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arXiv:1402.4245 · New Journal of Physics

Irreducible many-body correlations in topologically ordered systems

Yang Liu, Bei Zeng, D L Zhou

Published Feb 4, 2016• 11 citations
arXivHTMLPDF

Abstract

Topologically ordered systems exhibit large-scale correlation in their ground states, which may be characterized by quantities such as topological entanglement entropy. We propose that the concept of irreducible many-body correlation (IMC), the correlation that cannot be implied by all local correlations, may also be used as a signature of topological order. In a topologically ordered system, we demonstrate that for a part of the system with holes, the reduced density matrix exhibits IMCs which become reducible when the holes are removed. The appearance of these IMCs then represents a key feature of topological phase. We analyze the many-body correlation structures in the ground state of the toric code model in external magnetic fields, and show that the topological phase transition is signaled by the IMCs.

DOI
10.1088/1367-2630/18/2/023024

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Topologically ordered systems exhibit large-scale correlation in their ground states, which may be characterized by quantities such as topological entanglement entropy. We propose that the concept of irreducible many-body correlation (IMC), the correlation that cannot be implied by all local correlations, may also be used as a signature of topological order. In a topologically ordered system, we demonstrate that for a part of the system with holes, the reduced density matrix exhibits IMCs which become reducible when the holes are removed. The appearance of these IMCs then represents a key feature of topological phase. We analyze the many-body correlation structures in the ground state of the toric code model in external magnetic fields, and show that the topological phase transition is signaled by the IMCs.
A special feature of the ground state in a topologically ordered phase is the existence of large-scale correlations depending only on the topology of the regions. These correlations can be detected by the topological entanglement entropy or by a measure called irreducible correlation. We show that these two measures coincide for states obeying an area law and having zero correlation length. Moreover, we provide an operational meaning for these measures by proving its equivalence to the optimal rate of a particular class of secret sharing protocols. This establishes an information-theoretical approach to multipartite correlations in topologically ordered systems.
We analyze the computational aspects of detecting topological order in a quantum many-body system. We contrast the widely used topological entanglement entropy with a recently introduced operational definition for topological order based on error correction properties, finding exponential scaling with the system size for the former and polynomial scaling for the latter. We exemplify our approach for a variant of the paradigmatic toric code model with mobile particles, finding that the error correction method allows to treat substantially larger system sizes. In particular, the phase diagram of the model can be successfully computed using error correction, while the topological entanglement entropy is too severely limited by finite size effects to obtain conclusive results. While we mainly focus on one-dimensional systems whose ground states can be expressed in terms of matrix product states, our strategy can be readily generalized to higher dimensions and systems out of equilibrium, even allowing for an efficient detection of topological order in current quantum simulation experiments.
Abstract We analyze the computational aspects of detecting topological order in a quantum many-body system. We contrast the widely used topological entanglement entropy with a recently introduced operational definition for topological order based on error correction properties, finding exponential scaling with the system size for the former and polynomial scaling for the latter. We exemplify our approach for a variant of the paradigmatic toric code model with mobile particles, finding that the error correction method allows to treat substantially larger system sizes. In particular, the phase diagram of the model can be successfully computed using error correction, while the topological entanglement entropy is too severely limited by finite size effects to obtain conclusive results. While we mainly focus on one-dimensional systems whose ground states can be expressed in terms of matrix product states, our strategy can be readily generalized to higher dimensions and systems out of equilibrium, even allowing for an efficient detection of topological order in current quantum simulation experiments.
The interplay of the constituents of interacting many-body systems may reveal emergent properties on the macroscopic scale which are not inherent to the individual constituents. These properties are expressed in macroscopic observables \ndescribing the state — denoted as the phase of the system. Continuous phase transition between phases are generically manifested in critical behavior, for example, a divergence of a macroscopic observable. The identification of the present phase of a system and the classification of critical phenomena into universality classes are exciting challenges of condensed-matter physics. \nIn this thesis, we use entanglement entropies as macroscopic quantities for the characterization of phases of quantum matter and critical theories. For groundstates of quantum many-body systems the entanglement entropy is a measure of \nthe amount of entanglement between two subsystems. The generic dependence of the entanglement entropy on the size and shape of the subsystems is contained in the well-known boundary law — stating a scaling of the entanglement entropy with the boundary between the two subsystems. We numerically investigate how the coefficient of this dependence reflects quantum phase transitions in simple spin-half bilayer models. \nSubleading terms to the boundary law such as a logarithmic contribution provide universal numbers for the criticality of field theories. We examine free and interacting theories from an entanglement entropy perspective in order to assess the role of the coefficient of the logarithmic correction induced by corners in the subsystems. Beyond its universality, this coefficient also quantifies degrees of freedom of low-lying excitations in the conformal field theory describing a critical point. \nA constant contribution to the boundary law indicates the presence of so-called topological order in the ground state of a many-body system. This extremely useful property can also be identified in classical counterparts of entanglement entropy which we study at the example of various toric code models. To this endeavor, we have designed Monte Carlo techniques which allow for an efficient numerical computation of the constant contribution. In particular, we analyze via entanglement entropies under which conditions remnants of topological order are present in the quantum system at finite temperature and at perturbations from a magnetic field. The major motivation behind this effort is to use topological order for the robust storage of a quantum information — a basic need for the construction of quantum \ncomputers.
Topological order has become a new paradigm to distinguish ground states of interacting many-body systems without conventional long-range order. Here, we discuss possible extensions of this concept to density matrices describing statistical ensembles. For a large class of quasithermal states, which can be realized as thermal states of some quasilocal Hamiltonian, we generalize earlier definitions of density-matrix topology to generic many-body systems with strong correlations. We point out that the robustness of topological order, defined as a pattern of long-range entanglement, depends crucially on the perturbations under consideration. While it is intrinsically protected against local perturbations of arbitrary strength in an infinite closed quantum system, purely local perturbations can destroy topological order in open systems coupled to baths if the coupling is sufficiently strong. We discuss our classification scheme using the finite-temperature quantum Hall states and point out that the classical Hall effect can be understood as a finite-temperature topological phase.
A quantum Ising model in a transverse field is of the simplest quantum many-body systems used for studying universal properties of quantum phase transitions. Interestingly, it is well known that such phase transitions can be mapped to topological phase transitions in toric code models. Therefore, one can expect that well-known properties of the transverse Ising model are used for characterizing topological phase transitions in toric code models. In this paper, we consider the magnetization of the Ising model and show, while it is a local order parameter, it is mapped to a nonlocal order parameter on the topological side. Consequently, we introduce a string order parameter for topological phase transitions in toric code models defined on different lattices with different dimensions in the presence of a uniform magnetic field. Since such a string order parameter is dual of the Ising order parameter with well-known properties, it leads to a topological (nonlocal) characterization of phase transition in the toric code model in uniform field. Our results show that although topological phase transitions do not follow a symmetry-breaking mechanism, it might still be possible to use concepts of symmetry-breaking phase transitions for topological ones by finding suitable mappings.