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arXiv:2101.01018 · Physical Review Research

Anomaly constraint on chiral central charge of (2+1)d topological order

Ryohei Kobayashi

Published May 10, 2021• 5 citations
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Abstract

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.

DOI
10.1103/physrevresearch.3.023107

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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.
In this short paper, we argue that the chiral central charge c−c_-c−​ of a (2+1)d topological ordered state is sometimes strongly constrained by 't Hooft anomaly of anti-unitary global symmetry. For example, if a (2+1)d fermionic TQFT has a time reversal anomaly with T2=(−1)FT^2=(-1)^FT2=(−1)F labeled as ν∈Z16\nu\in\mathbb{Z}_{16}ν∈Z16​, the TQFT must have c−=1/4c_-=1/4c−​=1/4 mod 1/21/21/2 for odd ν\nuν, while c−=0c_-=0c−​=0 mod 1/21/21/2 for even ν\nuν. This generalizes the fact that the bosonic TQFT with TTT anomaly in a particular class must carry c−=4c_-=4c−​=4 mod 888 to fermionic cases. We also study such a constraint for fermionic TQFT with U(1)×CTU(1)\times CTU(1)×CT symmetry, which is regarded as a gapped surface of the topological superconductor in class AIII.
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 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 (1+1)-dimensional topological phases, unpaired Majorana zero modes (MZMs) can arise only if the internal symmetry group Gf{G}_{f}Gf​ of the ground state splits as ${G}_{f}={G}_{b}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{f}$, where Z2f{\mathbb{Z}}_{2}^{f}Z2f​ is generated by fermion parity ${(\ensuremath{-}1)}^{F}$. In contrast, (2+1)-dimensional [(2+1)D] topological superconductors (TSC) can host unpaired MZMs at defects even when Gf{G}_{f}Gf​ is not of the form ${G}_{b}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{f}$. In this paper we study how Gf{G}_{f}Gf​ together with the chiral central charge ${c}_{\ensuremath{-}}$ strongly constrain the existence of unpaired MZMs and the quantum numbers of symmetry defects. Our results utilize a recent algebraic characterization of (2+1)D invertible fermionic topological states, which provides a nonperturbative approach based on topological quantum field theory, beyond free fermions. We study physically relevant groups such as $\mathrm{U}{(1)}^{f}\ensuremath{\rtimes}H,\phantom{\rule{0.28em}{0ex}}\mathrm{SU}{(2)}^{f}\ifmmode\times\else\texttimes\fi{}H,\phantom{\rule{0.28em}{0ex}}\mathrm{U}{(2)}^{f}\ensuremath{\rtimes}H$, generic Abelian groups, as well as more general compact Lie groups, antiunitary symmetries, and crystalline symmetries. We present an algebraic formula for the fermionic crystalline equivalence principle, which gives an equivalence between states with crystalline and internal symmetries. In light of our theory, we discuss several previously proposed realizations of unpaired MZMs in TSC materials such as Sr2RuO4{\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}Sr2​RuO4​, transition metal dichalcogenides, and iron superconductors, in which crystalline symmetries are often important; in some cases we present additional predictions for the properties of these models.
By definition, the physics of the ddd-dimensional (dim) boundary of a (d+1)(d+1)(d+1)-dim symmetry protected topological (SPT) state cannot be realized as itself on a ddd-dim lattice. If the symmetry of the system is unitary, then a formal way to determine whether a ddd-dim theory must be a boundary or not, is to couple this theory to a gauge field (or to ``gauge'' its symmetry), and check if there is a gauge anomaly. In this paper we discuss the following question: Can the boundary of a SPT state be driven into a fully gapped topological order which preserves all the symmetries? We argue (conjecture) that if the gauge anomaly of the boundary is ``perturbative,'' then the boundary must remain gapless; while if the boundary only has global gauge anomaly but no perturbative anomaly, then it is possible to gap out the boundary by driving it into a topological state, when $d\ensuremath{\ge}2$. We will demonstrate this conjecture with two examples: (1) the 3d3d3d spin-1/2 chiral fermion with the well-known Witten's global anomaly [Phys. Lett. 117, 324 (1982)], which can be realized on the boundary of a 4d4d4d topological superconductor with SU(2) or $U(1)\ensuremath{\rtimes}{Z}_{2}$ symmetry; and (2) the 4d4d4d boundary of a 5d5d5d topological superconductor with the same symmetry. We show that these boundary systems can be driven into a fully gapped Z2N{\mathbb{Z}}_{2N}Z2N​ topological order with topological degeneracy, but this Z2N{\mathbb{Z}}_{2N}Z2N​ topological order cannot be future driven into a trivial confined phase that preserves all the symmetries due to some special properties of its topological defects. Our study also leads to exotic states of matter in pure 3d3d3d space.
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.