One of the central ideas regarding anomalies in topological phases of matter is that they imply the existence of higher-dimensional physics, with an anomaly in a D-dimensional theory typically being canceled by a bulk (D+1)-dimensional symmetry-protected topological phase (SPT). We demonstrate that for some topological phases with reflection symmetry, anomalies may actually be canceled by a D-dimensional SPT, provided that it comes embedded in an otherwise trivial (D+1)-dimensional bulk. We illustrate this for the example of
ZN topological order enriched with reflection symmetry in (2+1)D, and along the way establish a classification of anomalous reflection symmetry fractionalization patterns. In particular, we show that anomalies occur if and only if both electric and magnetic quasiparticle excitations possess nontrivial fractional reflection quantum numbers.
We study a class of anomalies associated with time-reversal and spatial-reflection symmetry in (2+1)-dimensional 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 time reversal (denoted
Z2T) 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
Z2T 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
Z2T anomalies can all be obtained by gauging the unitary
Z2 subgroup of a different TQFT with a
Z4T symmetry. We further show that the anomaly can be resolved in several distinct ways: (1) the true symmetry of the theory is
Z4T, or (2) the theory can be considered to be a theory of fermions, with ${\mathbf{T}}^{2}={(\ensuremath{-}1)}^{{N}_{f}}$ corresponding to fermion parity. Finally, we demonstrate that theories with the
Z2T localization anomaly can be compatible with
Z2T if they are ``pseudorealized'' at the surface of a (3+1)D symmetry-enriched topological phase. The ``pseudorealization'' 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 quasiparticles are truly confined to the surface.
We study anomalies in time-reversal-(
Z2T) and
U(1)-symmetric topological orders. In this context, an anomalous topological order is one that cannot be realized in a strictly
(2+1)-dimensional system but can be realized on the surface of a
(3+1)-dimensional [(
3+1)D] symmetry-protected topological (SPT) phase. To detect these anomalies we propose several anomaly indicators, functions that take as input the algebraic data of a symmetric topological order and that output a number indicating the presence or absence of an anomaly. We construct such indicators for both structures of the full symmetry group, i.e., $U(1)\ensuremath{\rtimes}{\mathbb{Z}}_{2}^{T}$ and $U(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{T}$, and for both bosonic and fermionic topological orders. In all cases we conjecture that our indicators are complete in the sense that the anomalies they detect are in one-to-one correspondence with the known classification of
(3+1)D SPT phases with the same symmetry. We also show that one of our indicators for bosonic topological orders has a mathematical interpretation as a partition function for the bulk
(3+1)D SPT phase on a particular manifold and in the presence of a particular background gauge field for the
U(1) symmetry.
We present the correspondence between symmetry-protected topological (SPT) phases and their anomalous boundary states, based on examples in various spacetime dimensions. Through the study of the effect of interactions on these SPT phases, we discovered a new formalism of quantum anomalies, associated with discrete spacetime (such as time-reversal and spatial reflection) symmetries in particular, to classify distinct interacting topological phases. An example is the Z2 classification of the (2+1)d topological insulator protected by charge U(1) and time-reversal (or CP) symmetries, which can be deduced by the form of the global U(1) gauge anomalies on its edge theories defined on closed unorientable manifolds. In this case, the nontrivial phase (in free systems) is robust against electron interactions. Another example is the (3+1)d topological superconductor protected by only time-reversal or reflection symmetry. For this system, we identified the bulk phase by studying the global gravitational anomalies of the surface theories formulated on unorientable spacetime manifolds, and also discussed its connection to the collapse of the non-interacting classification by an integer Z to Z16, in the presence of interactions. We also revisit the problem of gauging a discrete internal symmetry in theories of chiral (Weyl) fermions in 3+1 dimensions – which have not been fully understood so far – from the perspective of fermionic SPT phases in 4+1 dimensions. Comparing with the previous results, we give a complete answer for the anomalies constraints on the discrete symmetry, as our approach is based on purely geometrical considerations, namely, our assumption is more fundamental and general. Furthermore, our result also provides an understanding of gapped states of fermions with anomalous discrete symmetries, and we present a model, based on weak coupling, for constructing these anomalous gapped states.
In addition to possessing fractional statistics, anyon excitations of a 2D topologically ordered state can realize symmetry in distinct ways, leading to a variety of symmetry-enriched topological (SET) phases.While the symmetry fractionalization must be consistent with the fusion and braiding rules of the anyons, not all ostensibly consistent symmetry fractionalizations can be realized in 2D systems.Instead, certain "anomalous" SETs can only occur on the surface of a 3D symmetry-protected topological (SPT) phase.In this paper, we describe a procedure for determining whether a SET of a discrete, on-site, unitary symmetry group G is anomalous or not.The basic idea is to gauge the symmetry and expose the anomaly as an obstruction to a consistent topological theory combining both the original anyons and the gauge fluxes.Utilizing a result of Etingof, Nikshych, and Ostrik, we point out that a class of obstructions is captured by the fourth cohomology group H 4 ðG; Uð1ÞÞ, which also precisely labels the set of 3D SPT phases, with symmetry group G.An explicit procedure for calculating the cohomology data from a SET is given, with the corresponding physical intuition explained.We thus establish a general bulk-boundary correspondence between the anomalous SET and the 3D bulk SPT whose surface termination realizes it.We illustrate this idea using the chiral spin liquid [Uð1Þ 2 ] topological order with a reduced symmetry Z 2 × Z 2 ⊂ SOð3Þ, which can act on the semion quasiparticle in an anomalous way.We construct exactly solved 3D SPT models realizing the anomalous surface terminations and demonstrate that they are nontrivial by computing three-loop braiding statistics.Possible extensions to antiunitary symmetries are also discussed.
We introduce a method, dubbed the flux-fusion anomaly test, to detect certain anomalous symmetry fractionalization patterns in two-dimensional symmetry-enriched topological (SET) phases. We focus on bosonic systems with Z2 topological order and a symmetry group of the form G=U(1)⋊G′, where G′ is an arbitrary group that may include spatial symmetries and/or time reversal. The anomalous fractionalization patterns we identify cannot occur in strictly d=2 systems but can occur at surfaces of d=3 symmetry-protected topological (SPT) phases. This observation leads to examples of d=3 bosonic topological crystalline insulators (TCIs) that, to our knowledge, have not previously been identified. In some cases, these d=3 bosonic TCIs can have an anomalous superfluid at the surface, which is characterized by nontrivial projective transformations of the superfluid vortices under symmetry. The basic idea of our anomaly test is to introduce fluxes of the U(1) symmetry and to show that some fractionalization patterns cannot be extended to a consistent action of G′ symmetry on the fluxes. For some anomalies, this can be described in terms of dimensional reduction to d=1 SPT phases. We apply our method to several different symmetry groups with nontrivial anomalies, including G=U(1)×ZT2 and G=U(1)×ZP2, where ZT2 and ZP2 are time-reversal and d=2 reflection symmetry, respectively.