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arXiv:1112.2215

Quantum Entanglement and Detection of Topological Order in Numerics

Tarun Grover

Published Dec 9, 2011• 3 citations
arXivHTMLPDF

Abstract

"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.

DOI
10.48550/arxiv.1112.2215

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"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 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.
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.
To perform topological quantum computation we first need to experimentally realise anyons in a topological system. These systems are characterised by intriguing non-local quantum correlations that give rise to the anyonic statistics. What are the diagnostic tools we have to identify if a given system is indeed topological or not? Different phases of matter are characterised by their symmetries. This information is captured by order parameters. Usually, order parameters are defined in terms of local operators that can be measured in the laboratory. For example, the magnetisation of a spin system is given as the expectation value of a single spin with respect to the ground state. Such local properties can describe fascinating physical phenomena efficiently, such as ferromagnetism, and can pinpoint quantum phase transitions. But what about topological systems? Experimentally, we usually identify the topological character of systems, such as the fractional quantum Hall liquids, by probing the anyonic properties of their excitations (Miller et al. , 2007). However, topological order should be a characteristic of the ground state (Thouless et al. , 1982; Wen, 1995). The natural question arises: is it possible to identify a property of the ground state of a system that implies anyonic excitations? The theoretical background that made it possible to answer this question came from entropic considerations of simple topological models. Hamma et al . (2005) studied the entanglement entropy of the toric code ground state and noticed an unusual behaviour.
Topological ordered phases are gapped states of matter that are characterized by non-local entanglement in their ground state wave functions instead of a local order parameter. In this paper, we review some of the basic results on the entanglement structure of topologically ordered phases. In particular, we focus on the notion and uses of "topological entanglement entropy" in two and higher dimensions, and also briefly review the relation between entanglement spectrum and the spectrum of the physical edge states for chiral topological states. Furthermore, we discuss a curvature expansion for the entanglement entropy which sharpens the nonlocality of topological entanglement entropy.
Z2 spin liquids have topological order. One manifestation of this is that a Z2 spin liquid on a torus exhibits a four-fold degeneracy. Recent numerical evidence has argued for the existence of a spin liquid ground state in the Hubbard model on a honeycomb lattice near U approximately 4. The evidence for this claim involves the presence of a gapped state that lacks any identifiable order. This argument relies on being able to distinguish small order from no order which is notoriously difficult. In this paper we demonstrate an approach which uses quantum Monte Carlo to search for one of the key features that positively identify the topological spin liquid: the topological degeneracy. For any finite system, this topological degeneracy is split where the splitting decays exponentially with system size. We search for low lying states in the energy spectrum that could be identified as these topologically degenerate states. We show that, for system sizes up to N=162 sites, there is no evidence for these states being significantly below the first excited state giving evidence against the existence of a topological phase. We discuss the possible options for the Hubbard model on the honeycomb lattice in the absence of such degeneracy.
Abstract Entanglement spectroscopy has stimulated an extensive range of studies. The entanglement spectrum is the spectrum of the reduced density matrix when the system is partitioned into two. For many quantum systems, when computed from the bulk ground-state wavefunction, it gives access to the physics of edge excitations. It is thus a valuable tool to diagnose topological ordering. These lecture notes provide an overview of entanglement spectroscopy, mostly in the context of the fractional quantum Hall effect. The basic concepts are introduced through the example of quantum spin chains. The connection with entanglement entropy and the matrix product state representation is discussed. The entanglement spectrum can be computed for non-interacting topological phases, revealing edge excitation from the ground state. Application to fractional quantum Hall phases shows how much information is encoded within the ground state and how different partitions probe different types of excitations. Application to fractional Chern insulators is discussed.