In addition to possessing fractional statistics, anyon excitations of a 2D topologically ordered state can realize symmetry in distinct ways, leading to a variety of symmetry-enriched topological (SET) phases.While the symmetry fractionalization must be consistent with the fusion and braiding rules of the anyons, not all ostensibly consistent symmetry fractionalizations can be realized in 2D systems.Instead, certain "anomalous" SETs can only occur on the surface of a 3D symmetry-protected topological (SPT) phase.In this paper, we describe a procedure for determining whether a SET of a discrete, on-site, unitary symmetry group G is anomalous or not.The basic idea is to gauge the symmetry and expose the anomaly as an obstruction to a consistent topological theory combining both the original anyons and the gauge fluxes.Utilizing a result of Etingof, Nikshych, and Ostrik, we point out that a class of obstructions is captured by the fourth cohomology group H 4 ðG; Uð1ÞÞ, which also precisely labels the set of 3D SPT phases, with symmetry group G.An explicit procedure for calculating the cohomology data from a SET is given, with the corresponding physical intuition explained.We thus establish a general bulk-boundary correspondence between the anomalous SET and the 3D bulk SPT whose surface termination realizes it.We illustrate this idea using the chiral spin liquid [Uð1Þ 2 ] topological order with a reduced symmetry Z 2 × Z 2 ⊂ SOð3Þ, which can act on the semion quasiparticle in an anomalous way.We construct exactly solved 3D SPT models realizing the anomalous surface terminations and demonstrate that they are nontrivial by computing three-loop braiding statistics.Possible extensions to antiunitary symmetries are also discussed.In addition to possessing fractional statistics, anyon excitations of a 2D topologically ordered state can realize symmetry in distinct ways, leading to a variety of symmetry-enriched topological (SET) phases.While the symmetry fractionalization must be consistent with the fusion and braiding rules of the anyons, not all ostensibly consistent symmetry fractionalizations can be realized in 2D systems.Instead, certain "anomalous" SETs can only occur on the surface of a 3D symmetry-protected topological (SPT) phase.In this paper, we describe a procedure for determining whether a SET of a discrete, on-site, unitary symmetry group G is anomalous or not.The basic idea is to gauge the symmetry and expose the anomaly as an obstruction to a consistent topological theory combining both the original anyons and the gauge fluxes.Utilizing a result of Etingof, Nikshych, and Ostrik, we point out that a class of obstructions is captured by the fourth cohomology group H 4 ðG; Uð1ÞÞ, which also precisely labels the set of 3D SPT phases, with symmetry group G.An explicit procedure for calculating the cohomology data from a SET is given, with the corresponding physical intuition explained.We thus establish a general bulk-boundary correspondence between the anomalous SET and the 3D bulk SPT whose surface termination realizes it.We illustrate this idea using the chiral spin liquid [Uð1Þ 2 ] topological order with a reduced symmetry Z 2 × Z 2 ⊂ SOð3Þ, which can act on the semion quasiparticle in an anomalous way.We construct exactly solved 3D SPT models realizing the anomalous surface terminations and demonstrate that they are nontrivial by computing three-loop braiding statistics.Possible extensions to antiunitary symmetries are also discussed.

arXiv:1403.6491Physical Review XOct 23, 2015144 citations

Bulk-boundary correspondence of (1+1)D symmetric gapped phases

Yizhou Ma, Gen Yue, Tian Lan

We develop an operator-algebraic framework for boundary conditions and bulk-boundary correspondence in one-dimensional gapped phases with categorical symmetry. Working directly in the thermodynamic limit, we construct half-infinite fusion spin chains and commuting-projector boundary Hamiltonians from a unitary fusion category $\mathcal{C}$ , an indecomposable semisimple right $\mathcal{C}$ -module category $\mathcal{M}$ , a Q-system $Q\in\mathcal{C}$ specifying the bulk phase, and a right $Q$ -module $K\in\mathcal{M}_{Q}$ , regarded as an object of $\mathcal{M}_{Q}^{\mathrm{op}}$ , specifying the boundary. We prove that these Hamiltonians have unique ground states and that the resulting realization functor $\mathcal{M}_{Q}^{\mathrm{op}}\to\mathrm{BCond}$ is an equivalence, so simple boundary conditions are classified by simple objects of $\mathcal{M}_{Q}$ and general boundary conditions by their finite direct sums. We also give a microscopic formulation of the boundary symmetry topological field theory using DHR bimodules of the boundary quasi-local algebra. For a half-infinite fusion spin chain, the boundary DHR category is monoidally equivalent to $(\mathcal{C}_{\mathcal{M}}^{\vee})^{\mathrm{rev}}$ , and the canonical action of the bulk DHR category on it agrees with the categorical action of $Z_1(\mathcal{C}^{\mathrm{rev}})$ . Finally, we identify the action of the boundary DHR category on boundary conditions with the categorical action of $(\mathcal{C}_{\mathcal{M}}^{\vee})^{\mathrm{rev}}$ on $\mathcal{M}_{Q}^{\mathrm{op}}$ . This yields a one-dimensional bulk-boundary correspondence: the enriched monoidal category describing the bulk is the enriched center of the enriched category describing the boundary.We develop an operator-algebraic framework for boundary conditions and bulk-boundary correspondence in one-dimensional gapped phases with categorical symmetry. Working directly in the thermodynamic limit, we construct half-infinite fusion spin chains and commuting-projector boundary Hamiltonians from a unitary fusion category $\mathcal{C}$, an indecomposable semisimple right $\mathcal{C}$-module category $\mathcal{M}$, a Q-system $Q\in\mathcal{C}$ specifying the bulk phase, and a right $Q$-module $K\in\mathcal{M}_{Q}$, regarded as an object of $\mathcal{M}_{Q}^{\mathrm{op}}$, specifying the boundary. We prove that these Hamiltonians have unique ground states and that the resulting realization functor $\mathcal{M}_{Q}^{\mathrm{op}}\to\mathrm{BCond}$ is an equivalence, so simple boundary conditions are classified by simple objects of $\mathcal{M}_{Q}$ and general boundary conditions by their finite direct sums. We also give a microscopic formulation of the boundary symmetry topological field theory using DHR bimodules of the boundary quasi-local algebra. For a half-infinite fusion spin chain, the boundary DHR category is monoidally equivalent to $(\mathcal{C}_{\mathcal{M}}^{\vee})^{\mathrm{rev}}$, and the canonical action of the bulk DHR category on it agrees with the categorical action of $Z_1(\mathcal{C}^{\mathrm{rev}})$. Finally, we identify the action of the boundary DHR category on boundary conditions with the categorical action of $(\mathcal{C}_{\mathcal{M}}^{\vee})^{\mathrm{rev}}$ on $\mathcal{M}_{Q}^{\mathrm{op}}$. This yields a one-dimensional bulk-boundary correspondence: the enriched monoidal category describing the bulk is the enriched center of the enriched category describing the boundary.

arXiv:2606.19137Jun 17, 20260 citations

Boundary Physics and Bulk-Boundary Correspondence in Topological Phases of Matter

Abhijeet Alase

Springer thesesJan 1, 201911 citations

\mathcal{C}

, an indecomposable semisimple right

\mathcal{C}

-module category

\mathcal{M}

, a Q-system

Q\in\mathcal{C}

specifying the bulk phase, and a right

Q

-module

K\in\mathcal{M}_{Q}

, regarded as an object of

\mathcal{M}_{Q}^{\mathrm{op}}

, specifying the boundary. We prove that these Hamiltonians have unique ground states and that the resulting realization functor

\mathcal{M}_{Q}^{\mathrm{op}}\to\mathrm{BCond}

is an equivalence, so simple boundary conditions are classified by simple objects of

\mathcal{M}_{Q}

and general boundary conditions by their finite direct sums. We also give a microscopic formulation of the boundary symmetry topological field theory using DHR bimodules of the boundary quasi-local algebra. For a half-infinite fusion spin chain, the boundary DHR category is monoidally equivalent to

(\mathcal{C}_{\mathcal{M}}^{\vee})^{\mathrm{rev}}

, and the canonical action of the bulk DHR category on it agrees with the categorical action of

Z_1(\mathcal{C}^{\mathrm{rev}})

. Finally, we identify the action of the boundary DHR category on boundary conditions with the categorical action of

(\mathcal{C}_{\mathcal{M}}^{\vee})^{\mathrm{rev}}

\mathcal{M}_{Q}^{\mathrm{op}}

. This yields a one-dimensional bulk-boundary correspondence: the enriched monoidal category describing the bulk is the enriched center of the enriched category describing the boundary.