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arXiv:1905.03276 · Physical Review Research

Fractionalization and anomalies in symmetry-enriched U(1) gauge theories

Shang-Qiang Ning, Liujun Zou, Meng Cheng

Published Oct 8, 2020• 16 citations
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Abstract

We classify symmetry fractionalization and anomalies in a (3+1)d U(1) gauge theory enriched by a global symmetry group G.We find that, in general, a symmetry-enrichment pattern is specified by four pieces of data: ρ, a map from G to the duality symmetry group of this U(1) gauge theory which physically encodes how the symmetry permutes the fractional excitations, ν ∈ H 2 ρ [G, U T (1)], the symmetry actions on the electric charge, p ∈ H 1 [G, Z T ], indication of certain domain wall decoration with bosonic integer quantum Hall (BIQH) states, and a torsor n over H 3 ρ [G, Z], the symmetry actions on the magnetic monopole.However, certain choices of (ρ, ν, p, n) are not physically realizable, i.e., they are anomalous.We find that there are two levels of anomalies.The first level of anomalies obstruct the fractional excitations being deconfined, thus are referred to as the deconfinement anomaly.States with these anomalies can be realized on the boundary of a (4+1)d long-range entangled state.If a state does not suffer from a deconfinement anomaly, there can still be the second level of anomaly, the more familiar 't Hooft anomaly, which forbids certain types of symmetry fractionalization patterns to be implemented in an on-site fashion.States with these anomalies can be realized on the boundary of a (4+1)d short-range entangled state.We apply these results to some interesting physical examples.

DOI
10.1103/physrevresearch.2.043043

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We classify symmetry fractionalization and anomalies in a (3+1)d U(1) gauge theory enriched by a global symmetry group G.We find that, in general, a symmetry-enrichment pattern is specified by four pieces of data: ρ, a map from G to the duality symmetry group of this U(1) gauge theory which physically encodes how the symmetry permutes the fractional excitations, ν ∈ H 2 ρ [G, U T (1)], the symmetry actions on the electric charge, p ∈ H 1 [G, Z T ], indication of certain domain wall decoration with bosonic integer quantum Hall (BIQH) states, and a torsor n over H 3 ρ [G, Z], the symmetry actions on the magnetic monopole.However, certain choices of (ρ, ν, p, n) are not physically realizable, i.e., they are anomalous.We find that there are two levels of anomalies.The first level of anomalies obstruct the fractional excitations being deconfined, thus are referred to as the deconfinement anomaly.States with these anomalies can be realized on the boundary of a (4+1)d long-range entangled state.If a state does not suffer from a deconfinement anomaly, there can still be the second level of anomaly, the more familiar 't Hooft anomaly, which forbids certain types of symmetry fractionalization patterns to be implemented in an on-site fashion.States with these anomalies can be realized on the boundary of a (4+1)d short-range entangled state.We apply these results to some interesting physical examples.
A bstract We use the intrinsic one-form and two-form global symmetries of (3+1) d bosonic field theories to classify quantum phases enriched by ordinary (0-form) global symmetry. Different symmetry-enriched phases correspond to different ways of coupling the theory to the background gauge field of the ordinary symmetry. The input of the classification is the higher-form symmetries and a permutation action of the 0-form symmetry on the lines and surfaces of the theory. From these data we classify the couplings to the background gauge field by the 0-form symmetry defects constructed from the higher-form symmetry defects. For trivial two-form symmetry the classification coincides with the classification for symmetry fractionalizations in (2 + 1) d . We also provide a systematic method to obtain the symmetry protected topological phases that can be absorbed by the coupling, and we give the relative ’t Hooft anomaly for different couplings. We discuss several examples including the gapless pure U(1) gauge theory and the gapped Abelian finite group gauge theory. As an application, we discover a tension with a conjectured duality in (3 + 1) d for SU(2) gauge theory with two adjoint Weyl fermions.
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.
In a phase with fractional excitations, topological properties are enriched in the presence of global symmetry. In particular, fractional excitations can transform under symmetry in a fractionalized manner, resulting in different symmetry enriched topological (SET) phases. While a good deal is now understood in 2D regarding what symmetry fractionalization patterns are possible, the situation in 3D is much more open. A new feature in 3D is the existence of loop excitations, so to study 3D SET phases, first we need to understand how to properly describe the fractionalized action of symmetry on loops. Using a dimensional reduction procedure, we show that these loop excitations exist as the boundary between two 2D SET phases, and the symmetry action is characterized by the corresponding difference in SET orders. Moreover, similar to the 2D case, we find that some seemingly possible symmetry fractionalization patterns are actually anomalous and cannot be realized strictly in 3D. We detect such anomalies using the flux fusion method we introduced previously in 2D. To illustrate these ideas, we use the 3DZ23\mathrm{D}\phantom{\rule{4pt}{0ex}}{Z}_{2}3DZ2​ gauge theory with Z2{Z}_{2}Z2​ global symmetry as an example, and enumerate and describe the corresponding SET phases. In particular, we find four nonanomalous SET phases and one anomalous SET phase, which we show can be realized as the surface of a 4D system with symmetry protected topological order.
We investigate the gauging of higher-form finite Abelian symmetries and their sub-groups in quantum spin models in spatial dimensions d=2 d = 2 and 3. Doing so, we naturally uncover gauged models with dual higher-group symmetries and potential mixed ‘t Hooft anomalies. We demonstrate that the mixed anomalies manifest as the symmetry fractionalization of higher-form symmetries participating in the mixed anomaly. Gauging is realized as an isomorphism or duality between the bond algebras that generate the space of quantum spin models with the dual generalized symmetry structures. We explore the mapping of gapped phases under such gauging related dualities for 0-form and 1-form symmetries in spatial dimension d=2 d = 2 and 3. In d=2 d = 2 , these include several non-trivial dualities between short-range entangled gapped phases with 0-form symmetries and 0-form symmetry enriched Higgs and (twisted) deconfined phases of the gauged theory with possible symmetry fractionalizations. Such dualities also imply strong constraints on several unconventional, i.e., deconfined or topological transitions. In d=3 d = 3 , among others, we find, dualities between topological orders via gauging of 1-form symmetries. Hamiltonians self-dual under gauging of 1-form symmetries host emergent non-invertible symmetries, realizing higher-categorical generalizations of the Tambara-Yamagami fusion category.
Topological order in two-dimensions can be described in terms of deconfined quasiparticle excitations—anyons—and their braiding statistics. However, it has recently been realized that this data does not completely describe the situation in the presence of an unbroken global symmetry. In this case, there can be multiple distinct quantum phases with the same anyons and statistics, but with different patterns of symmetry fractionalization—termed symmetry enriched topological order. When the global symmetry group G , which we take to be discrete, does not change topological superselection sectors—i.e. does not change one type of anyon into a different type of anyon—one can imagine a local version of the action of G around each anyon. This leads to projective representations and a group cohomology description of symmetry fractionalization, with the second cohomology group H 2 ( G , A a b e l i a n ) being the relevant group. In this paper, we treat the general case of a symmetry group G possibly permuting anyon types. We show that despite the lack of a local action of G , one can still make sense of a so-called twisted group cohomology description of symmetry fractionalization, and show how this data is encoded in the associativity of fusion rules of the extrinsic ‘twist’ defects of the symmetry. Furthermore, building on work of Hermele (2014 Phys. Rev. B 90 184418 ), we construct a wide class of exactly-solvable models which exhibit this twisted symmetry fractionalization, and connect them to our formal framework.
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.