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:0905.2726 · Physical Review Letters

Splitting the Topological Degeneracy of Non-Abelian Anyons

Parsa Bonderson

Published Sep 9, 2009• 49 citations
arXivHTMLPDF

Abstract

We examine tunneling of topological charge between non-Abelian anyons as a perturbation of the long-range effective theory of a topologically ordered system. We obtain energy corrections in terms of the anyons' universal algebraic structure and nonuniversal tunneling amplitudes. We find that generic tunneling completely lifts the topological degeneracy of non-Abelian anyons. This degeneracy splitting is exponentially suppressed for long distances between anyons, but leaves no topological protection at shorter distances. We also show that general interactions of anyons can be expressed in terms of the transfer of topological charge, and thus can be treated effectively as tunneling interactions.

DOI
10.1103/physrevlett.103.110403

Related papers

Nonadiabatic effects in the braiding of non-Abelian anyons in topological superconductors

Meng Cheng, Victor M. Galitski, S. Das Sarma

Qubits in topological quantum computation are built from non-Abelian anyons. Adiabatic braiding of anyons is exploited as topologically protected logical gate operations. Thus, the adiabaticity upon which the notion of quantum statistics is defined plays a fundamental role in defining the non-Abelian anyons. We study the nonadiabatic effects in braidings of Ising-type anyons, namely, Majorana fermions in topological superconductors, using the formalism of time-dependent Bogoliubov-de Gennes equations. Using this formalism, we consider nonadiabatic corrections to non-Abelian statistics from (1) tunneling splitting of anyons imposing an additional dynamical phase to the transformation of ground states and (2) transitions to excited states that are potentially destructive to non-Abelian statistics since the nonlocal fermion occupation can be spoiled by such processes. However, if the bound states are localized and being braided together with the anyons, non-Abelian statistics can be recovered once the definition of Majorana operators is appropriately generalized taking into account the fermion parity in these states. On the other hand, if the excited states are extended over the whole system and form a continuum, the notion of local fermion parity no longer holds. We then quantitatively characterize the errors introduced in this situation.

arXiv:1106.2549•Physical Review B•Sep 30, 2011•93 citations

Entanglement Spectroscopy of non-Abelian Anyons

Eyal Cornfeld, L. Aviad Landau, Kirill Shtengel +1 more

We study the entanglement spectrum of topological systems hosting non-Abelian anyons. Similar to energy levels of a Hamiltonian, the entanglement spectrum is organized into symmetry multiplets. We find that the ratio between different eigenvalues within one multiplet is universal and is determined by the anyonic quantum dimensions. This result is a consequence of the conservation of the total topological charge. For systems with non-Abelian topological order, this generalizes known degeneracies of the entanglement spectrum, which are hallmarks of topological states. Experimental detection of these entanglement spectrum signatures may become possible using multi-copy schemes allowing to measure quantum entanglement and its symmetry-resolution. We discuss such a protocol explicitly in the context of Majorana wires.

•Oct 3, 2018•0 citations

Braiding of non-Abelian anyons using pairwise interactions

Michele Burrello, Bernard van Heck, Anton Akhmerov

The common approach to topological quantum computation is to implement quantum gates by adiabatically moving non-Abelian anyons around each other. Here we present an alternative perspective based on the possibility of realizing the exchange (braiding) operators of anyons by adiabatically varying pairwise interactions between them rather than their positions. We analyze a system composed by four anyons whose couplings define a T junction, and we show that the braiding operator of two of them can be obtained through a particular adiabatic cycle in the space of the coupling parameters. We also discuss how to couple this scheme with anyonic chains in order to recover the topological protection.

arXiv:1210.5452•Physical Review A•Feb 27, 2013•24 citations

Entanglement spectroscopy of non-Abelian anyons: Reading off quantum dimensions of individual anyons

Eyal Cornfeld, L. Aviad Landau, Kirill Shtengel +1 more

We study the entanglement spectrum of topological systems hosting non-Abelian anyons. Akin to energy levels of a Hamiltonian, the entanglement spectrum is composed of symmetry multiplets. We find that the ratio between different eigenvalues within one multiplet is universal and is determined by the anyonic quantum dimensions. This result is a consequence of the conservation of the total topological charge. For systems with non-Abelian topological order, this generalizes known degeneracies of the entanglement spectrum, which are hallmarks of topological states. Experimental detection of these entanglement spectrum signatures may become possible in Majorana wires using multicopy schemes, allowing the measurement of quantum entanglement and its symmetry resolution.

arXiv:1810.01853•Physical review. B./Physical review. B•Mar 20, 2019•69 citations

Synthetic non-Abelian statistics by Abelian anyon condensation

Yi‐Zhuang You, Chao‐Ming Jian, Xiao-Gang Wen

Topological degeneracy is the degeneracy of the ground states in a many-body system in the large-system-size limit. Topological degeneracy cannot be lifted by any local perturbation of the Hamiltonian. The topological degeneracies on closed manifolds have been used to discover/define topological order in many-body systems, which contain excitations with fractional statistics. In this paper, we study a new type of topological degeneracy induced by condensing anyons along a line in two-dimensional topological ordered states. Such topological degeneracy can be viewed as carried by each end of the line defect, which is a generalization of Majorana zero modes. The topological degeneracy can be used as a quantum memory. The ends of line defects carry projective non-Abelian statistics even though they are produced by the condensation of Abelian anyons, and braiding them allows us to perform fault tolerant quantum computations.

arXiv:1208.4109•Physical Review B•Jan 8, 2013•68 citations

Transport properties of anyons in random topological environments

Václav Zatloukal, Lauri Lehman, Sukhwinder Singh +2 more

The quasi-one-dimensional transport of Abelian and non-Abelian anyons is studied in the presence of a random topological background. In particular, we consider the quantum walk of an anyon that braids around islands of randomly filled static anyons of the same type. Two distinct behaviors are identified. We analytically demonstrate that all types of Abelian anyons localize purely due to the statistical phases induced by their random anyonic environment. In contrast, we numerically show that non-Abelian Ising anyons do not localize. This is due to their entanglement with the anyonic environment, which effectively induces dephasing. Our study demonstrates that localization properties strongly depend on nonlocal topological interactions, and it provides a clear distinction in the transport properties of Abelian and non-Abelian anyons.

arXiv:1207.5000•Physical Review B•Oct 9, 2014•5 citations
We examine tunneling of topological charge between non-Abelian anyons as a perturbation of the long-range effective theory of a topologically ordered system. We obtain energy corrections in terms of the anyons' universal algebraic structure and nonuniversal tunneling amplitudes. We find that generic tunneling completely lifts the topological degeneracy of non-Abelian anyons. This degeneracy splitting is exponentially suppressed for long distances between anyons, but leaves no topological protection at shorter distances. We also show that general interactions of anyons can be expressed in terms of the transfer of topological charge, and thus can be treated effectively as tunneling interactions.
Qubits in topological quantum computation are built from non-Abelian anyons. Adiabatic braiding of anyons is exploited as topologically protected logical gate operations. Thus, the adiabaticity upon which the notion of quantum statistics is defined plays a fundamental role in defining the non-Abelian anyons. We study the nonadiabatic effects in braidings of Ising-type anyons, namely, Majorana fermions in topological superconductors, using the formalism of time-dependent Bogoliubov-de Gennes equations. Using this formalism, we consider nonadiabatic corrections to non-Abelian statistics from (1) tunneling splitting of anyons imposing an additional dynamical phase to the transformation of ground states and (2) transitions to excited states that are potentially destructive to non-Abelian statistics since the nonlocal fermion occupation can be spoiled by such processes. However, if the bound states are localized and being braided together with the anyons, non-Abelian statistics can be recovered once the definition of Majorana operators is appropriately generalized taking into account the fermion parity in these states. On the other hand, if the excited states are extended over the whole system and form a continuum, the notion of local fermion parity no longer holds. We then quantitatively characterize the errors introduced in this situation.
We study the entanglement spectrum of topological systems hosting non-Abelian anyons. Similar to energy levels of a Hamiltonian, the entanglement spectrum is organized into symmetry multiplets. We find that the ratio between different eigenvalues within one multiplet is universal and is determined by the anyonic quantum dimensions. This result is a consequence of the conservation of the total topological charge. For systems with non-Abelian topological order, this generalizes known degeneracies of the entanglement spectrum, which are hallmarks of topological states. Experimental detection of these entanglement spectrum signatures may become possible using multi-copy schemes allowing to measure quantum entanglement and its symmetry-resolution. We discuss such a protocol explicitly in the context of Majorana wires.
The common approach to topological quantum computation is to implement quantum gates by adiabatically moving non-Abelian anyons around each other. Here we present an alternative perspective based on the possibility of realizing the exchange (braiding) operators of anyons by adiabatically varying pairwise interactions between them rather than their positions. We analyze a system composed by four anyons whose couplings define a T junction, and we show that the braiding operator of two of them can be obtained through a particular adiabatic cycle in the space of the coupling parameters. We also discuss how to couple this scheme with anyonic chains in order to recover the topological protection.
We study the entanglement spectrum of topological systems hosting non-Abelian anyons. Akin to energy levels of a Hamiltonian, the entanglement spectrum is composed of symmetry multiplets. We find that the ratio between different eigenvalues within one multiplet is universal and is determined by the anyonic quantum dimensions. This result is a consequence of the conservation of the total topological charge. For systems with non-Abelian topological order, this generalizes known degeneracies of the entanglement spectrum, which are hallmarks of topological states. Experimental detection of these entanglement spectrum signatures may become possible in Majorana wires using multicopy schemes, allowing the measurement of quantum entanglement and its symmetry resolution.
Topological degeneracy is the degeneracy of the ground states in a many-body system in the large-system-size limit. Topological degeneracy cannot be lifted by any local perturbation of the Hamiltonian. The topological degeneracies on closed manifolds have been used to discover/define topological order in many-body systems, which contain excitations with fractional statistics. In this paper, we study a new type of topological degeneracy induced by condensing anyons along a line in two-dimensional topological ordered states. Such topological degeneracy can be viewed as carried by each end of the line defect, which is a generalization of Majorana zero modes. The topological degeneracy can be used as a quantum memory. The ends of line defects carry projective non-Abelian statistics even though they are produced by the condensation of Abelian anyons, and braiding them allows us to perform fault tolerant quantum computations.
The quasi-one-dimensional transport of Abelian and non-Abelian anyons is studied in the presence of a random topological background. In particular, we consider the quantum walk of an anyon that braids around islands of randomly filled static anyons of the same type. Two distinct behaviors are identified. We analytically demonstrate that all types of Abelian anyons localize purely due to the statistical phases induced by their random anyonic environment. In contrast, we numerically show that non-Abelian Ising anyons do not localize. This is due to their entanglement with the anyonic environment, which effectively induces dephasing. Our study demonstrates that localization properties strongly depend on nonlocal topological interactions, and it provides a clear distinction in the transport properties of Abelian and non-Abelian anyons.