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A categorical perspective on symmetry, topological order, and quantum information

Colleen Delaney

Published Jan 1, 2019• 1 citation
OpenAlex

Abstract

We investigate the algebraic theory of symmetry-enriched topological (SET) order in (2+1)D bosonic topological phases of matter and its applications to topological quantum computing. Our goal is twofold: first, to demonstrate how an abstract categorical approach can be applied to understand phenomena in (2+1)D topological phases of matter, and second, to show how ideas from physics can be useful for categorification.After reviewing modular tensor categories (MTCs) and their role as algebraic theories of anyons in topological phases of matter, we recall their associated quantum representations and their interpretation as quantum gates for a topological quantum computer. Next we recall the characterization of SET order in terms of $G$-crossed braided extensions of MTCs and the mathematical formalism of topological quantum computing (TQC) with anyons and symmetry defects.We then apply modular tensor category theory to construct algebraic models of symmetry defects in multi-layer (2+1)D topological order with layer-exchange permutation symmetry. Our main result frames a correspondence between bilayer symmetry enriched topological order and monolayer topological order on surfaces with genus, illuminating a connection between quantum symmetry and topological order that first appeared in the work of condensed matter theorists Barkeshi, Ji, and Qian.

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Symmetry defects and their application to topological quantum computing

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We describe the mathematical theory of topological quantum computing with symmetry defects in the language of fusion categories and unitary representations. Symmetry defects together with anyons are modeled by G-crossed braided extensions of unitary modular tensor categories. The algebraic data of these categories afford projective unitary representations of the braid group. Elements in the image of such representations correspond to quantum gates arising from exchanging anyons and symmetry defects in topological phases of matter with symmetry. We provide some small examples that highlight features of practical interest for quantum computing. In particular, symmetry defects show the potential to generate non-abelian statistics from abelian topological phases and to be used as as a tool to enlarge the set of quantum gates accessible to an anyonic device.

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Characterization and Classification of Fermionic Symmetry Enriched Topological Phases

David Aasen, Parsa Bonderson, Christina Knapp

We examine the interplay of symmetry and topological order in $2+1$ 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 ${\cal G}^{\rm f}$ 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 $G= {\cal G}^{\rm f}/\mathbb{Z}_2^{\rm f}$ on the physical fermions is essentially the central extension of $G$ by the $\mathbb{Z}_2^{\rm f}$ fermion parity conservation that yields the fermionic symmetry ${\cal G}^{\rm f}$. We develop an algebraic theory of ${\cal G}^{\rm f}$ symmetry defects for fermionic topological phases using $G$-crossed braided tensor category theory. This formalism allows us to fully characterize and classify $2+1$ dimensional fermionic symmetry enriched topological phases with on-site unitary fermionic symmetry group ${\cal G}^{\rm f}$. 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 ${\cal G}^{\rm f}$ 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.

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Symmetry fractionalization, defects, and gauging of topological phases

Maissam Barkeshli, Parsa Bonderson, Meng Cheng +1 more

This paper develops a comprehensive theory of symmetry fractionalization together with the properties of symmetry defects in topologically ordered phases of matter in two spatial dimensions. To do this, the authors also introduce the full set of data, consistency conditions, and equivalences for a mathematical theory, known as a G-crossed braided tensor category, that characterizes the algebraic braiding and fusion properties of symmetry defects. This theoretical framework can completely characterize and classify symmetry-enriched topological phases of matter in the presence of arbitrarily strong interactions in quantum many-body systems in two spatial dimensions.

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Symmetry-enriched topological order in tensor networks: Defects, gauging and anyon condensation

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We study symmetry-enriched topological order in two-dimensional tensor network states by using graded matrix product operator algebras to represent symmetry-induced domain walls. A close connection to the theory of graded unitary fusion categories is established. Tensor network representations of the topological defect superselection sectors are constructed for all domain walls. The emergent symmetry-enriched topological order is extracted from these representations, including the symmetry action on the underlying anyons. Dual phase transitions, induced by gauging a global symmetry, and condensation of a bosonic subtheory, are analyzed and the relationship between topological orders on either side of the transition is derived. Several examples are worked through explicitly.

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Boundary symmetries of (2+1)D topological orders

Kylan Schatz

Abstract For a G-crossed braided extension of a unitary modular tensor category $$\mathcal {C}$$ C —as in one representing a (2+1)D symmetry enriched topological order (SETO)—preservation of global on-site group symmetry after condensation by a commutative Q-system object $$A \in \mathcal {C}$$ A ∈ C necessitates the existence of a G-equivariant structure on A. When interpreted spatially, the condensation boundary has its own internal topological symmetries. We elaborate an algebraic framework for describing the internal topological symmetries of compatible (1+1)D gapped boundaries for (2+1)D topologically ordered systems in terms of hypergroup actions. Then, we investigate the coherence of global on-site bulk symmetries and boundary symmetries. We present a categorical obstruction to the preservation of symmetry in a way which is coherent in terms of lifts of categorical actions to a certain 2-group of bulk symmetries. We give a characterization of this obstruction in the case of condensation by a Lagrangian algebra and boundary symmetries given by subalgebras of the convolution algebra associated with a Lagrangian algebra object.

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Symmetries in topological tensor network states: classification, construction and detection

José Garre-Rubio

This thesis contributes to the understanding of symmetry-enriched topological phases focusing on their descriptions in terms of tensor network states. The Projected Entangled Pair State (PEPS) formalism allows us to locally encode the main properties of the models (like topological order, symmetries and their interplay) in the tensors. We have used that encoding to classify, construct and detect some classes of symmetry-enriched topological phases in 2D PEPS. For that purpose, we have studied what the allowed freedom in the tensors generating the same tensor network state is. These results are the so-called 'fundamental theorems' and they give the necessary knowledge to properly study symmetries (actions that leave the states invariant). We focus our study on the family of PEPS describing quantum double models of $G$, the so-called $G$-injective PEPS, and on global on-site symmetries coming from a finite group $Q$: as a result of the classification we have a finite number of phases closely related to the theory of group extensions. We also provide the maps that appear in group extension theory with physical meaning and we characterize their actions on the ground subspace and on the anyons. We construct a representative of each class of our classification. Moreover, we connect our construction with an interesting physical phenomenon, the so-called anyon condensation, which describes topological phase transitions. We have also proposed a family of gauge invariant quantities and their corresponding order parameters (in the form of expectation values of a local operator in 2D) to detect the corresponding quantum phases, in particular their symmetry fractionalization patterns. We conclude this thesis by touching on some mathematical open problems in PEPS.

arXiv:1912.08597•Dec 18, 2019•1 citation
We investigate the algebraic theory of symmetry-enriched topological (SET) order in (2+1)D bosonic topological phases of matter and its applications to topological quantum computing. Our goal is twofold: first, to demonstrate how an abstract categorical approach can be applied to understand phenomena in (2+1)D topological phases of matter, and second, to show how ideas from physics can be useful for categorification.After reviewing modular tensor categories (MTCs) and their role as algebraic theories of anyons in topological phases of matter, we recall their associated quantum representations and their interpretation as quantum gates for a topological quantum computer. Next we recall the characterization of SET order in terms of GGG-crossed braided extensions of MTCs and the mathematical formalism of topological quantum computing (TQC) with anyons and symmetry defects.We then apply modular tensor category theory to construct algebraic models of symmetry defects in multi-layer (2+1)D topological order with layer-exchange permutation symmetry. Our main result frames a correspondence between bilayer symmetry enriched topological order and monolayer topological order on surfaces with genus, illuminating a connection between quantum symmetry and topological order that first appeared in the work of condensed matter theorists Barkeshi, Ji, and Qian.
We describe the mathematical theory of topological quantum computing with symmetry defects in the language of fusion categories and unitary representations. Symmetry defects together with anyons are modeled by G-crossed braided extensions of unitary modular tensor categories. The algebraic data of these categories afford projective unitary representations of the braid group. Elements in the image of such representations correspond to quantum gates arising from exchanging anyons and symmetry defects in topological phases of matter with symmetry. We provide some small examples that highlight features of practical interest for quantum computing. In particular, symmetry defects show the potential to generate non-abelian statistics from abelian topological phases and to be used as as a tool to enlarge the set of quantum gates accessible to an anyonic device.
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.
This paper develops a comprehensive theory of symmetry fractionalization together with the properties of symmetry defects in topologically ordered phases of matter in two spatial dimensions. To do this, the authors also introduce the full set of data, consistency conditions, and equivalences for a mathematical theory, known as a G-crossed braided tensor category, that characterizes the algebraic braiding and fusion properties of symmetry defects. This theoretical framework can completely characterize and classify symmetry-enriched topological phases of matter in the presence of arbitrarily strong interactions in quantum many-body systems in two spatial dimensions.
We study symmetry-enriched topological order in two-dimensional tensor network states by using graded matrix product operator algebras to represent symmetry-induced domain walls. A close connection to the theory of graded unitary fusion categories is established. Tensor network representations of the topological defect superselection sectors are constructed for all domain walls. The emergent symmetry-enriched topological order is extracted from these representations, including the symmetry action on the underlying anyons. Dual phase transitions, induced by gauging a global symmetry, and condensation of a bosonic subtheory, are analyzed and the relationship between topological orders on either side of the transition is derived. Several examples are worked through explicitly.
Abstract For a G-crossed braided extension of a unitary modular tensor category C\mathcal {C}C C —as in one representing a (2+1)D symmetry enriched topological order (SETO)—preservation of global on-site group symmetry after condensation by a commutative Q-system object A∈CA \in \mathcal {C}A∈C A ∈ C necessitates the existence of a G-equivariant structure on A. When interpreted spatially, the condensation boundary has its own internal topological symmetries. We elaborate an algebraic framework for describing the internal topological symmetries of compatible (1+1)D gapped boundaries for (2+1)D topologically ordered systems in terms of hypergroup actions. Then, we investigate the coherence of global on-site bulk symmetries and boundary symmetries. We present a categorical obstruction to the preservation of symmetry in a way which is coherent in terms of lifts of categorical actions to a certain 2-group of bulk symmetries. We give a characterization of this obstruction in the case of condensation by a Lagrangian algebra and boundary symmetries given by subalgebras of the convolution algebra associated with a Lagrangian algebra object.
This thesis contributes to the understanding of symmetry-enriched topological phases focusing on their descriptions in terms of tensor network states. The Projected Entangled Pair State (PEPS) formalism allows us to locally encode the main properties of the models (like topological order, symmetries and their interplay) in the tensors. We have used that encoding to classify, construct and detect some classes of symmetry-enriched topological phases in 2D PEPS. For that purpose, we have studied what the allowed freedom in the tensors generating the same tensor network state is. These results are the so-called 'fundamental theorems' and they give the necessary knowledge to properly study symmetries (actions that leave the states invariant). We focus our study on the family of PEPS describing quantum double models of GGG, the so-called GGG-injective PEPS, and on global on-site symmetries coming from a finite group QQQ: as a result of the classification we have a finite number of phases closely related to the theory of group extensions. We also provide the maps that appear in group extension theory with physical meaning and we characterize their actions on the ground subspace and on the anyons. We construct a representative of each class of our classification. Moreover, we connect our construction with an interesting physical phenomenon, the so-called anyon condensation, which describes topological phase transitions. We have also proposed a family of gauge invariant quantities and their corresponding order parameters (in the form of expectation values of a local operator in 2D) to detect the corresponding quantum phases, in particular their symmetry fractionalization patterns. We conclude this thesis by touching on some mathematical open problems in PEPS.