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arXiv:1610.07010 · Progress of Theoretical and Experimental Physics

On time-reversal anomaly of 2+1d topological phases

Yuji Tachikawa, Kazuya Yonekura

Published Jan 30, 2017• 64 citations
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Abstract

We describe a method to find the anomaly of the time-reversal symmetry of 2+1d topological quantum field theories, by computing the fractional anomalous momentum on the cross-cap background. This allows us, for example, to identify the parameter $ν$ mod 16 of the bulk 3+1d topological superconductor with $\mathsf{T}^2=(-1)^F$ on whose boundary a given 2+1d time-reversal-invariant topological phase can appear.

DOI
10.1093/ptep/ptx010

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We describe a method to find the anomaly of the time-reversal symmetry of 2+1d topological quantum field theories, by computing the fractional anomalous momentum on the cross-cap background. This allows us, for example, to identify the parameter ννν mod 16 of the bulk 3+1d topological superconductor with T2=(−1)F\mathsf{T}^2=(-1)^FT2=(−1)F on whose boundary a given 2+1d time-reversal-invariant topological phase can appear.
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 ${\mathbf{T}}^{2}={(\ensuremath{-}1)}^{{N}_{f}}$ 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.
In this short paper, we argue that the chiral central charge c -of a (2 + 1)d topological ordered state is sometimes strongly constrained by 't Hooft anomaly of antiunitary global symmetry.For example, if a (2 + 1)d fermionic topological quantum field theory (TQFT) has a time-reversal anomaly with T 2 = (-1) F labeled as ν ∈ Z 16 , the TQFT must have c -= 1/4 mod 1/2 for odd ν, while c -= 0 mod 1/2 for even ν.This generalizes the fact that the bosonic TQFT with T anomaly in a particular class must carry c -= 4 mod 8 to fermionic cases.We also study such a constraint for fermionic TQFT with U (1) × CT symmetry, which is regarded as a gapped surface of the topological superconductor in class AIII.
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.
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 $\text{U}(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{T,R}$.