Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1702.08759 · Physical review. B./Physical review. B

Symmetry reduction induced by anyon condensation: A tensor network approach

José Garre-Rubio, S. Iblisdir, David Pérez-Garcı́a

Published Oct 16, 2017• 15 citations
arXivHTMLPDF

Abstract

Topological ordered phases are related to changes in the properties of their quasiparticle excitations (anyons). We study these relations in the framework of projected entanglement pair states and show how condensing and confining anyons reduces a local gauge symmetry to a global on-site symmetry. We also study the action of this global symmetry over the quasiparticle excitations. As a byproduct, we observe that this symmetry reduction effect can be applied to one-dimensional systems as well and brings about appealing physical interpretations on the classification of phases with symmetries using matrix product states. The case of ${\mathbb{Z}}_{2}$ on-site symmetry is studied in detail.

DOI
10.1103/physrevb.96.155123

Related papers

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

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

Dominic J. Williamson, Nick Bultinck, Frank Verstraete

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.

arXiv:1711.07982•Quantum•Sep 1, 2026•40 citations

Complete characterization of non-Abelian topological phase transitions and detection of anyon splitting with projected entangled pair states

Wen-Tao Xu, José Garre-Rubio, Norbert Schuch

Topological phases - systems with a global ordering in the entanglement - exhibit exotic excitations known as anyons which can display highly nontrivial (non-Abelian) statistics. Through transitions between such phases, these anyons can undergo rather surprising changes. Perhaps most surprisingly, one anyon can split into two or more anyons, some of which might condense or become confined. The authors propose here a complete set of order parameters which allow one to extract the full information about how the properties of anyons change through phase transitions, and to apply them to a range of topological phase transitions.

arXiv:2208.13200•Physical review. B./Physical review. B•Nov 23, 2022•19 citations

Shadows of anyons and the entanglement structure of topological phases

Jutho Haegeman, Valentin Zauner-Stauber, Norbert Schuch +1 more

The low-temperature dynamics of quantum systems are dominated by the low-energy eigenstates. For two-dimensional systems in particular, exotic phenomena such as topological order and anyon excitations can emerge. While a complete low-energy description of strongly correlated systems is hard to obtain, essential information about the elementary excitations is encoded in the eigenvalue structure of the quantum transfer matrix. Here we study the transfer matrix of topological quantum systems using the tensor network formalism and demonstrate that topological quantum order requires a particular type of 'symmetry breaking' for the fixed point subspace. We also relate physical anyon excitations to domain-wall excitations at the level of the transfer matrix. This formalism enables us to determine the structure of the topological sectors in two-dimensional gapped phases very efficiently, therefore opening novel avenues for studying fundamental questions related to anyon condensation and confinement.

arXiv:1410.5443•Nature Communications•Oct 6, 2015•83 citations

Symmetry fractionalization and twist defects

Nicolas Tarantino, Netanel H Lindner, Lukasz Fidkowski

Topological order in two-dimensions can be described in terms of deconfined quasiparticle excitations—anyons—and their braiding statistics. However, it has recently been realized that this data does not completely describe the situation in the presence of an unbroken global symmetry. In this case, there can be multiple distinct quantum phases with the same anyons and statistics, but with different patterns of symmetry fractionalization—termed symmetry enriched topological order. When the global symmetry group G , which we take to be discrete, does not change topological superselection sectors—i.e. does not change one type of anyon into a different type of anyon—one can imagine a local version of the action of G around each anyon. This leads to projective representations and a group cohomology description of symmetry fractionalization, with the second cohomology group H 2 ( G , A a b e l i a n ) being the relevant group. In this paper, we treat the general case of a symmetry group G possibly permuting anyon types. We show that despite the lack of a local action of G , one can still make sense of a so-called twisted group cohomology description of symmetry fractionalization, and show how this data is encoded in the associativity of fusion rules of the extrinsic ‘twist’ defects of the symmetry. Furthermore, building on work of Hermele (2014 Phys. Rev. B 90 184418 ), we construct a wide class of exactly-solvable models which exhibit this twisted symmetry fractionalization, and connect them to our formal framework.

arXiv:1506.06754•New Journal of Physics•Mar 7, 2016•145 citations

Phase diagram of a symmetry-enriched toric code

José Garre-Rubio, Mohsin Iqbal, David T. Stephen

We analyze the effect of perturbations on a toric code-like Hamiltonian enriched with global symmetries. The ground states of this Hamiltonian are superpositions of loops decorated with cluster states, resulting in anyonic excitations that fractionalize a global on-site $\mathbb{Z}_2\times\mathbb{Z}_2$, inversion and time reversal symmetry. We use a modified version of the string order parameters proposed in [New Journal of Physics 21, 113016 (2019)], adapted to work beyond the framework of tensor networks, to capture the symmetry fractionalization pattern of the anyons. We show that this order parameter accurately captures the symmetry fractionalization in a direct Hamiltonian interpolation between the decorated and trivial toric codes, and in the presence of symmetry-preserving local magnetic fields and Ising interactions. We also observe how the condensation of an anyon that fractionalizes the symmetry forces the symmetry to spontaneously break at the phase transition. This notably changes the phase diagram of this Hamiltonian under parallel magnetic fields compared to the regular toric code since the condensation of charge and flux anyons drives the system to two distinct trivial phases, a symmetry broken phase and symmetric phase. Moreover, we provide a proof of spontaneous symmetry breaking from anyon condensation in the framework of projected entangled pair states.

•Nov 5, 2020•0 citations
Topological ordered phases are related to changes in the properties of their quasiparticle excitations (anyons). We study these relations in the framework of projected entanglement pair states and show how condensing and confining anyons reduces a local gauge symmetry to a global on-site symmetry. We also study the action of this global symmetry over the quasiparticle excitations. As a byproduct, we observe that this symmetry reduction effect can be applied to one-dimensional systems as well and brings about appealing physical interpretations on the classification of phases with symmetries using matrix product states. The case of Z2{\mathbb{Z}}_{2}Z2​ on-site symmetry is studied in detail.
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.
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.
Topological phases - systems with a global ordering in the entanglement - exhibit exotic excitations known as anyons which can display highly nontrivial (non-Abelian) statistics. Through transitions between such phases, these anyons can undergo rather surprising changes. Perhaps most surprisingly, one anyon can split into two or more anyons, some of which might condense or become confined. The authors propose here a complete set of order parameters which allow one to extract the full information about how the properties of anyons change through phase transitions, and to apply them to a range of topological phase transitions.
The low-temperature dynamics of quantum systems are dominated by the low-energy eigenstates. For two-dimensional systems in particular, exotic phenomena such as topological order and anyon excitations can emerge. While a complete low-energy description of strongly correlated systems is hard to obtain, essential information about the elementary excitations is encoded in the eigenvalue structure of the quantum transfer matrix. Here we study the transfer matrix of topological quantum systems using the tensor network formalism and demonstrate that topological quantum order requires a particular type of 'symmetry breaking' for the fixed point subspace. We also relate physical anyon excitations to domain-wall excitations at the level of the transfer matrix. This formalism enables us to determine the structure of the topological sectors in two-dimensional gapped phases very efficiently, therefore opening novel avenues for studying fundamental questions related to anyon condensation and confinement.
Topological order in two-dimensions can be described in terms of deconfined quasiparticle excitations—anyons—and their braiding statistics. However, it has recently been realized that this data does not completely describe the situation in the presence of an unbroken global symmetry. In this case, there can be multiple distinct quantum phases with the same anyons and statistics, but with different patterns of symmetry fractionalization—termed symmetry enriched topological order. When the global symmetry group G , which we take to be discrete, does not change topological superselection sectors—i.e. does not change one type of anyon into a different type of anyon—one can imagine a local version of the action of G around each anyon. This leads to projective representations and a group cohomology description of symmetry fractionalization, with the second cohomology group H 2 ( G , A a b e l i a n ) being the relevant group. In this paper, we treat the general case of a symmetry group G possibly permuting anyon types. We show that despite the lack of a local action of G , one can still make sense of a so-called twisted group cohomology description of symmetry fractionalization, and show how this data is encoded in the associativity of fusion rules of the extrinsic ‘twist’ defects of the symmetry. Furthermore, building on work of Hermele (2014 Phys. Rev. B 90 184418 ), we construct a wide class of exactly-solvable models which exhibit this twisted symmetry fractionalization, and connect them to our formal framework.
We analyze the effect of perturbations on a toric code-like Hamiltonian enriched with global symmetries. The ground states of this Hamiltonian are superpositions of loops decorated with cluster states, resulting in anyonic excitations that fractionalize a global on-site Z2×Z2\mathbb{Z}_2\times\mathbb{Z}_2Z2​×Z2​, inversion and time reversal symmetry. We use a modified version of the string order parameters proposed in [New Journal of Physics 21, 113016 (2019)], adapted to work beyond the framework of tensor networks, to capture the symmetry fractionalization pattern of the anyons. We show that this order parameter accurately captures the symmetry fractionalization in a direct Hamiltonian interpolation between the decorated and trivial toric codes, and in the presence of symmetry-preserving local magnetic fields and Ising interactions. We also observe how the condensation of an anyon that fractionalizes the symmetry forces the symmetry to spontaneously break at the phase transition. This notably changes the phase diagram of this Hamiltonian under parallel magnetic fields compared to the regular toric code since the condensation of charge and flux anyons drives the system to two distinct trivial phases, a symmetry broken phase and symmetric phase. Moreover, we provide a proof of spontaneous symmetry breaking from anyon condensation in the framework of projected entangled pair states.