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arXiv:1111.2342 · Physical Review B

Quasiparticle statistics and braiding from ground-state entanglement

Yi Zhang, Tarun Grover, Ari M. Turner, Masaki Oshikawa, Ashvin Vishwanath

Published Jun 29, 2012• 343 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.85.235151

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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.
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 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.
Abstract We use the topological entanglement entropy (TEE) as an efficient tool to fully characterize the Abelian phase of a Z 2 × Z 2 spin liquid emerging as the ground state of topological color code (TCC), which is a class of stabilizer states on the honeycomb lattice. We provide the fusion rules of the quasiparticle (QP) excitations of the model by introducing single- or two-body operators on physical spins for each fusion process which justify the corresponding fusion outcome. Besides this, we extract the TEE from Renyi entanglement entropy (EE) of the TCC, analytically and numerically by finite size exact diagonalization on the disk shape regions with contractible boundaries. We obtain that the EE has a local contribution, which scales linearly with the boundary length in addition to a topological term, i.e. the TEE, arising from the condensation of closed strings in the ground state. We further investigate the ground state dependence of the TEE on regions with non-contractible boundaries, i.e. by cutting the torus to half cylinders, from which we further identify multiple independent minimum entropy states (MES) of the TCC and then extract the U and S modular matrices of the system, which contain the self and mutual statistics of the anyonic QPs and fully characterize the topological phase of the TCC. Eventually, we show that, in spite of the lack of a local order parameter, TEE and other physical quantities obtained from the ground state wave function such as the entanglement spectrum (ES) and ground state fidelity are sensitive probes to study the robustness of a topological phase. We find that the topological order in the presence of a magnetic field persists until the vicinity of the transition point, where the TEE and fidelity drops to zero and the ES splits severely, signaling breakdown of the topological phase of the TCC.
Gapped spin liquid is a topological phase of matter that hosts deconfined fractionalized quasiparticles. Quasiparticle statistics are an intrinsic property of spin liquid. However, it is difficult to identify explicitly in a microscopic model. Here we perform an exact diagonalization study of the Z2 quasiparticle statistics in the 'easy-axis' S = 1/2 Heisenberg J1 − J2 − J3 model on the kagome lattice, which hosts a robust Z2 spin liquid. Using the entanglement entropy measurement, we identify multiple independent minimal entangled states (MESs) on the torus geometry, which represents the eigenstates of the Wilson loop operators encircling the torus that measure the Z2 flux threading it. Based on the MESs, we obtain that: (1) The scaling of the finite size entropy leads to a topological entanglement entropy γ ≈ 0.637, indicating a Z2 topological spin liquid. (2) The modular transformation of MESs gives rise to the modular matrices and , which unambiguously fixes the Z2 quasiparitcle statistics including quasiparticle quantum dimensions, fusion rules and topological spins. (3) Through adiabatical threading a 2π flux in the hole of the torus, spinon state ψs (vacuum state ψ1) evolves into ψ1 (ψs), as predicted by Z2 gauge theory. Away from the 'easy-axis' limit with intermediate spin-z coupling, we identify some signature evidence of the chiral spin liquid based on the entanglement spectrum of the state using density matrix renormalization group simulation.
"Topological ordered" phases such as gapped quantum spin-liquids and fractional quantum Hall states possess ground state degeneracy on a torus. We show that the topological nature of this degeneracy has interesting consequences for the entanglement structure of the degenerate ground states. This leads to a simple method for detecting topological order, which is ready-made for numerical schemes such as density matrix renormalization group. The method is valid for phases that do or do not possess edge states alike. We demonstrate it by calculating the entanglement spectrum and the topological entanglement entropy for a chiral spin-liquid state on a 18 site system.
We use entanglement entropy signatures to establish non-Abelian topological order in projected Chern-insulator wave functions. The simplest instance is obtained by Gutzwiller projecting a filled band with Chern number C=2C=2C=2, whose wave function may also be viewed as the square of the Slater determinant of a band insulator. We demonstrate that this wave function is captured by the SU(2)2\text{SU}{(2)}_{2}SU(2)2​ Chern-Simons theory coupled to fermions. This is established most persuasively by calculating the modular S\mathcal{S}S-matrix from the candidate ground-state wave functions, following a recent entanglement-entropy-based approach. This directly demonstrates the peculiar non-Abelian braiding statistics of Majorana fermion quasiparticles in this state. We also provide microscopic evidence for the field theoretic generalization, that the NNNth power of a Chern number CCC Slater determinant realizes the topological order of the SU(N)C\text{SU}{(N)}_{C}SU(N)C​ Chern-Simons theory coupled to fermions, by studying the SU(2)3\text{SU}{(2)}_{3}SU(2)3​ (Read-Rezayi--type state) and the SU(3)2\text{SU}{(3)}_{2}SU(3)2​ wave functions. An advantage of our projected Chern-insulator wave functions is the relative ease with which physical properties, such as entanglement entropy and modular S\mathcal{S}S-matrix, can be numerically calculated using Monte Carlo techniques.
We show how to numerically calculate several quantities that characterize topological order starting from a microscopic fractional quantum Hall Hamiltonian. To find the set of degenerate ground states, we employ the infinite density matrix renormalization group method based on the matrix-product state representation of fractional quantum Hall states on an infinite cylinder. To study localized quasiparticles of a chosen topological charge, we use pairs of degenerate ground states as boundary conditions for the infinite density matrix renormalization group. We then show that the wave function obtained on the infinite cylinder geometry can be adapted to a torus of arbitrary modular parameter, which allows us to explicitly calculate the non-Abelian Berry connection associated with the modular T transformation. As a result, the quantum dimensions, topological spins, quasiparticle charges, chiral central charge, and Hall viscosity of the phase can be obtained using data contained entirely in the entanglement spectrum of an infinite cylinder.