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arXiv:1212.1827 · Physical Review B

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

Ling-Yan Hung, Xiao-Gang Wen

Published Apr 5, 2013• 94 citations
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Abstract

We study the quantized topological terms in a weak-coupling gauge theory with gauge group ${G}_{g}$ and a global symmetry ${G}_{s}$ in $d$ space-time dimensions. We show that the quantized topological terms are classified by a pair $(G,{\ensuremath{\nu}}_{d})$, where $G$ is an extension of ${G}_{s}$ by ${G}_{g}$ and ${\ensuremath{\nu}}_{d}$ an element in group cohomology ${\mathcal{H}}^{d}(G,\mathbb{R}/\mathbb{Z})$. When $d=3$ and/or when ${G}_{g}$ is finite, the weak-coupling gauge theories with quantized topological terms describe gapped symmetry enriched topological (SET) phases (i.e., gapped long-range-entangled phases with symmetry). Thus, those SET phases are classified by ${\mathcal{H}}^{d}(G,\mathbb{R}/\mathbb{Z})$, where $G/{G}_{g}={G}_{s}$. We also apply our theory to a simple case ${G}_{s}={G}_{g}={Z}_{2}$, which leads to 12 different SET phases in $2+1$ dimensions [(2+1)D], where quasiparticles have different patterns of fractional ${G}_{s}={Z}_{2}$ quantum numbers and fractional statistics. If the weak-coupling gauge theories are gapless, then the different quantized topological terms may describe different gapless phases of the gauge theories with a symmetry ${G}_{s}$, which may lead to different fractionalizations of ${G}_{s}$ quantum numbers and different fractional statistics [if in ($2+1$)D].

DOI
10.1103/physrevb.87.165107

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We study the quantized topological terms in a weak-coupling gauge theory with gauge group G[subscript g] and a global symmetry G[subscript s] in d space-time dimensions. We show that the quantized topological terms are classified by a pair (G,ν[subscript d]), where G is an extension of G[subscript s] by G[subscript g] and ν[subscript d] an element in group cohomology H[superscript d](G,R/Z). When d = 3 and/or when G[subscript g] is finite, the weak-coupling gauge theories with quantized topological terms describe gapped symmetry enriched topological (SET) phases (i.e., gapped long-range-entangled phases with symmetry). Thus, those SET phases are classified by H[superscript d](G,R/Z), where G/G[subscript g] = G[subscript s]. We also apply our theory to a simple case G[subscript s] = G[subscript g] = Z[subscript 2], which leads to 12 different SET phases in 2 + 1 dimensions [(2 + 1)D], where quasiparticles have different patterns of fractional G[subscript s] = Z[subscript 2] quantum numbers and fractional statistics. If the weak-coupling gauge theories are gapless, then the different quantized topological terms may describe different gapless phases of the gauge theories with a symmetry G[subscript s], which may lead to different fractionalizations of G[subscript s] quantum numbers and different fractional statistics [if in (2 + 1)D].
We consider a weakly coupled gauge theory where charged particles all have large gaps (ie no Higgs condensation to break the gauge "symmetry") and the field strength fluctuates only weakly. We ask what kind of topological terms can be added to the Lagrangian of such a weakly coupled gauge theory. In this paper, we systematically construct quantized topological terms which are generalization of the Chern-Simons terms and F∧FF\wedge FF∧F terms, in space-time dimensions ddd and for any gauge groups (continuous or discrete), using each element of the topological cohomology classes Hd+1(BG,Z)H^{d+1}(BG,\Z)Hd+1(BG,Z) on the classifying space BGBGBG of the gauge group GGG. In 3ddd or for finite gauge groups above 3ddd, the weakly coupled gauge theories are gapped. So our results on topological terms can be viewed as a systematic construction of gapped topologically ordered phases of weakly coupled gauge theories. In other cases, the weakly coupled gauge theories are gapless. So our results can be viewed as an attempt to systematically construct different gapless phases of weakly coupled gauge theories. Amazingly, the bosonic symmetry protected topological (SPT) phases with a finite on-site symmetry group GGG are also classified by the same Hd+1(BG,Z)H^{d+1}(BG,\Z)Hd+1(BG,Z). (SPT phases are gapped quantum phases with a symmetry and trivial topological order.) In this paper, we show an explicit duality relation between topological gauge theories with the quantized topological terms and the bosonic SPT phases, for any finite group GGG and in any dimensions; a result first obtained by Levin and Gu. We also study the relation between topological lattice gauge theory and the string-net states with non-trivial topological order and no symmetry.
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.
Recently, a new class of quantum phases of matter---symmetry protected topological states, such as topological insulators---has attracted much attention. In presence of interactions, group cohomology provides a classification of these [Chen et al., Phys. Rev. B 87, 155114 (2013)]. These phases have short-ranged entanglement and no topological order in the bulk. However, when long-range entangled topological order is present, it is much less understood how to classify quantum phases of matter in the presence of global symmetries. Here we present a classification of bosonic gapped quantum phases with or without long-range entanglement in the presence or absence of on-site global symmetries. In 2+12+12+1 dimensions, the quantum phases in the presence of a global symmetry group SGSGSG, and with topological order described by a finite gauge group GGGGGG, are classified by the cohomology group ${H}^{3}(SG\ifmmode\times\else\texttimes\fi{}GG,U(1))$. Generally, in d+1d+1d+1 dimensions, such quantum phases are classified by ${H}^{d+1}(SG\ifmmode\times\else\texttimes\fi{}GG,U(1))$. Although we only partially understand to what extent our classification is complete, we present an exactly solvable local bosonic model, in which the topological order is emergent, for each given class in our classification. When the global symmetry is absent, the topological order in our models is described by the general Dijkgraaf-Witten discrete gauge theories. When the topological order is absent, our models become the exactly solvable models for symmetry protected topological phases [Chen et al., Phys. Rev. B 87, 155114 (2013)]. When both the global symmetry and the topological order are present, our models describe symmetry enriched topological phases. Our classification includes, but goes beyond, the previously discussed projective symmetry group classification. Measurable signatures of these symmetry enriched topological phases and generalizations of our classification are discussed.
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 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.