventional Landau-Ginzburg-Wilson phases, which are classified on the basis of the spontaneous breaking of an underlying symmetry, topologically ordered phases, such as those occurring in the fractional quantum Hall effect, are absolutely stable, not requiring any such symmetry. Recently, though, it has been realized that symmetries, which may still be present in such systems, can give rise to a host of new, distinct, many-body phases, all of which share the same underlying topological order. These ``symmetry enriched'' topological (SET) phases are distinguished not on the basis of anyon braiding statistics alone, but also by the symmetry properties of the anyons, such as their fractional charges, or the way that different anyons are permuted by the symmetry. Thus a useful approach to classifying SETs is to determine all possible such symmetry actions on the anyons that are algebraically consistent with the anyon statistics. Remarkably, however, there exist symmetry actions that, despite being algebraically consistent, cannot be realized in any physical system, and hence do not lead to valid 2d SETs. One class of such ``anomalous'' SETs, characterized by certain disallowed symmetry fractionalization patterns, finds a physical interpretation as an allowed surface state of certain three-dimensional (3d) short-range entangled phases, but another, characterized by some seemingly valid but anomalous permutation actions of the symmetry on the anyons and encoded in an H3(G,A){H}^{3}(G,\mathcal{A})H3(G,A) group cohomology class, has so far eluded a physical interpretation. In this work, we find a way to physically realize these anomalously permuting SETs at the surfaces of certain 3d long-range entangled phases, expanding our understanding of general anomalous SETs in two dimensions.