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arXiv:2210.02452 · Physical review. B./Physical review. B

Nonperturbative constraints from symmetry and chirality on Majorana zero modes and defect quantum numbers in (2+1) dimensions

Naren Manjunath, Vladimir Calvera, Maissam Barkeshli

Published Apr 13, 2023• 20 citations
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Abstract

In (1+1)-dimensional topological phases, unpaired Majorana zero modes (MZMs) can arise only if the internal symmetry group ${G}_{f}$ of the ground state splits as ${G}_{f}={G}_{b}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{f}$, where ${\mathbb{Z}}_{2}^{f}$ 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 ${G}_{f}$ is not of the form ${G}_{b}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{f}$. In this paper we study how ${G}_{f}$ 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 ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$, 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.

DOI
10.1103/physrevb.107.165126

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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 Gf=Gb\ifmmode×\else\texttimes\fiZ2f{G}_{f}={G}_{b}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{f}Gf​=Gb​\ifmmode×\else\texttimes\fiZ2f​, where Z2f{\mathbb{Z}}_{2}^{f}Z2f​ is generated by fermion parity (\ensuremath−1)F{(\ensuremath{-}1)}^{F}(\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 Gb\ifmmode×\else\texttimes\fiZ2f{G}_{b}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{f}Gb​\ifmmode×\else\texttimes\fiZ2f​. In this paper we study how Gf{G}_{f}Gf​ together with the chiral central charge c\ensuremath−{c}_{\ensuremath{-}}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 U(1)f\ensuremath⋊H,SU(2)f\ifmmode×\else\texttimes\fiH,U(2)f\ensuremath⋊H\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}HU(1)f\ensuremath⋊H,SU(2)f\ifmmode×\else\texttimes\fiH,U(2)f\ensuremath⋊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.
In a two-dimensional topological superconductor hosting vortices with Majorana zero modes (MZMs), the exchange of vortices yields a projective representation of the braid group. We investigate whether the full anyonic exchange matrix is uniquely determined by the MZM parity operators and whether consistency with the hexagon equation of the Ising modular tensor category necessitates chiral edge modes. We show that the MZM algebra γiγj+γjγi=2δij\gamma_i\gamma_j + \gamma_j\gamma_i = 2\delta_{ij}γi​γj​+γj​γi​=2δij​ fixes the non-Abelian structure of the exchange operator up to an overall Abelian phase. Completing the modular data and imposing the Ising hexagon equation requires a topological spin θσ=eiπ/8\theta_\sigma = e^{i\pi/8}θσ​=eiπ/8 for the vortices, which via the Gauss-Milgram sum implies a chiral central charge c=1/2(mod8)c = 1/2 \pmod{8}c=1/2(mod8). Non-chiral systems with c=0c=0c=0 yield incompatible topological spins, demonstrating that the MZM parity operators alone do not uniquely fix the full anyonic exchange matrix. We discuss implications for extrinsic defects and anomaly inflow. ## 1. Introduction Non-Abelian anyons in two-dimensional topological phases offer a route to topologically protected quantum computation. In a class of topological superconductors—exemplified by the chiral px+ipyp_x + ip_ypx​+ipy​ state—vortices trap Majorana zero modes (MZMs) whose algebraic exchange realizes a projective representation of the braid group (Ivanov, 2001). The resulting anyon theory is the Ising theory, with anyon types {1,σ,ψ}\{1, \sigma, \psi\}{1,σ,ψ}, where σ\sigmaσ labels the vortex and ψ\psiψ labels the fermion. A central question is whether the MZM parity operators—the only local algebraic data available from the zero-mode subspace—uniquely determine the full anyonic exchange matrix, including all phases. Moreover, the consistency of braiding with the hexagon equation, a defining axiom of modular tensor categories (MTCs), imposes constraints relating the topological spin to the chiral central charge. This raises the question of whether non-chira
We develop a systematic theory of symmetry fractionalization for fermionic topological phases of matter in (2+1)D(2+1)\mathrm{D}(2+1)D with a general fermionic symmetry group Gf{G}_{f}Gf​. In general, Gf{G}_{f}Gf​ is a central extension of the bosonic symmetry group Gb{G}_{b}Gb​ by fermion parity, ${(\ensuremath{-}1)}^{F}$, characterized by a nontrivial cohomology class $[{\ensuremath{\omega}}_{2}]\ensuremath{\in}{\mathcal{H}}^{2}({G}_{b},{\mathbb{Z}}_{2})$. We show how the presence of local fermions places a number of constraints on the algebraic data that defines the action of the symmetry on the supermodular tensor category that characterizes the anyon content. We find two separate obstructions to defining symmetry fractionalization, which we refer to as the bosonic and fermionic symmetry localization obstructions. The former is valued in H3(Gb,K(C)){\mathcal{H}}^{3}({G}_{b},K(\mathcal{C}))H3(Gb​,K(C)), while the latter is valued in either ${\mathcal{H}}^{3}({G}_{b},\mathcal{A}/{1,\ensuremath{\psi}})$ or Z2(Gb,Z2){Z}^{2}({G}_{b},{\mathbb{Z}}_{2})Z2(Gb​,Z2​) depending on additional details of the theory. K(C)K(\mathcal{C})K(C) is the Abelian group of functions from anyons to U(1)\mathrm{U}(1)U(1) phases obeying the fusion rules, A\mathcal{A}A is the Abelian group defined by fusion of Abelian anyons, and $\ensuremath{\psi}$ is the fermion. When these obstructions vanish, we show that distinct symmetry fractionalization patterns form a torsor over ${\mathcal{H}}^{2}({G}_{b},\mathcal{A}/{1,\ensuremath{\psi}})$. We study a number of examples in detail; in particular, we provide a characterization of fermionic Kramers degeneracy arising in symmetry class DIII within this general framework, and we discuss fractional quantum Hall and Z2{\mathbb{Z}}_{2}Z2​ quantum spin liquid states of electrons.
Majorana zero modes (MZMs) localized on vortices of a two-dimensional topological superconductor carry a projective representation of the braid group, and the associated anyon theory is encoded in a modular tensor category whose fusion rules and modular data (SSS and TTT matrices) determine the topological order. We address two questions: whether the braid group representation on NNN anyons uniquely determines the modular data of the underlying non-Abelian topological order, and whether a representation-theoretic framework can classify all possible fusion rules for MZMs in 2D topological superconductors. We develop a classification pipeline that (i) fixes the local MZM algebra as the Clifford algebra generated by 2n2n2n Majorana operators, (ii) imposes the pentagon and hexagon coherence equations on candidate braided fusion structures, and (iii) filters candidates by modular-data consistency conditions, including the Verlinde formula and Frobenius–Schur indicators. As a fully worked example we compute, with explicit arithmetic, the modular SSS matrix and total quantum dimension for the Ising-type theory relevant to chiral ppp-wave superconductors, obtaining D=2\mathcal{D} = 2D=2 and verifying unitarity of SSS to machine precision. We then analyze metaplectic categories SO(N)2SO(N)_2SO(N)2​, which generalize the Ising/MZM fusion pattern, and show how Property F (finiteness of braid images) constrains the classification. We argue that the framework classifies MZM fusion rules up to gauging and extension operations, but that non-semisimple and non-unitary candidates escape the semisimple machinery; we state falsification criteria and open problems. ## 1. Introduction A two-dimensional topological superconductor (2D TSC) supports vortices that bind Majorana zero modes (MZMs): self-adjoint fermionic operators γi=γi†\gamma_i = \gamma_i^\daggerγi​=γi†​ obeying the Clifford algebra {γi,γj}=2δij\{\gamma_i,\gamma_j\} = 2\delta_{ij}{γi​,γj​}=2δij​. Exchanging vortices iii and jjj acts on the ground-state manifold by the operato Full text and updates: papers.qnfo.org/papers/braid-group-representations-modular-data-and-the-classification-of-majo-5099abb8/
We investigate whether the braid group representation on NNN anyons uniquely determines the modular data (SSS and TTT matrices) of a non-Abelian topological order, and whether this framework classifies all fusion rules for Majorana zero modes (MZMs) in 2D topological superconductors. We show that the local NNN-anyon braid group representation constrains the modular data up to finite gauge ambiguity but does not uniquely fix the global SSS-matrix without specifying the underlying spin structure. For MZMs, the braid generators satisfy the Ising anyon RRR-matrix with the asymptotic Hecke relation σi2=1\sigma_i^2 = 1σi2​=1 (up to phase), which rigidly fixes the local FFF and RRR symbols of the Ising category. Consequently, the fusion rules are strictly of Ising type (σ×σ=1+ψ\sigma \times \sigma = 1 + \psiσ×σ=1+ψ), precluding alternative non-Abelian fusion rules. The TTT-matrix is determined by topological spins, but the full SSS-matrix depends on the spin structure of the underlying manifold, reflecting the super-modular structure of the theory. ## 1. Introduction Topological quantum computation exploits the non-Abelian statistics of anyons—quasiparticle excitations in two-dimensional systems whose braiding operations are represented by non-commuting unitary matrices on a degenerate ground-state manifold [1]. A central question is whether the local braiding data of NNN anyons suffices to reconstruct the full modular data of the underlying topological order (TO), or whether additional global information is required. This question has direct physical consequences for 2D topological superconductors (TSCs) hosting Majorana zero modes (MZMs), where the anyonic excitations are vortices binding MZMs and the fermionic quasiparticle is a transparent (local) fermion [2,3]. The modular data of a TO consists of the SSS-matrix, which encodes the mutual braiding statistics of anyons, and the TTT-matrix, which encodes the topological spins. Together, these matrices form a projective representation of t
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 examine the interplay of symmetry and topological order in 2+12+12+1 dimensional fermionic topological phases of matter. We define fermionic topological symmetries acting on the emergent topological effective theory described using braided tensor category theory. Connecting this to the Gf{\cal G}^{\rm f}Gf fermionic symmetry of the microscopic physical system, we characterize and classify symmetry fractionalization in fermionic topological phases. We find that the physical fermion provides constraints that result in a tiered structure of obstructions and classification of fractionalization with respect to the physical fermions, the quasiparticles, and the vortices. The fractionalization of the (bosonic) symmetry G=Gf/Z2fG= {\cal G}^{\rm f}/\mathbb{Z}_2^{\rm f}G=Gf/Z2f​ on the physical fermions is essentially the central extension of GGG by the Z2f\mathbb{Z}_2^{\rm f}Z2f​ fermion parity conservation that yields the fermionic symmetry Gf{\cal G}^{\rm f}Gf. We develop an algebraic theory of Gf{\cal G}^{\rm f}Gf symmetry defects for fermionic topological phases using GGG-crossed braided tensor category theory. This formalism allows us to fully characterize and classify 2+12+12+1 dimensional fermionic symmetry enriched topological phases with on-site unitary fermionic symmetry group Gf{\cal G}^{\rm f}Gf. We first apply this formalism to extract the minimal data specifying a general fermionic symmetry protected topological phase, and demonstrate that such phases with fixed Gf{\cal G}^{\rm f}Gf form a group under fermionic stacking. Then we analyze general fermionic symmetry enriched topological phases and find their classification is given torsorially by the classification of the symmetry fractionalization of quasiparticles combined with the classification of fermionic symmetry protected topological phases. We illustrate our results by detailing a number of examples, including all the invertible fermionic topological phases.