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arXiv:1611.01601 · Physical Review Letters

Derivation of the Time-Reversal Anomaly for ( 2+1 )-Dimensional Topological Phases

Yuji Tachikawa, Kazuya Yonekura

Published Sep 13, 2017• 47 citations
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Abstract

We prove an explicit formula conjectured recently by Wang and Levin for the anomaly of time-reversal symmetry in (2+1)-dimensional fermionic topological quantum field theories. The crucial step is to determine the cross-cap state in terms of the modular S matrix and T^{2} eigenvalues, generalizing the recent analysis by Barkeshli et al. in the bosonic case.

DOI
10.1103/physrevlett.119.111603

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We prove an explicit formula conjectured recently by Wang and Levin for the anomaly of time-reversal symmetry in (2+1)-dimensional fermionic topological quantum field theories. The crucial step is to determine the cross-cap state in terms of the modular S matrix and T^{2} eigenvalues, generalizing the recent analysis by Barkeshli et al. in the bosonic case.
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 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.
Abstract We provide a mathematical proposal for the anomaly indicators of symmetries of (2+1)-D fermionic topological orders, and work out the consequences of our proposal in several nontrivial examples. Our proposal is an invariant of a super modular tensor category with a fermionic group action, which gives a (3+1)-D topological field theory (TFT) that we conjecture to be invertible; the anomaly indicators are partition functions of this TFT on 4-manifolds generating the corresponding twisted spin bordism group. Our construction relies on a bosonization construction due to Gaiotto–Kapustin and Tata–Kobayashi–Bulmash–Barkeshli, together with a “bosonization conjecture” which we explain in detail. In the second half of the paper, we discuss several examples of our invariants relevant to condensed-matter physics. The most important example we consider is Z/4T×Z/2f\mathbb {Z}/4^T\times \mathbb {Z}/2^fZ/4T×Z/2f Z / 4 T × Z / 2 f time-reversal symmetry with symmetry algebra T2=(−1)FC{\mathcal {T}}^2 = (-1)^FCT2=(−1)FC T 2 = ( - 1 ) F C , which many fermionic topological orders enjoy, including the U(1)5\textrm{U}(1)_5U(1)5​ U ( 1 ) 5 spin Chern–Simons theory. Using newly developed tools involving the Smith long exact sequence, we calculate the cobordism group that classifies its anomaly, present the generating manifold, and calculate the partition function on the generating manifold which serves as our anomaly indicator. Our approach allows us to reproduce anomaly indicators known in the literature with simpler proofs, including Z/4Tf\mathbb {Z}/4^{Tf}Z/4Tf Z / 4 Tf time-reversal symmetry with symmetry algebra T2=(−1)F\mathcal T^2 = (-1)^FT2=(−1)F T 2 = ( - 1 ) F , and other symmetry groups in the 10-fold way involving Lie group symmetries.
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.
Some time-reversal symmetric topological orders are anomalous in that they cannot be realized in strictly two-dimensional systems; instead, they can only be realized on the surface of three-dimensional symmetry-protected topological phases. We propose two quantities, which we call anomaly indicators, that can detect if a time-reversal symmetric topological order is anomalous in this sense. Both anomaly indicators are expressed in terms of the quantum dimensions, topological spins, and time-reversal properties of the anyons in the given topological order. The first indicator, η_{2}, applies to bosonic systems while the second indicator, η_{f}, applies to fermionic systems in the DIII class. We conjecture that η_{2}, together with a previously known indicator η_{1}, can detect the two known Z_{2} anomalies in the bosonic case, while η_{f} can detect the Z_{16} anomaly in the fermionic case.
We study anomalies in time-reversal-(Z2T{\mathbb{Z}}_{2}^{T}Z2T​) and U(1)U(1)U(1)-symmetric topological orders. In this context, an anomalous topological order is one that cannot be realized in a strictly (2+1)(2+1)(2+1)-dimensional system but can be realized on the surface of a (3+1)(3+1)(3+1)-dimensional [(3+13+13+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(3+1)\mathrm{D}(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(3+1)\mathrm{D}(3+1)D SPT phase on a particular manifold and in the presence of a particular background gauge field for the U(1)U(1)U(1) symmetry.