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:1412.0677 · Physical Review B

General procedure for determining braiding and statistics of anyons using entanglement interferometry

Yi Zhang, Tarun Grover, Ashvin Vishwanath

Published Jan 26, 2015• 25 citations
arXivHTMLPDF

Abstract

Recently, it was argued that the braiding and statistics of anyons in a two-dimensional topological phase can be extracted by studying the quantum entanglement of the degenerate ground states on the torus. This construction either required a lattice symmetry (such as $\ensuremath{\pi}/2$ rotation) or tacitly assumed that the minimum entanglement states (MESs) for two different bipartitions can be uniquely assigned quasiparticle labels. Here we describe a procedure to obtain the modular $\mathcal{S}$ matrix, which encodes the braiding statistics of anyons, which does not require making any of these assumptions. Our strategy is to coherently compare MESs of three independent entanglement bipartitions of the torus, which leads to a unique modular $\mathcal{S}$. This procedure also puts strong constraints on the modular $\mathcal{T}$ and $\mathcal{U}$ matrices without requiring any symmetries, and in certain special cases, completely determines it. Our method applies equally to Abelian and non-Abelian topological phases.

DOI
10.1103/physrevb.91.035127

Related papers

Detecting topological order from modular transformations of ground states on the torus

Zhuan Li, Roger S. K. Mong

The ground states encode the information of the topological phases of a 2-dimensional system, which makes them crucial in determining the associated topological quantum field theory (TQFT). Most numerical methods for detecting the TQFT relied on the use of minimum entanglement states (MESs), extracting the anyon mutual statistics and self statistics via overlaps and/or the entanglement spectra. The MESs are the eigenstates of the Wilson loop operators, and are labeled by the anyons corresponding to their eigenvalues. Here we revisit the definition of the Wilson loop operators and MESs. We derive the modular transformation of the ground states purely from the Wilson loop algebra, and as a result, the modular $S$ and $T$ matrices naturally show up in the overlap of MESs. Importantly, we show that due to the phase degree of freedom of the Wilson loop operators, the MES-anyon assignment is not unique. This ambiguity obstructs our attempt to detect the topological order, that is, there exist different TQFTs that cannot be distinguished solely by the overlap of MESs. In this paper, we provide the upper limit of the information one may obtain from the overlap of MESs without other additional structure. Finally, we show that if the phase is enriched by rotational symmetry, there may be additional TQFT information that can be extracted from overlap of MESs.

arXiv:2203.04329•Physical review. B./Physical review. B•Dec 12, 2022•6 citations

Quasiparticle statistics and braiding from ground-state entanglement

Yi Zhang, Tarun Grover, Ari M. Turner +2 more

Topologically ordered phases are gapped states, defined by the properties of excitations when taken around one another. Here we demonstrate a method to extract the statistics and braiding of excitations, given just the set of ground-state wave functions on a torus. This is achieved by studying the topological entanglement entropy (TEE) upon partitioning the torus into two cylinders. In this setting, general considerations dictate that the TEE generally differs from that in trivial partitions and depends on the chosen ground state. Central to our scheme is the identification of ground states with minimum entanglement entropy, which reflect the quasiparticle excitations of the topological phase. The transformation of these states allows for the determination of the modular $\mathcal{S}$ and $\mathcal{U}$ matrices which encode quasiparticle properties. We demonstrate our method by extracting the modular $\mathcal{S}$ matrix of a chiral spin liquid phase using a Monte Carlo scheme to calculate the TEE and prove that the quasiparticles obey semionic statistics. This method offers a route to nearly complete determination of the topological order in certain cases.

arXiv:1111.2342•Physical Review B•Jun 29, 2012•343 citations

Braid Group Representations, Modular Data, and the Classification of Majorana Zero Mode Fusion Rules in 2D Topological Superconductors

Rowan Brad Quni-Gudzinas

We investigate whether the braid group representation on $N$ anyons uniquely determines the modular data ($S$ and $T$ matrices) of a non-Abelian topological order, and whether this framework classifies all fusion rules for Majorana zero modes (MZMs) in 2D topological superconductors. We show that the local $N$-anyon braid group representation constrains the modular data up to finite gauge ambiguity but does not uniquely fix the global $S$-matrix without specifying the underlying spin structure. For MZMs, the braid generators satisfy the Ising anyon $R$-matrix with the asymptotic Hecke relation $\sigma_i^2 = 1$ (up to phase), which rigidly fixes the local $F$ and $R$ symbols of the Ising category. Consequently, the fusion rules are strictly of Ising type ($\sigma \times \sigma = 1 + \psi$), precluding alternative non-Abelian fusion rules. The $T$-matrix is determined by topological spins, but the full $S$-matrix depends on the spin structure of the underlying manifold, reflecting the super-modular structure of the theory. ## 1. Introduction Topological quantum computation exploits the non-Abelian statistics of anyons—quasiparticle excitations in two-dimensional systems whose braiding operations are represented by non-commuting unitary matrices on a degenerate ground-state manifold [1]. A central question is whether the local braiding data of $N$ anyons suffices to reconstruct the full modular data of the underlying topological order (TO), or whether additional global information is required. This question has direct physical consequences for 2D topological superconductors (TSCs) hosting Majorana zero modes (MZMs), where the anyonic excitations are vortices binding MZMs and the fermionic quasiparticle is a transparent (local) fermion [2,3]. The modular data of a TO consists of the $S$-matrix, which encodes the mutual braiding statistics of anyons, and the $T$-matrix, which encodes the topological spins. Together, these matrices form a projective representation of t

•Sep 7, 2026•0 citations

Identifying Non-Abelian Topological Order through Minimal Entangled States

W. Zhu, Shou-Shu Gong, F. D. M. Haldane +1 more

The topological order is encoded in the pattern of long-range quantum entanglements, which cannot be measured by any local observable. Here we perform an exact diagonalization study to establish the non-Abelian topological order for topological band models through entanglement entropy measurement. We focus on the quasiparticle statistics of the non-Abelian Moore-Read and Read-Rezayi states on the lattice models with bosonic particles. We identify multiple independent minimal entangled states (MESs) in the ground state manifold on a torus. The extracted modular S matrix from MESs faithfully demonstrates the Ising anyon or Fibonacci quasiparticle statistics, including the quasiparticle quantum dimensions and the fusion rules for such systems. These findings unambiguously demonstrate the topological nature of the quantum states for these flatband models without using the knowledge of model wave functions.

arXiv:1308.3878•Physical Review Letters•Mar 4, 2014•28 citations

Braid Group Representations and Modular Data: Non-Uniqueness, Finite Images, and the Ising Test Case

Rowan Brad Quni-Gudzinas

A central question in the theory of topological order asks whether the braid group representation $B_N$ carried by $N$ non-Abelian anyons uniquely determines the modular data — the $S$ and $T$ matrices — of the underlying topological quantum field theory. We reconcile three lines of analysis into a single assessment. Structurally, we formalize the question as a reduction map $\Phi$ from modular tensor category data to projective braid group representations, whose kernel is the locus of potential counterexamples. Two rigorous finiteness results constrain the problem: braid representations from twisted quantum doubles of finite groups factor through finite groups, and the $SO(N)_2$/$O(N)_2$ braid representations are Gaussian with finite image. We then perform an explicit computation on the Ising anyon model — the canonical non-Abelian order of Majorana zero modes — reconstructing the topological spins $\theta_\sigma = e^{i\pi/8}$, $\theta_\psi = -1$, the total quantum dimension $\mathcal{D}=2$, and the full $S$ matrix from the braid eigenvalues, with all arithmetic shown. We exhibit a Galois-conjugate deformation $\mathrm{Ising}(7)$ with identical fusion rules but distinct $S$ matrix, and verify that its braid representation is inequivalent to the Ising representation on $B_3$ by an explicit trace computation, while remaining indistinguishable at the level of fusion rules. We conclude that braid representations constrain but do not provably uniquely determine modular data in general: uniqueness can hold only under additional hypotheses, and the Majorana fusion-rule classification is not closed by braid data alone. The affirmative claim that braiding is a complete invariant is restricted to a narrower regime than sometimes asserted, and we document this divergence explicitly. ## 1. Introduction Non-Abelian anyons in two spatial dimensions are characterized by two layers of algebraic structure. The first is combinatorial: fusion rules $N_{ab}^c$ specifying how anyon c Full text and updates: papers.qnfo.org/papers/braid-group-representations-and-modular-data-non-uniqueness-finite-images-and-th/

•Oct 1, 2026•0 citations

Simulation of braiding anyons using matrix product states

Babatunde M. Ayeni, Sukhwinder Singh, Robert N. C. Pfeifer +1 more

Anyons exist as pointlike particles in two dimensions and carry braid statistics, which enable interactions that are independent of the distance between the particles. Except for a relatively few number of models, which are analytically tractable, much of the physics of anyons remains still unexplored. In this paper, we show how U(1) symmetry can be combined with the previously proposed anyonic matrix product states to simulate ground states and dynamics of anyonic systems on a lattice at any rational particle number density. We provide proof of principle by studying itinerant anyons on a one-dimensional chain where no natural notion of braiding arises and also on a two-leg ladder where the anyons hop between sites and possibly braid. We compare the result of the ground-state energies of Fibonacci anyons against hardcore bosons and spinless fermions. In addition, we report the entanglement entropies of the ground states of interacting Fibonacci anyons on a fully filled two-leg ladder at different interaction strength, identifying gapped or gapless points in the parameter space. As an outlook, our approach can also prove useful in studying the time dynamics of a finite number of non-Abelian anyons on a finite two-dimensional lattice.

arXiv:1509.00903•Physical review. B./Physical review. B•Apr 20, 2016•9 citations
Recently, it was argued that the braiding and statistics of anyons in a two-dimensional topological phase can be extracted by studying the quantum entanglement of the degenerate ground states on the torus. This construction either required a lattice symmetry (such as \ensuremathπ/2\ensuremath{\pi}/2\ensuremathπ/2 rotation) or tacitly assumed that the minimum entanglement states (MESs) for two different bipartitions can be uniquely assigned quasiparticle labels. Here we describe a procedure to obtain the modular S\mathcal{S}S matrix, which encodes the braiding statistics of anyons, which does not require making any of these assumptions. Our strategy is to coherently compare MESs of three independent entanglement bipartitions of the torus, which leads to a unique modular S\mathcal{S}S. This procedure also puts strong constraints on the modular T\mathcal{T}T and U\mathcal{U}U matrices without requiring any symmetries, and in certain special cases, completely determines it. Our method applies equally to Abelian and non-Abelian topological phases.
The ground states encode the information of the topological phases of a 2-dimensional system, which makes them crucial in determining the associated topological quantum field theory (TQFT). Most numerical methods for detecting the TQFT relied on the use of minimum entanglement states (MESs), extracting the anyon mutual statistics and self statistics via overlaps and/or the entanglement spectra. The MESs are the eigenstates of the Wilson loop operators, and are labeled by the anyons corresponding to their eigenvalues. Here we revisit the definition of the Wilson loop operators and MESs. We derive the modular transformation of the ground states purely from the Wilson loop algebra, and as a result, the modular SSS and TTT matrices naturally show up in the overlap of MESs. Importantly, we show that due to the phase degree of freedom of the Wilson loop operators, the MES-anyon assignment is not unique. This ambiguity obstructs our attempt to detect the topological order, that is, there exist different TQFTs that cannot be distinguished solely by the overlap of MESs. In this paper, we provide the upper limit of the information one may obtain from the overlap of MESs without other additional structure. Finally, we show that if the phase is enriched by rotational symmetry, there may be additional TQFT information that can be extracted from overlap of MESs.
Topologically ordered phases are gapped states, defined by the properties of excitations when taken around one another. Here we demonstrate a method to extract the statistics and braiding of excitations, given just the set of ground-state wave functions on a torus. This is achieved by studying the topological entanglement entropy (TEE) upon partitioning the torus into two cylinders. In this setting, general considerations dictate that the TEE generally differs from that in trivial partitions and depends on the chosen ground state. Central to our scheme is the identification of ground states with minimum entanglement entropy, which reflect the quasiparticle excitations of the topological phase. The transformation of these states allows for the determination of the modular S\mathcal{S}S and U\mathcal{U}U matrices which encode quasiparticle properties. We demonstrate our method by extracting the modular S\mathcal{S}S matrix of a chiral spin liquid phase using a Monte Carlo scheme to calculate the TEE and prove that the quasiparticles obey semionic statistics. This method offers a route to nearly complete determination of the topological order in certain cases.
We investigate whether the braid group representation on NNN anyons uniquely determines the modular data (SSS and TTT matrices) of a non-Abelian topological order, and whether this framework classifies all fusion rules for Majorana zero modes (MZMs) in 2D topological superconductors. We show that the local NNN-anyon braid group representation constrains the modular data up to finite gauge ambiguity but does not uniquely fix the global SSS-matrix without specifying the underlying spin structure. For MZMs, the braid generators satisfy the Ising anyon RRR-matrix with the asymptotic Hecke relation σi2=1\sigma_i^2 = 1σi2​=1 (up to phase), which rigidly fixes the local FFF and RRR symbols of the Ising category. Consequently, the fusion rules are strictly of Ising type (σ×σ=1+ψ\sigma \times \sigma = 1 + \psiσ×σ=1+ψ), precluding alternative non-Abelian fusion rules. The TTT-matrix is determined by topological spins, but the full SSS-matrix depends on the spin structure of the underlying manifold, reflecting the super-modular structure of the theory. ## 1. Introduction Topological quantum computation exploits the non-Abelian statistics of anyons—quasiparticle excitations in two-dimensional systems whose braiding operations are represented by non-commuting unitary matrices on a degenerate ground-state manifold [1]. A central question is whether the local braiding data of NNN anyons suffices to reconstruct the full modular data of the underlying topological order (TO), or whether additional global information is required. This question has direct physical consequences for 2D topological superconductors (TSCs) hosting Majorana zero modes (MZMs), where the anyonic excitations are vortices binding MZMs and the fermionic quasiparticle is a transparent (local) fermion [2,3]. The modular data of a TO consists of the SSS-matrix, which encodes the mutual braiding statistics of anyons, and the TTT-matrix, which encodes the topological spins. Together, these matrices form a projective representation of t
The topological order is encoded in the pattern of long-range quantum entanglements, which cannot be measured by any local observable. Here we perform an exact diagonalization study to establish the non-Abelian topological order for topological band models through entanglement entropy measurement. We focus on the quasiparticle statistics of the non-Abelian Moore-Read and Read-Rezayi states on the lattice models with bosonic particles. We identify multiple independent minimal entangled states (MESs) in the ground state manifold on a torus. The extracted modular S matrix from MESs faithfully demonstrates the Ising anyon or Fibonacci quasiparticle statistics, including the quasiparticle quantum dimensions and the fusion rules for such systems. These findings unambiguously demonstrate the topological nature of the quantum states for these flatband models without using the knowledge of model wave functions.
A central question in the theory of topological order asks whether the braid group representation BNB_NBN​ carried by NNN non-Abelian anyons uniquely determines the modular data — the SSS and TTT matrices — of the underlying topological quantum field theory. We reconcile three lines of analysis into a single assessment. Structurally, we formalize the question as a reduction map Φ\PhiΦ from modular tensor category data to projective braid group representations, whose kernel is the locus of potential counterexamples. Two rigorous finiteness results constrain the problem: braid representations from twisted quantum doubles of finite groups factor through finite groups, and the SO(N)2SO(N)_2SO(N)2​/O(N)2O(N)_2O(N)2​ braid representations are Gaussian with finite image. We then perform an explicit computation on the Ising anyon model — the canonical non-Abelian order of Majorana zero modes — reconstructing the topological spins θσ=eiπ/8\theta_\sigma = e^{i\pi/8}θσ​=eiπ/8, θψ=−1\theta_\psi = -1θψ​=−1, the total quantum dimension D=2\mathcal{D}=2D=2, and the full SSS matrix from the braid eigenvalues, with all arithmetic shown. We exhibit a Galois-conjugate deformation Ising(7)\mathrm{Ising}(7)Ising(7) with identical fusion rules but distinct SSS matrix, and verify that its braid representation is inequivalent to the Ising representation on B3B_3B3​ by an explicit trace computation, while remaining indistinguishable at the level of fusion rules. We conclude that braid representations constrain but do not provably uniquely determine modular data in general: uniqueness can hold only under additional hypotheses, and the Majorana fusion-rule classification is not closed by braid data alone. The affirmative claim that braiding is a complete invariant is restricted to a narrower regime than sometimes asserted, and we document this divergence explicitly. ## 1. Introduction Non-Abelian anyons in two spatial dimensions are characterized by two layers of algebraic structure. The first is combinatorial: fusion rules NabcN_{ab}^cNabc​ specifying how anyon c Full text and updates: papers.qnfo.org/papers/braid-group-representations-and-modular-data-non-uniqueness-finite-images-and-th/
Anyons exist as pointlike particles in two dimensions and carry braid statistics, which enable interactions that are independent of the distance between the particles. Except for a relatively few number of models, which are analytically tractable, much of the physics of anyons remains still unexplored. In this paper, we show how U(1) symmetry can be combined with the previously proposed anyonic matrix product states to simulate ground states and dynamics of anyonic systems on a lattice at any rational particle number density. We provide proof of principle by studying itinerant anyons on a one-dimensional chain where no natural notion of braiding arises and also on a two-leg ladder where the anyons hop between sites and possibly braid. We compare the result of the ground-state energies of Fibonacci anyons against hardcore bosons and spinless fermions. In addition, we report the entanglement entropies of the ground states of interacting Fibonacci anyons on a fully filled two-leg ladder at different interaction strength, identifying gapped or gapless points in the parameter space. As an outlook, our approach can also prove useful in studying the time dynamics of a finite number of non-Abelian anyons on a finite two-dimensional lattice.