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arXiv:2109.15307 · Physical Review Research

Enforced symmetry breaking by invertible topological order

Shang-Qiang Ning, Yang Qi, Zheng‐Cheng Gu, Chenjie Wang

Published Jun 6, 2023• 7 citations
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Abstract

It is well known that two-dimensional fermionic systems with a nonzero Chern number must break the time-reversal symmetry, manifested by the appearance of chiral edge modes on an open boundary. Such an incompatibility between topology and symmetry can occur more generally. We will refer to this phenomenon as enforced symmetry breaking (ESB) by topological orders. In this work, we systematically study ESB of a finite symmetry group ${G}_{f}$ by fermionic invertible topological orders. Mathematically, the group ${G}_{f}$ is a central extension over a bosonic symmetry group $G$ by the fermion parity group ${\mathbb{Z}}_{2}^{f}$, characterized by a 2-cocycle $\ensuremath{\lambda}\ensuremath{\in}{\mathcal{H}}^{2}(G,{\mathbb{Z}}_{2})$. For given $G$ and $\ensuremath{\lambda}$, we are able to obtain a series of criteria on the existence or nonexistence of ESB by the corresponding fermionic invertible topological orders. For 2D systems, we define a set of physical quantities to describe symmetry-enriched invertible topological orders and derive obstruction functions using both fermionic and bosonic languages. The study in the bosonic language is performed after gauging the fermion parity, and we find that some obstruction functions are consequences of conditional anomalies of the bosonic symmetry-enriched topological states, with the conditions inherited from the original fermionic system. The obstruction functions are crucial components of the ESB criteria that we derive. With these criteria, we discover many new ESB examples, e.g., we find that the quaternion group ${Q}_{8}$ is incompatible with two copies of $p+ip$ superconductors. We also obtain explicit results on the ESB phenomena of the continuous group ${\mathrm{SU}}_{f}(N)$ by 2D invertible topological orders through a different argument.

DOI
10.1103/physrevresearch.5.023153

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It is well known that two-dimensional fermionic systems with a nonzero Chern number must break the time-reversal symmetry, manifested by the appearance of chiral edge modes on an open boundary. Such an incompatibility between topology and symmetry can occur more generally. We will refer to this phenomenon as enforced symmetry breaking (ESB) by topological orders. In this work, we systematically study ESB of a finite symmetry group Gf{G}_{f}Gf​ by fermionic invertible topological orders. Mathematically, the group Gf{G}_{f}Gf​ is a central extension over a bosonic symmetry group GGG by the fermion parity group Z2f{\mathbb{Z}}_{2}^{f}Z2f​, characterized by a 2-cocycle \ensuremathλ\ensuremath∈H2(G,Z2)\ensuremath{\lambda}\ensuremath{\in}{\mathcal{H}}^{2}(G,{\mathbb{Z}}_{2})\ensuremathλ\ensuremath∈H2(G,Z2​). For given GGG and \ensuremathλ\ensuremath{\lambda}\ensuremathλ, we are able to obtain a series of criteria on the existence or nonexistence of ESB by the corresponding fermionic invertible topological orders. For 2D systems, we define a set of physical quantities to describe symmetry-enriched invertible topological orders and derive obstruction functions using both fermionic and bosonic languages. The study in the bosonic language is performed after gauging the fermion parity, and we find that some obstruction functions are consequences of conditional anomalies of the bosonic symmetry-enriched topological states, with the conditions inherited from the original fermionic system. The obstruction functions are crucial components of the ESB criteria that we derive. With these criteria, we discover many new ESB examples, e.g., we find that the quaternion group Q8{Q}_{8}Q8​ is incompatible with two copies of p+ipp+ipp+ip superconductors. We also obtain explicit results on the ESB phenomena of the continuous group SUf(N){\mathrm{SU}}_{f}(N)SUf​(N) by 2D invertible topological orders through a different argument.
Symmetry can be intrinsically broken in topological phases due to inherent incompatibilities, a phenomenon known as enforced symmetry breaking (ESB) in the framework of topological order. In our previous work, we developed a systematic framework to understand ESB within 2D invertible topological order. Meanwhile, the origin of parity violation in the Standard Model remains one of the most profound mysteries in physics, with no clear explanation to date. In this study, we explore the ESB of parity symmetry by three-dimensional fermionic topological order (fTO), offering potential insights into the origins of parity violation. As the simplest example, here we consider an fTO related to the intrinsic interacting fermionic SPT phase protected by Z2f×Z2×Z8Z_2^f\times Z_2\times Z_8Z2f​×Z2​×Z8​ symmetry in three dimensions. We show that time-reversal symmetry (TRS) with T2=1{T}^2=1T2=1 on physical fermions is incompatible with such fTO; then, through the so-called crystalline equivalence principle, we show that the parity symmetry is also incompatible with it. In comparison, conventional TRS with T2=Pf{T}^2={P}_fT2=Pf​ remains compatible to this fTO. We also discuss a general framework to study the ESB phenomenon for 3D fTO.
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.
Symmetry-protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry GGG, which can all be smoothly connected to the trivial product states if we break the symmetry. It has been shown that a large class of interacting bosonic SPT phases can be systematically described by group cohomology theory. In this paper, we introduce a (special) group supercohomology theory which is a generalization of the standard group cohomology theory. We show that a large class of short-range interacting fermionic SPT phases can be described by the group supercohomology theory. Using the data of supercocycles, we can obtain the ideal ground state wave function for the corresponding fermionic SPT phase. We can also obtain the bulk Hamiltonian that realizes the SPT phase, as well as the anomalous (i.e., non-onsite) symmetry for the boundary effective Hamiltonian. The anomalous symmetry on the boundary implies that the symmetric boundary must be gapless for (1+1)-dimensional [(1+1)D] boundary, and must be gapless or topologically ordered beyond (1+1)D. As an application of this general result, we construct a new SPT phase in three dimensions, for interacting fermionic superconductors with coplanar spin order (which have T2=1{T}^{2}=1T2=1 time-reversal Z2T{Z}_{2}^{T}Z2T​ and fermion-number-parity Z2f{Z}_{2}^{f}Z2f​ symmetries described by a full symmetry group ${Z}_{2}^{T}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{f}$). Such a fermionic SPT state can neither be realized by free fermions nor by interacting bosons (formed by fermion pairs), and thus are not included in the KKK-theory classification for free fermions or group cohomology description for interacting bosons. We also construct three interacting fermionic SPT phases in two dimensions (2D) with a full symmetry group ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{f}$. Those 2D fermionic SPT phases all have central-charge c=1c=1c=1 gapless edge excitations, if the symmetry is not broken.
We develop a theory of anomalies of fermionic topological phases of matter in (2+12+12+1)D with a general fermionic symmetry group Gf{G}_{f}Gf​. In general, Gf{G}_{f}Gf​ can be a nontrivial central extension of the bosonic symmetry group Gb{G}_{b}Gb​ by fermion parity ${(\ensuremath{-}1)}^{F}$. We encounter four layers of obstructions to gauging the Gf{G}_{f}Gf​ symmetry, which we dub the anomaly cascade: (i) An H1(Gb,ZT){\mathcal{H}}^{1}({G}_{b},{\mathbb{Z}}_{\mathbf{T}})H1(Gb​,ZT​) obstruction to extending the symmetry permutations on the anyons to the fermion parity gauged theory, (ii) An H2(Gb,kerr){\mathcal{H}}^{2}({G}_{b},kerr)H2(Gb​,kerr) obstruction to extending the Gb{G}_{b}Gb​ group structure of the symmetry permutations to the fermion parity gauged theory, where rrr is a map that restricts symmetries of the fermion parity gauged theory to the anyon theory, (iii) An H3(Gb,Z2){\mathcal{H}}^{3}({G}_{b},{\mathbb{Z}}_{2})H3(Gb​,Z2​) obstruction to extending the symmetry fractionalization class to the fermion parity gauged theory, and (iv) the well-known H4(Gb,U(1[))]{\mathcal{H}}^{4}({G}_{b},U(1[))]H4(Gb​,U(1[))] obstruction to developing a consistent theory of Gb{G}_{b}Gb​ symmetry defects for the fermion parity gauged theory. We describe how the H2{\mathcal{H}}^{2}H2 obstruction can be canceled by anomaly inflow from a bulk (3+13+13+1)D symmetry-protected topological state (SPT) and also its relation to the Arf invariant of spin structures on a torus. If any anomaly in the above sequence is nontrivial, the subsequent ones become relative anomalies. A number of conjectures regarding symmetry actions on super-modular categories, guided by general expectations of anomalies in physics, are also presented.
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.