# Emergence and Entanglement

Conference Date:
Tuesday, May 25, 2010 (All day) to Saturday, May 29, 2010 (All day)
Scientific Areas:
Condensed Matter

In recent years, there has been considerable interest in quantum systems which show novel behavior of the whole system emerging from the collective behavior and interaction of the constituent local degrees of freedom.  The novel phenomena include fractional quantum number, fractional statistics, topologically protected gapless bulk/edge excitations, etc. It appears that the long-range quantum entanglement in those system plays a critical role in generating those novel phenomena.  The goal of this workshop is the study of the emergent and entanglement properties of quantum many-body systems. Possible topics of the workshop will be the emergence of gauge fields, fractionalized particles and symmetries as low energy properties of condensed matter systems, the study of entanglement in many body systems and its relationships with novel quantum phase transitions, the study of the AdS/CFT correspondence to understand properties of systems of strongly interacting fermions, the study of topological order, topological insulators, and quantum spin liquids.

Ian Affleck, University of British Columbia

Sougato Bose, University College of London

Claudio Chamon, Boston University

Ignacio Cirac, Max Planck Institute

Jens Eisert, University of Potsdam

Paul Fendley, University of Virginia

Matthew Fisher, California Institute of Technology

Eduardo Fradkin,  University of Illinois at Urbana-Champaign

Michel Gingras, University of Waterloo

Taylor Hughes, University of Illinois at Urbana-Champaign

Shamit Kachru, KITP/Stanford

Andreas Karch, University of Washington

Yong Baek Kim, University of Toronto

Israel Klich, University of Virginia

Karyn Le Hur, Yale University

Sung-Sik Lee, McMaster University

Anthony Leggett, University of Illinois at Urbana-Champaign

John McGreevy, Massachusetts Institute of Technology

Roger Melko, University of Waterloo

Max Metlitski, Harvard University

Chetan Nayak, Microsoft Station Q and University of California, Santa Barbara

Gil Refael, California Institute of Technology

Subir Sachdev, Harvard University

Omid Saremi, McGill University

Barbara Terhal, IBM

Frank Verstraete, Universität Wien

Guifre Vidal, University of Queensland

Xiao-Gang Wen, Massachusetts Institute of Technology

Cenke Xu, Harvard University

Paolo Zanardi, University of Southern California

Dorit Aharonov, Hebrew University*

Eva Silverstein, KITP/Stanford University*

*participation to be confirmed

 Ian Affleck, University of British Columbia Impurity entanglement entropy and the Kondo model The entanglement entropy in conformal field theory was predicted to include a boundary term which depends on the choice of conformally invariant boundary condition. We have studied this effect in the Kondo model of a magnetic impurity in a metal, which exhibits a renormalization group flow between conformally invariant fixed points. Sougato Bose, University College of London Entanglement across a separation in spin chains: statics & dynamics In this talk, I will present two schemes which would result in substantial entanglement between distant individual spins of a spin chain. One relies on a global quench of the couplings of a spin chain, while the other relies on a bond quenching at one end. Both of the schemes result in substantial entanglement between the ends of a chain so that such chains could be used as a quantum wire to connect quantum registers. I will also examine the resource of entanglement already existing between separated parts of a many-body system at criticality as the size of the parts and their separation is varied. This form of entanglement displays an interesting scale invariance. Claudio Chamon, Boston University Quantum mechanical and information theoretic view on classical glass transitions Using the mapping of the Fokker-Planck description of classical stochastic dynamics onto a quantum Hamiltonian, we argue that a dynamical glass transition in the former must have a precise definition in terms of a quantum phase transition in the latter. At the dynamical level, the transition corresponds to a collapse of the excitation spectrum at a critical point. At the static level, the transition affects the ground state wavefunction: while in some cases it could be picked up by the expectation value of a local operator, in others the order may be non-local, and impossible to be determined with any local probe. Here we propose instead to use concepts from quantum information theory that are not centered around local order parameters, such as fidelity and entanglement measures. We show that for systems derived from the mapping of classical stochastic dynamics, singularities in the fidelity susceptibility translate directly into singularities in the heat capacity of the classical system. In classical glassy systems with an extensive number of metastable states, we find that the prefactor of the area law term in the entanglement entropy jumps across the transition. We also discuss how entanglement measures can be used to detect a growing correlation length that diverges at the transition.  Finally, we illustrate how static order can be hidden in systems with a macroscopically large number of degenerate equilibrium states by constructing a three dimensional lattice gauge model with only short-range interactions but with a finite temperature continuous phase transition into a massively degenerate phase. Ignacio Cirac, Max Planck Institute Description of many-body systems using MPS, PEPS, and other families of states Matrix Product States (MPS) and their higher dimensional extensions, the Projected Entangled-Pair States (PEPS) can efficiently describe the ground and thermal states of interacting systems with short-range interactions. We will describe some mathematical properties of this families of states, as well as possible extensions. Work in collaboration with N. Schuch, D. Perez-Garcia, M. Sanz, M. Wolf, F. Verstraete and G. Sierra. Jens Eisert, University of Potsdam Grasping quantum many-body systems in terms of tensor networks This talk will be concerned with three new results (or a subset thereof) on the idea of grasping quantum many-body systems in terms of suitable tensor networks, such as finitely correlated states (FCS), tree tensor networks (TTN), projected entangled pair states (PEPS) or entanglement renormalization (MERA). We will first briefly introduce some basic ideas and relate the feasibility of such approaches to entanglement properties and area laws. We will then see that (a) surprisingly, any MERA can be efficiently encoded in a PEPS, hence in a sense unifying these approaches. (b) We will also find that the ground state-manifold of any frustration-free spin-1/2 nearest neighbor Hamiltonian can be completely characterized in terms of tensor networks, how all such ground states satisfy an area law, and in which way such states serve as ansatz states for simulating almost frustration-free systems. (c) The last part will be concerned with using flow techniques to simulate interacting quantum fields with finitely correlated state approaches, and with simulating interacting fermions using efficiently contractible tensor networks. Paul Fendley, University of Virginia From few to many I discuss a class of systems with a very special property: exact results for physical quantities can be found in the many-body limit in terms of the original (bare) parameters in the Hamiltonian. A classic result of this type is Onsager and Yang's formula for the magnetization in the Ising model. I show how analogous results occur in a fermion chain with strong interactions, closely related to the XXZ spin chain. This is done by exploiting a supersymmetry, and noting that certain quantites are independent of finite-size effects. I also discuss how these ideas are related to an interacting generalization of the Kitaev honeycomb model. Matthew Fisher, California Institute of Technology Gapless Spin Liquids in Two Dimensions Many crystalline materials predicted by band theory to be metals are insulators due to strong electron interactions. Both experiment and theory suggest that such Mott-insulators can exhibit exotic gapless spin-liquid ground states, having no magnetic or any other order. Such “critical spin liquids” will possess power law spin correlations which oscillate at various wavevectors. In a sub-class dubbed “Spin Bose-Metals” the singularities reside along surfaces in momentum space, analogous to a Fermi surface but without long-lived quasiparticle excitations. I will describe recent theoretical progress in accessing such states via controlled numerical and analytical studies on quasi-1d model systems. Eduardo Fradkin,  University of Illinois at Urbana-Champaign Quantum Entanglement and Quantum Criticality The entanglement entropy of a pure quantum state of a bipartite system is defined as the von Neumann entropy of the reduced density matrix obtained by tracing over one of the two parts. Critical ground states of local Hamiltonians in one dimension have entanglement that diverges logarithmically in the subsystem size, with a universal coefficient that is is related to the central charge of the associated conformal field theory. In this talk I will discuss the extension of these ideas to two dimensional systems, either at a special quantum critical point or in a topological phase. We find the entanglement entropy for a standard class of z=2 quantum critical points in two spatial dimensions with scale invariant ground state wave functions: in addition to a nonuniversal area law'' contribution proportional to the size of the boundary of the region under observation, there is generically a universal logarithmically divergent correction, and in its absence a universal finite piece is found. This logarithmic term is completely determined by the geometry of the partition into subsystems and the central charge of the field theory that describes the equal-time correlations of the critical wavefunction. On the other hand, in a topological phase there is no such logarithmic term but instead a universal constant term. We will discuss the connection between this universal entanglement entropy and the nature of the topological phase.   Michel Gingras, University of Waterloo Recent developments in the physics of spin ice and related quantum cousin. In the Ho2Ti2O7 and Dy2Ti2O7 magnetic pyrochlore oxides, the Ho and Dy Ising magnetic moments interact via geometrically frustrated effective ferromagnetic coupling. These systems possess and extensive zero entropy related to the extensive entropy of ice water -- hence the name spin ice. The classical ground states of spin ice obey a constraint on each individual tetrahedron of interacting spins -- the so-called "ice rules".  At large distance, the ice-rules can be described by an effective divergent-free field and, therefore, by an emergent classical gauge theory.  In contrast, while it would appear at first sight to relate to the spin ices, the Tb2Ti2O7 material displays properties that much differ from spin ices and the behaviour of that system has largely remained unexplained for over ten years. In this talk, I will review the key features of the (Ho,Dy)2Ti2O7 spin ice materials, discuss the recent experimental results that support the emergent gauge theory description of spin ices and discuss how Tb2Ti2O7 is perhaps a "quantum melted" spin ice. Taylor Hughes, University of Illinois at Urbana-Champaign Entanglement Spectra and Topological Insulators Recent work has explored some aspects of entanglement in topological insulators. Notably, the entanglement spectrum has been shown to mimic certain properties of the low-energy fermionic modes found on real spatial boundaries. I will discuss the many-body entanglement spectrum of topological insulators and show that it matches the expected CFT character structure that has been previously shown to hold in fractional quantum Hall effect ground states. I also present the analysis of a disorder-driven Anderson localization transition in a Chern-Insulator from an analysis of the entanglement spectrum. Interestingly, the disorder-averaged level-spacing statistics of the entanglement spectrum characterizes the system just as well as the statistics of the real energy spectrum, but with the advantage that only the ground state, and not the entire spectrum of excited states, needs to be calculated.  Shamit Kachru, KITP/Stanford An isotropic to anisotropic transition in a fractional quantum Hall state I describe a novel abelian gauge theory in 2+1 dimensions which has surprising theoretical and phenomenological features. The theory has a vanishing coefficient for the square of the electric field $e_i^2$, characteristic of a quantum critical point with dynamical critical exponent $z=2$, and a level-$k$ Chern-Simons coupling, which is marginal at this critical point. For $k=0$, this theory is dual to a free $z=2$ scalar field theory describing a quantum Lifshitz transition, but $k \neq 0$ renders the scalar description non-local. The $k \neq 0$ theory exhibits properties intermediate between the (topological) pure Chern-Simons theory and the scalar theory. For instance, the Chern-Simons term does not make the gauge field massive. Nevertheless, there are chiral edge modes when the theory is placed on a space with boundary, and a non-trivial ground state degeneracy $k^g$ when it is placed on a finite-size Riemann surface of genus $g$. The coefficient of $e_i^2$ is the only relevant coupling; it tunes the system through a quantum phase transition between an isotropic fractional quantum Hall state and an anisotropic fractional quantum Hall state. I describe zero-temperature transport coefficients in both phases and at the critical point, and comment briefly on the relevance of the results to recent experiments. Andreas Karch, University of Washington A particle physicist's perspective on topological insulators The theory of topological insulators will be reviewed in terms familiar to particle theorists. Yong Baek Kim, University of Toronto Spin Liquid and Topological Insulator in Frustrated Magnets Recently several proposals are made for possible spin liquid and topological insulator phases in frustrated magnets. I will review some of these efforts and present some new results. Implications to real materials will also be made. Israel Klich, University of Virginia Cluster expansions and the stability of topological phases. Anyons are a special kind of excitations which are allowed in two dimensional systems, along with fermions and bosons. The topological nature of braiding of non-abelian anyons may allow a realization of quantum computing gates which is immune to noise. While the insensitivity of the such systems to a localized noise source is a built-in feature, an issue of great importance is more subtle: the robustness to slight deformations of the amiltonian describing the phase by perturbations which are locally tiny but are spread over through the entire system. Such will always arise if the realization of the Hamiltonian in a particular system is not quite perfect. The subject of the talk will be a proof of such stability, and the cluster expansion representation of deformed topological states. Karyn Le Hur, Yale University Entanglement and Fluctuations in Many-Body Quantum Systems TBA Sung-Sik Lee, McMaster University Holographic description of quantum field theory The AdS/CFT correspondence has opened the door to understand a class of strongly coupled quantum field theories. Although the original correspondence has been conjectured based on string theory, it is possible that the underlying principle is more general, and a wider class of quantum field theories can be understood through holographic descriptions. In this talk, I will discuss about a prescription to construct holographic theories for general quantum field theories. As an example, I will present a holographic dual theory for the D-dimensional O(N) vector model. The phase transition and critical behaviors of the model are reproduced through the holographic theory. Tony Leggett, University of Illinois at Urbana-Champaign Does entanglement persist at the macroscopic level? The quantum states postulated to occur in situations of the "Schroedinger's Cat" type are essentially N-particle GHZ states with N very large compared to 1,and their observation would thus be particularly compelling evidence for the ubiquity of the phenomenon of entanglement.However,in the traditional quantum measurement literature considerable scepticism has been expresssed about the observability of this kind of "macroscopically entangled" state,primarily because of the puatatively disastrous effect on it of decoherence.In this talk I first examine why much of the literature has grossly overestimated the effects of decoherence,and then review the current experimental situation with respect to such states,as they (may) occur in fullerene diffraction,magnetic biomolecules,quantum-optical systems and Josephson devices;I also consider the prospects for their observation in nanomechanical systems.I conclude by reviewing and the theoretical implications of the experiments of the last decade in this area. John McGreevy, Massachusetts Institute of Technology Strange metal from holography This talk is about a class of non-Fermi liquid metals, identified using the AdS/CFT correspondence. Roger Melko, University of Waterloo Computing Entanglement in Simulations of Quantum Condensed Matter Condensed matter theorists have recently begun exploiting the properties of entanglement as a resource for studying quantum materials.  At the forefront of current efforts is the question of how the entanglement of two subregions in a quantum many-body groundstate scales with the subregion size.  The general belief is that typical groundstates obey the so-called "area law", with entanglement entropy scaling as the boundary between regions.  This has lead theorists to propose that sub-leading corrections to the area law provide new universal quantities at quantum critical points and in exotic quantum phases (i.e. topological Mott insulators).  However, away from one dimension, entanglement entropy is difficult or impossible to calculate exactly, leaving the community in the dark about scaling in all but the simplest non-interacting systems.  In this talk, I will discuss recent breakthroughs in calculating entanglement entropy in two dimensions and higher using advanced quantum Monte Carlo simulation techniques.  We show, for the first time, evidence of leading-order area law scaling in a prototypical model of strongly-interacting quantum spins.  This paves the way for future work in calculating new universal quantities derived from entanglement, in the plethora of real condensed matter systems amenable to numerical simulation. Max Metlitski, Harvard University Entanglement entropy in the O(N) model In recent years the characterization of many-body ground states via the entanglement of their wave-function has attracted a lot of attention. One useful measure of entanglement is provided by the entanglement entropy S. The interest in this quantity has been sparked, in part, by the result that at one dimensional quantum critical points (QCPs) S scales logarithmically with the subsystem size with a universal coefficient related to the central charge of the conformal field theory describing the QCP. On the other hand, in spatial dimension d > 1 the leading contribution to the entanglement entropy scales as the area of the boundary of the subsystem. The coefficient of this "area law" is non-universal. However, in the neighbourhood of a QCP, S is believed to possess subleading universal corrections. In this talk, I will present the first field-theoretic study of entanglement entropy in dimension d > 1 at a stable interacting QCP - the quantum O(N) model. Our results confirm the presence of universal corrections to the entanglement entropy and exhibit a number of surprises such as different epsilon -> 0 limits of the Wilson-Fisher and Gaussian fixed points, violation of large N counting and subtle dependence on replica index. Gil Refael, California Institute of Technology Entanglement entropy and infinite randomness fixed points in disordered magnetic and non-abelian quasi-particle chains Many one dimensional random quantum systems exhibit infinite randomness phases, such as the random singlet phase of the spin-1/2 Heisenberg model. These phases are typically the result of destabilizing systems described by a conformal field theory with disorder.  Interestingly, entanglement entropy in 1d infinite randomness phases also exhibits a universal log scaling with length. In my talk I will touch upon calculating the entanglement entropy for inifinite-randomness phases, as well as describe the exotic infinite randomness phases realized in chains of non-abelian anyon chains. It was speculated that the entanglement entropy of an infinite-randomness phase is associated with the direction of RG flow, just as the c-theorem dictates the direction of RG flows for CFT's. I will also show that the entanglement entropy in disordered non-abelian chains provide the only known counter example. Subir Sachdev, Harvard University Quantum "disordering" magnetic order in insulators, superconductors, and metals The S=1/2 square lattice Heisenberg spin model with nearest neighbor interactions is known to have long-range magnetic order (Neel order) in its ground state. Now imagine perturbing the model so that there is a quantum phase transition to a "disordered" ground state without Neel order. What is the nature of the resulting quantum state ? I will describe a general approach to answering such questions. Similar questions can be asked for wide classes of models of insulators, superconductors, and metals, and I will give a survey of the answers obtained. Omid Saremi, McGill University Mode-Coupling, Hydrodynamic Long-time tails from Anti de Sitter Space TBA Eva Silverstein, KITP/Stanford University *tentatively attending Towards strange metallic holography TBA Barbara Terhal, IBM How to make a low-dimensional thermally stable quantum memory I will discuss the question of thermal stability of a passive quantum memory, or finite-temperature topological order, in two or three spatial dimensions. We will analyze the criteria for thermal stability.  We will present new results on Majorana fermion codes and a new extension of the 2D surface code to three dimensions. Frank Verstraete, Universität Wien A variational wave-function based method for simulating quantum field theories TBA Guifre Vidal, University of Queensland Entanglement renormalization and gauge symmetry A lattice gauge theory is described by a redundantly large vector space that is subject to local constraints, and it can be regarded as the low energy limit of a lattice model with a local symmetry. I will describe a coarse-graining scheme capable of exactly preserving local symmetries. The approach results in a variational ansatz for the ground state(s) and low energy excitations of a lattice gauge theory. This ansatz has built-in local symmetries, which are exploited to significantly reduce simulation costs. I will describe benchmark results in the context of Kitaev’s toric code with a magnetic field or, equivalently, Z2 lattice gauge theory, for lattices with up to 16 x 16 sites (16^2 x 2 = 512 spins) on a torus. Xiao-Gang Wen, Massachusetts Institute of Technology Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order Adiabatic evolutions connect two gapped quantum states in the same phase.  We argue that the adiabatic evolutions are closely related to local unitary transformations which define a equivalence relation.  So the equivalence classes of the local unitary transformations are the universality classes that define the different phases of quantum system. Since local unitary transformations can remove local entanglements, the above equivalence/universality classes correspond to pattern of long range entanglement, which is the essence of topological order.  The local unitary transformation also allows us to define wave function renormalization, where a wave function can flow to a simpler one within the same equivalence/universality class. Using such a setup, we find conditions on the possible fixed-point wave functions where the local unitary transformations have finite dimensions.  The solutions of the conditions allow us to classify this type of topological orders, which include all the string-net states. Cenke Xu, Harvard University Emergent gauge theories in Quantum spin Hall superconductor Josephson arrays and cold atom system We study a superconductor-ferromagnet-superconductor (SC-FM-SC) Josephson junction array deposited on top of a two-dimensional quantum spin Hall (QSH) insulator. The existence of Majorana bound states at the interface between SC and FM gives rise to charge-e tunneling, in addition to the usual charge-2e Cooper pair tunneling, between neighboring superconductor islands. Moreover, because Majorana fermions encode the information of charge number parity, an exact Z_2 gauge structure naturally emerges and leads to many new insulating phases, including a deconfined phase where electrons fractionalize into charge-e bosons and topological defects. We will also discuss the ultracold alkaline earth atoms trapped in optical lattice, which naturally has a SU(N) spin symmetry with N as large as 10. An SU(2)xU(1) gauge theory emerges in a large part of the phase diagram of this system. Paolo Zanardi, University of Southern California Unitary Equilibrations:  Temporal and Hilbert Space  Typicality of Loschmidt Echo Closed quantum systems evolve unitarily and therefore cannot converge in a strong sense to an equilibrium state starting out from a generic pure state. Nevertheless for large system size one observes temporal typicality. Namely, for the overwhelming majority of the time instants, the statistics of observables is practically indistinguishable from an effective equilibrium one. In this talk we will discuss he Loschmidt echo (LE) to study this sort of unitary equilibration after a quench. In particular we ll address the issue of typicality with respect the initial state preparation and the influence of quantum criticality of the long-time probability distribution of LE.