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arXiv:1706.09464 · Physical review. B./Physical review. B

Time-reversal and spatial-reflection symmetry localization anomalies in (2+1)-dimensional topological phases of matter

Maissam Barkeshli, Meng Cheng

Published Sep 17, 2018• 48 citations
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Abstract

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 ${\mathbb{Z}}_{2}$ symmetry which can be associated with either time reversal (denoted ${\mathbb{Z}}_{2}^{\mathbf{T}})$ 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 ${\mathbb{Z}}_{2}^{\mathbf{T}}$ symmetry localization anomalies exist in general. We present an infinite series of TQFTs with such anomalies, some examples of which include $\mathrm{USp}{(4)}_{2}$ Chern-Simons (CS) theory and $\mathrm{SO}{(4)}_{4}$ CS theory. The theories that we find with these ${\mathbb{Z}}_{2}^{\mathbf{T}}$ anomalies can all be obtained by gauging the unitary ${\mathbb{Z}}_{2}$ subgroup of a different TQFT with a ${\mathbb{Z}}_{4}^{\mathbf{T}}$ symmetry. We further show that the anomaly can be resolved in several distinct ways: (1) the true symmetry of the theory is ${\mathbb{Z}}_{4}^{\mathbf{T}}$, 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 ${\mathbb{Z}}_{2}^{\mathbf{T}}$ localization anomaly can be compatible with ${\mathbb{Z}}_{2}^{\mathbf{T}}$ 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 ${\mathbb{Z}}_{2}$ gauge theory and thus only a subset of the quasiparticles are truly confined to the surface.

DOI
10.1103/physrevb.98.115129

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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{\mathbb{Z}}_{2}Z2​ symmetry which can be associated with either time reversal (denoted Z2T){\mathbb{Z}}_{2}^{\mathbf{T}})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{\mathbb{Z}}_{2}^{\mathbf{T}}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\mathrm{USp}{(4)}_{2}USp(4)2​ Chern-Simons (CS) theory and SO(4)4\mathrm{SO}{(4)}_{4}SO(4)4​ CS theory. The theories that we find with these Z2T{\mathbb{Z}}_{2}^{\mathbf{T}}Z2T​ anomalies can all be obtained by gauging the unitary Z2{\mathbb{Z}}_{2}Z2​ subgroup of a different TQFT with a Z4T{\mathbb{Z}}_{4}^{\mathbf{T}}Z4T​ symmetry. We further show that the anomaly can be resolved in several distinct ways: (1) the true symmetry of the theory is Z4T{\mathbb{Z}}_{4}^{\mathbf{T}}Z4T​, or (2) the theory can be considered to be a theory of fermions, with T2=(\ensuremath−1)Nf{\mathbf{T}}^{2}={(\ensuremath{-}1)}^{{N}_{f}}T2=(\ensuremath−1)Nf​ corresponding to fermion parity. Finally, we demonstrate that theories with the Z2T{\mathbb{Z}}_{2}^{\mathbf{T}}Z2T​ localization anomaly can be compatible with Z2T{\mathbb{Z}}_{2}^{\mathbf{T}}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{\mathbb{Z}}_{2}Z2​ gauge theory and thus only a subset of the quasiparticles are truly confined to the surface.
Certain patterns of symmetry fractionalization in topologically ordered phases of matter are anomalous, in the sense that they can only occur at the surface of a higher dimensional symmetry-protected topological (SPT) state. An important question is to determine how to compute this anomaly, which means determining which SPT hosts a given symmetry-enriched topological order at its surface. While special cases are known, a general method to compute the anomaly has so far been lacking. In this paper we propose a general method to compute relative anomalies between different symmetry fractionalization classes of a given (2+1)D topological order. This method applies to all types of symmetry actions, including anyon-permuting symmetries and general space-time reflection symmetries. We demonstrate compatibility of the relative anomaly formula with previous results for diagnosing anomalies for \mathbb{Z}_2^{T} ℤ 2 T space-time reflection symmetry (e.g. where time-reversal squares to the identity) and mixed anomalies for U(1) \times \mathbb{Z}_2^{T} U ( 1 ) × ℤ 2 T and U(1) \rtimes \mathbb{Z}_2^{T} U ( 1 ) ⋊ ℤ 2 T symmetries. We also study a number of additional examples, including cases where space-time reflection symmetries are intertwined in non-trivial ways with unitary symmetries, such as \mathbb{Z}_4^{T} ℤ 4 T and mixed anomalies for \mathbb{Z}_2 \times \mathbb{Z}_2^{T} ℤ 2 × ℤ 2 T symmetry, and unitary \mathbb{Z}_2 \times \mathbb{Z}_2 ℤ 2 × ℤ 2 symmetry with non-trivial anyon permutations.
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.
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{\mathbb{Z}}_{N}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.
Abstract We study time-reversal symmetry in (2+1)(2+1)(2+1)D abelian bosonic topological phases. Time-reversal anomalies in such systems are classified by Z2×Z2\mathbb {Z}_2 \times \mathbb {Z}_2Z2​×Z2​ symmetry-protected topological (SPT) phases in (3+1)(3+1)(3+1)D, and can be diagnosed via partition functions on manifolds such as RP4\mathbb {RP}^4RP4 and CP2\mathbb {CP}^2CP2. These partition functions are related by the anomaly formula Z(RP4) Z(CP2)=θMZ(\mathbb {RP}^4)\, Z(\mathbb {CP}^2) = \theta _{\mathcal {M}}Z(RP4)Z(CP2)=θM​, where θM\theta _\mathcal {M}θM​ is the Dehn twist phase associated with the crosscap state. Meanwhile, the existence of gapped boundaries is constrained by so-called higher central charges ξn\xi _nξn​, which serve as computable invariants encoding obstruction data. Motivated by the known relation Z(CP2)=ξ1Z(\mathbb {CP}^2) = \xi _1Z(CP2)=ξ1​, we propose a generalization of the anomaly formula that involves both the higher central charges ξn\xi _nξn​ and a new time-reversal invariant ηn\eta _nηn​. Introducing a distinguished subset Mn⊂A\mathcal {M}^n \subset \mathcal {A}Mn⊂A of anyons, we establish the relation ηn⋅ξn=∑a∈Mnθ(a)n∣∑a∈Mnθ(a)n∣,\eta _n \cdot \xi _n = \frac{\sum _{a \in \mathcal {M}^n} \theta (a)^n}{\left| \sum _{a \in \mathcal {M}^n} \theta (a)^n \right|},ηn​⋅ξn​=∣∑a∈Mn​θ(a)n∣∑a∈Mn​θ(a)n​,which generalizes the known anomaly formula. We analyze the algebraic structure of Mn\mathcal {M}^nMn, derive consistency relations it satisfies, and clarify its connection to the original anomaly formula.
We analyze 2+1d2+1d2+1d and 3+1d3+1d3+1d bosonic symmetry protected topological (SPT) phases of matter protected by onsite symmetry group GGG by using dual bulk and boundary approaches. In the bulk, we study an effective field theory, which upon coupling to a background flat GGG gauge field furnishes a purely topological response theory. The response action evaluated on certain manifolds, with appropriate choice of background gauge field, defines a set of SPT topological invariants. Further, SPTs can be gauged by summing over all isomorphism classes of flat GGG gauge fields to obtain Dijkgraaf-Witten topological GGG gauge theories. These topological gauge theories can be ungauged by first introducing and then proliferating defects that spoil the gauge symmetry. This mechanism is related to anyon condensation in 2+1d2+1d2+1d and condensing bosonic gauge charges in 3+1d3+1d3+1d. In the dual boundary approach, we study 1+1d1+1d1+1d and 2+1d2+1d2+1d quantum field theories that have GGG 't-Hooft anomalies that can be precisely canceled by (the response theory of) the corresponding bulk SPT. We show how to construct/compute topological invariants for the bulk SPTs directly from the boundary theories. Further, we sum over boundary partition functions with different background gauge fields to construct GGG characters that generate topological data for the bulk topological gauge theory. Finally, we study a 2+1d2+1d2+1d quantum field theory with a mixed ${\mathbb{Z}}_{2}^{T/R}\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ anomaly where Z2T/R{\mathbb{Z}}_{2}^{T/R}Z2T/R​ is time-reversal/reflection symmetry, and the U(1)\text{U}(1)U(1) could be a 0-form or 1-form symmetry depending on the choice of time reversal/reflection action. We briefly discuss the bulk effective action and topological response for a theory in 3+1d3+1d3+1d that cancels this anomaly. This signals the existence of SPTs in 3+1d3+1d3+1d protected by 0,1-form $\text{U}(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{T,R}$.
We discuss an isomorphism between the possible anomalies of (d+1d+1d+1)-dimensional quantum field theories with Z2{\mathbb{Z}}_{2}Z2​ unitary global symmetry, and those of ddd-dimensional quantum field theories with time-reversal symmetry T\mathsf{T}T. This correspondence is an instance of symmetry defect decoration. The worldvolume of a Z2{\mathbb{Z}}_{2}Z2​ symmetry defect is naturally invariant under T\mathsf{T}T, and bulk Z2{\mathbb{Z}}_{2}Z2​ anomalies descend to T\mathsf{T}T anomalies on these defects. We illustrate this correspondence in detail for (1+1)d(1+1)d(1+1)d bosonic systems where the bulk Z2{\mathbb{Z}}_{2}Z2​ anomaly leads to a Kramers degeneracy in the symmetry defect Hilbert space and exhibits examples. We also discuss (1+1)d(1+1)d(1+1)d fermion systems protected by Z2{\mathbb{Z}}_{2}Z2​ global symmetry where interactions lead to a Z8{\mathbb{Z}}_{8}Z8​ classification of anomalies. Under the correspondence, this is directly related to the Z8{\mathbb{Z}}_{8}Z8​ classification of (0+1)d(0+1)d(0+1)d fermions protected by T\mathsf{T}T. Finally, we consider (3+1)d(3+1)d(3+1)d bosonic systems with Z2{\mathbb{Z}}_{2}Z2​ symmetry where the possible anomalies are classified by ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$. We construct topological field theories realizing these anomalies and show that their associated symmetry defects support anyons that can be either fermions or Kramers doublets.