New Forms of Ergodicity Breaking in Quantum Many-Body Systems

Total Page:16

File Type:pdf, Size:1020Kb

Load more

New Forms of Ergodicity Breaking in Quantum Many-Body Systems Sanjay Moudgalya a dissertation presented to the faculty of princeton university in candidacy for the degree of Doctor of Philosophy recommended for acceptance by the Department of Physics Adviser: B. Andrei Bernevig September 2020 © Copyright by Sanjay Moudgalya, 2020. All rights reserved. Abstract Generic isolated interacting quantum systems are believed to be ergodic, i.e. any simple initial state evolves to a thermal state at late-times, forming a barrier to the protection of quantum information. A sufficient condition for the thermalization of initial states in an isolated quantum system is the Eigenstate Thermalization Hypothesis (ETH). A complete breakdown of ETH is well-known in two kinds of systems: Integrable and Many-Body Localized, where none of the eigenstates satisfy it. In this dissertation, we introduce two new mechanisms of ergodicity breaking: Quantum Scars, and Hilbert Space Fragmentation. In both these cases, ETH-violating eigenstates coexist with ETH-satisfying ones, and thus the fate of an initial state under time-evolution depends on the properties of eigenstates it has weights on. We obtain the first analytical examples of quantum scars by solving for several excited states in a family of non-integrable quantum systems in one dimension: the AKLT models. These exact eigenstates include an infinite tower of states from the ground state to the highest excited state. The states in the middle of the spectrum obey a logarithmic scaling of entanglement entropy with system size, contrary to the volume-law scaling predicted by ETH. We further show the close connections between such quantum scarred models and Parent Hamiltonians of Matrix Product States, as well as connections to the phenomenon of eta-pairing known in the context of superconductivity in Hubbard models. Hilbert space fragmentation occurs in constrained systems with a center-of-mass or dipole moment conservation law, which naturally arises within a Landau level in quantum Hall systems or in systems subjected to a large electric field limit. We show that the Hilbert space of such systems fractures into several dynamically disconnected Krylov subspaces that are not distinguished by simple symmetries, and thus constitute a violation of conventional ETH. We show that ETH can be modified to apply to each connected subspace separately, and that large integrable and non-integrable subspaces can co-exist within the same system. iii Contents Abstract iii 0 Prelimanaries 1 0.1 Quantum Thermalization ....................... 6 0.2 Matrix Product States and Entanglement . 12 0.3 Outline of this thesis ......................... 18 1 Excited States of the AKLT Model 21 1.1 Introduction .............................. 21 1.2 The Spin-1 AKLT Model ....................... 23 1.3 Exact States .............................. 27 1.4 Exact Low energy Excited states of the Spin-1 AKLT Model . 29 1.5 Mid-Spectrum Exact States ..................... 35 1.6 1D Spin-S AKLT Models with S > 1 . 43 1.7 Conclusion ............................... 52 2 Entanglement of Quasiparticle Excited States 56 2.1 Introduction .............................. 56 2.2 MPS and MPO in the AKLT models . 59 2.3 MPO × MPS ............................. 66 2.4 Single-mode Excitations ....................... 74 2.5 Beyond Single-Mode Excitations ................... 81 2.6 Tower of States ............................ 87 2.7 Implications for ETH ......................... 92 2.8 Entanglement Spectra Degenarcies and Finite-Size Effects . 94 2.9 Conclusion ............................... 104 3 Quantum Scars from Matrix Product States 107 3.1 Introduction .............................. 107 3.2 MPS description of Quasiparticles . 109 3.3 Parent Hamiltonian . 113 3.4 Quantum Scarred Hamiltonians . 117 iv 3.5 New Families of AKLT-like Quantum Scarred Hamiltonians . 125 3.6 New Type of Quantum Scars: Two-Site Quasiparticle Operators . 129 3.7 Conclusions .............................. 139 4 Quantum Scars from Spectrum Generating Algebras 141 4.1 Introduction .............................. 141 4.2 Review of η-pairing in the Hubbard Model . 144 4.3 η-pairing on arbitrary graphs . 146 4.4 Examples of η-pairing . 149 4.5 Quantum Many-Body Scars from the Hubbard Model . 152 4.6 Connections to Quantum Scars . 157 4.7 D dimensions ............................. 159 4.8 RSGA and Quantum Scarred Models . 162 4.9 Conclusions .............................. 166 5 Hilbert Space Fragmentation 168 5.1 Introduction .............................. 168 5.2 Model and its symmetries . 170 5.3 Hamiltonian at 1/2 filling . 174 5.4 Krylov Fracture ............................ 178 5.5 Integrable subspaces . 184 5.6 Non-integrable subspaces and Krylov-Restricted ETH . 196 5.7 Quasilocalization from Thermalization . 200 5.8 Conclusions and Open Questions . 205 6 Approximate Quantum Scars in a Fractured Model 208 6.1 Introduction .............................. 208 6.2 Effective Spin-Chains and Constrained Hilbert spaces . 210 6.3 Symmetries and Non-integrability of the Effective Hamiltonians . 224 6.4 Quantum Many-Body Scars . 226 6.5 Forward Scattering Approximation . 228 6.6 Conclusions .............................. 236 Appendix A Algebra of dimers 238 A.1 Commutation relations . 238 A.2 Dimer basis states and scattering rules for the spin-1 AKLT model 240 A.3 Dimer basis scattering rules for spin-S AKLT basis states . 243 Appendix B Review of Matrix Product Operators 248 B.1 Matrix Product Operators . 248 B.2 Jordan normal form of block upper triangular matrices . 253 v Appendix C Embedding Quasiparticles using MPS Subspaces 259 C.1 Total Angular Momentum Eigenstates . 260 C.2 Examples of A and B subspaces for the AKLT-like MPS . 260 C.3 Single-Site Quasiparticle Exact Eigenstates in the MPS Language 262 C.4 SU(2) Multiplet of the Spin-2 Magnon for the AKLT chain . 265 C.5 Examples of A and Be subspaces for the Potts-like MPS . 267 Appendix D Eta pairing and RSGAs 269 D.1 Useful Identities ............................ 269 D.2 η-pairing with disorder and spin-orbit coupling . 270 D.3 Tower of States from (Restricted) Spectrum Generating Algebras 271 Appendix E Physical Origins of the Pair-Hopping Hamiltonian 275 E.1 Bloch Many-Body Localization . 276 E.2 Quantum Hall Effect . 287 References 315 vi Listing of figures 1.1 Level Statistics of the spin-1 AKLT Model . 25 1.2 AKLT Ground state. The big circles are physical spin-1 s and the smaller circles within the spin-1 s are spin-1/2 Schwinger bosons. Symmetric combinations of the Schwinger bosons on each site form the physical spin-1. The lines joining the Schwinger bosons repre- sent singlets. jGi with periodic boundary conditions . 25 1.3 Arovas A State Dimer Configuration . 32 1.4 Arovas B State Dimer Configurations . 33 1.5 Spin-2 Magnon Dimer Configuration . 35 1.6 Tower of States Dimer Configuration . 37 1.7 Destructive Interference of Spin-2 Magnons . 39 1.8 Position of the AKLT tower of states in the spectrum . 41 1.9 Spin-2 AKLT Ground State Dimer Configuration . 45 1.10 Spin-2 AKLT Arovas B State Dimer Configurations . 47 1.11 Spin-4 Magnon of Spin-2 AKLT Model . 51 1.12 Scattering Rules of the Spin-1 AKLT Model . 55 2.1 Spin-1 AKLT Ground State ..................... 59 2.2 Spin-2 AKLT Ground State ..................... 61 2.3 Entanglement entropy of AKLT eigenstates in the quantum number sectors with scar states ........................ 92 2.4 Entanglement Spectra of AKLT Towers of States . 102 5.1 Evidence of Krylov-Restricted ETH . 197 5.2 Quasilocalization from Thermalization . 201 6.1 Level Statistics of the ν = 2=5 Krylov subspace of the pair-hopping Hamiltonian .............................. 223 6.2 Hilbert Space Graph for the ν = 2=5 Krylov subspace of pair- hopping Hamiltonian . 224 6.3 Anomalous Dynamics of the Pair-Hopping Hamiltonian with ν = 2=5229 6.4 Approximate Scars of the Pair-Hopping Hamiltonian with ν = 2=5 231 vii 6.5 Stability of the ν = 2=5 scars . 233 A.1 Two types of singlet configurations around a bond fi; jg. 241 A.2 Types of non-singlet configurations around a bond fi; jg . 242 A.3 Scattering of S = 2 dimer configurations . 244 E.1 Bloch Many-Body Localization . 285 viii To my family, ix Acknowledgments This thesis would have not been possible without the enormous support of a lot of people, and I am woefully short of words to express my gratitude. Firstly, I would like to thank my advisor Andrei Bernevig for his guidance, encouragement, and support. Andrei’s vision and strong intuition shaped this thesis. His infectious enthusiasm and optimism taught me that physics could be a lot of fun, and his high standards of rigor constantly motivated me and greatly enhanced the clarity of my thoughts and scientific writing. I am also indebted to him for generously giving me the freedom and funding to pursue my own ideas and collaborations, due to which my grad school experience turned out to be exactly as I wanted it to be. Next, I would like to thank Nicolas Regnault, who played a role akin to my primary advisor’s. Nicolas’ humor and encouragement has made working on hard problems exciting. His patient comments have greatly improved my writing, pre- sentation, and computational skills. I am also extremely grateful for his invitation to spend a summer in ENS Paris, which has been one of the most fun and pro- ductive times during graduate school. I am also deeply indebted to Shivaji Sondhi for his guidance and for the many collaborations. Working closely with him on a wide variety of problems has been an incredible learning experience and helped me broaden my view of physics. In addition, his constant encouragement and close supervision right from the beginning has made navigating my way through grad school easier. I have further benefited a lot from several others in the physics community. I would particularly like to express my immense gratitude to Frank Pollmann for introducing me to research in condensed matter physics, and for his extraordinary support during graduate school applications without which this thesis would not exist.
Recommended publications
  • Ergodicity, Decisions, and Partial Information

    Ergodicity, Decisions, and Partial Information

    Ergodicity, Decisions, and Partial Information Ramon van Handel Abstract In the simplest sequential decision problem for an ergodic stochastic pro- cess X, at each time n a decision un is made as a function of past observations X0,...,Xn 1, and a loss l(un,Xn) is incurred. In this setting, it is known that one may choose− (under a mild integrability assumption) a decision strategy whose path- wise time-average loss is asymptotically smaller than that of any other strategy. The corresponding problem in the case of partial information proves to be much more delicate, however: if the process X is not observable, but decisions must be based on the observation of a different process Y, the existence of pathwise optimal strategies is not guaranteed. The aim of this paper is to exhibit connections between pathwise optimal strategies and notions from ergodic theory. The sequential decision problem is developed in the general setting of an ergodic dynamical system (Ω,B,P,T) with partial information Y B. The existence of pathwise optimal strategies grounded in ⊆ two basic properties: the conditional ergodic theory of the dynamical system, and the complexity of the loss function. When the loss function is not too complex, a gen- eral sufficient condition for the existence of pathwise optimal strategies is that the dynamical system is a conditional K-automorphism relative to the past observations n n 0 T Y. If the conditional ergodicity assumption is strengthened, the complexity assumption≥ can be weakened. Several examples demonstrate the interplay between complexity and ergodicity, which does not arise in the case of full information.
  • On the Continuing Relevance of Mandelbrot's Non-Ergodic Fractional Renewal Models of 1963 to 1967

    On the Continuing Relevance of Mandelbrot's Non-Ergodic Fractional Renewal Models of 1963 to 1967

    Nicholas W. Watkins On the continuing relevance of Mandelbrot's non-ergodic fractional renewal models of 1963 to 1967 Article (Published version) (Refereed) Original citation: Watkins, Nicholas W. (2017) On the continuing relevance of Mandelbrot's non-ergodic fractional renewal models of 1963 to 1967.European Physical Journal B, 90 (241). ISSN 1434-6028 DOI: 10.1140/epjb/e2017-80357-3 Reuse of this item is permitted through licensing under the Creative Commons: © 2017 The Author CC BY 4.0 This version available at: http://eprints.lse.ac.uk/84858/ Available in LSE Research Online: February 2018 LSE has developed LSE Research Online so that users may access research output of the School. Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. You may freely distribute the URL (http://eprints.lse.ac.uk) of the LSE Research Online website. Eur. Phys. J. B (2017) 90: 241 DOI: 10.1140/epjb/e2017-80357-3 THE EUROPEAN PHYSICAL JOURNAL B Regular Article On the continuing relevance of Mandelbrot's non-ergodic fractional renewal models of 1963 to 1967? Nicholas W. Watkins1,2,3 ,a 1 Centre for the Analysis of Time Series, London School of Economics and Political Science, London, UK 2 Centre for Fusion, Space and Astrophysics, University of Warwick, Coventry, UK 3 Faculty of Science, Technology, Engineering and Mathematics, Open University, Milton Keynes, UK Received 20 June 2017 / Received in final form 25 September 2017 Published online 11 December 2017 c The Author(s) 2017. This article is published with open access at Springerlink.com Abstract.
  • Ergodicity and Metric Transitivity

    Ergodicity and Metric Transitivity

    Chapter 25 Ergodicity and Metric Transitivity Section 25.1 explains the ideas of ergodicity (roughly, there is only one invariant set of positive measure) and metric transivity (roughly, the system has a positive probability of going from any- where to anywhere), and why they are (almost) the same. Section 25.2 gives some examples of ergodic systems. Section 25.3 deduces some consequences of ergodicity, most im- portantly that time averages have deterministic limits ( 25.3.1), and an asymptotic approach to independence between even§ts at widely separated times ( 25.3.2), admittedly in a very weak sense. § 25.1 Metric Transitivity Definition 341 (Ergodic Systems, Processes, Measures and Transfor- mations) A dynamical system Ξ, , µ, T is ergodic, or an ergodic system or an ergodic process when µ(C) = 0 orXµ(C) = 1 for every T -invariant set C. µ is called a T -ergodic measure, and T is called a µ-ergodic transformation, or just an ergodic measure and ergodic transformation, respectively. Remark: Most authorities require a µ-ergodic transformation to also be measure-preserving for µ. But (Corollary 54) measure-preserving transforma- tions are necessarily stationary, and we want to minimize our stationarity as- sumptions. So what most books call “ergodic”, we have to qualify as “stationary and ergodic”. (Conversely, when other people talk about processes being “sta- tionary and ergodic”, they mean “stationary with only one ergodic component”; but of that, more later. Definition 342 (Metric Transitivity) A dynamical system is metrically tran- sitive, metrically indecomposable, or irreducible when, for any two sets A, B n ∈ , if µ(A), µ(B) > 0, there exists an n such that µ(T − A B) > 0.
  • Dynamical Systems and Ergodic Theory

    Dynamical Systems and Ergodic Theory

    MATH36206 - MATHM6206 Dynamical Systems and Ergodic Theory Teaching Block 1, 2017/18 Lecturer: Prof. Alexander Gorodnik PART III: LECTURES 16{30 course web site: people.maths.bris.ac.uk/∼mazag/ds17/ Copyright c University of Bristol 2010 & 2016. This material is copyright of the University. It is provided exclusively for educational purposes at the University and is to be downloaded or copied for your private study only. Chapter 3 Ergodic Theory In this last part of our course we will introduce the main ideas and concepts in ergodic theory. Ergodic theory is a branch of dynamical systems which has strict connections with analysis and probability theory. The discrete dynamical systems f : X X studied in topological dynamics were continuous maps f on metric spaces X (or more in general, topological→ spaces). In ergodic theory, f : X X will be a measure-preserving map on a measure space X (we will see the corresponding definitions below).→ While the focus in topological dynamics was to understand the qualitative behavior (for example, periodicity or density) of all orbits, in ergodic theory we will not study all orbits, but only typical1 orbits, but will investigate more quantitative dynamical properties, as frequencies of visits, equidistribution and mixing. An example of a basic question studied in ergodic theory is the following. Let A X be a subset of O+ ⊂ the space X. Consider the visits of an orbit f (x) to the set A. If we consider a finite orbit segment x, f(x),...,f n−1(x) , the number of visits to A up to time n is given by { } Card 0 k n 1, f k(x) A .
  • Exploiting Ergodicity in Forecasts of Corporate Profitability

    Exploiting Ergodicity in Forecasts of Corporate Profitability

    Exploiting ergodicity in forecasts of corporate profitability Philipp Mundt, Simone Alfarano and Mishael Milakovic Working Paper No. 147 March 2019 b B A M BAMBERG E CONOMIC RESEARCH GROUP 0 k* k BERG Working Paper Series Bamberg Economic Research Group Bamberg University Feldkirchenstraße 21 D-96052 Bamberg Telefax: (0951) 863 5547 Telephone: (0951) 863 2687 [email protected] http://www.uni-bamberg.de/vwl/forschung/berg/ ISBN 978-3-943153-68-2 Redaktion: Dr. Felix Stübben [email protected] Exploiting ergodicity in forecasts of corporate profitability∗ Philipp Mundty Simone Alfaranoz Mishael Milaković§ Abstract Theory suggests that competition tends to equalize profit rates through the pro- cess of capital reallocation, and numerous studies have confirmed that profit rates are indeed persistent and mean-reverting. Recent empirical evidence further shows that fluctuations in the profitability of surviving corporations are well approximated by a stationary Laplace distribution. Here we show that a parsimonious diffusion process of corporate profitability that accounts for all three features of the data achieves better out-of-sample forecasting performance across different time horizons than previously suggested time series and panel data models. As a consequence of replicating the empirical distribution of profit rate fluctuations, the model prescribes a particular strength or speed for the mean-reversion of all profit rates, which leads to superior forecasts of individual time series when we exploit information from the cross-sectional collection of firms. The new model should appeal to managers, an- alysts, investors and other groups of corporate stakeholders who are interested in accurate forecasts of profitability.
  • STABLE ERGODICITY 1. Introduction a Dynamical System Is Ergodic If It

    STABLE ERGODICITY 1. Introduction a Dynamical System Is Ergodic If It

    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 41, Number 1, Pages 1{41 S 0273-0979(03)00998-4 Article electronically published on November 4, 2003 STABLE ERGODICITY CHARLES PUGH, MICHAEL SHUB, AND AN APPENDIX BY ALEXANDER STARKOV 1. Introduction A dynamical system is ergodic if it preserves a measure and each measurable invariant set is a zero set or the complement of a zero set. No measurable invariant set has intermediate measure. See also Section 6. The classic real world example of ergodicity is how gas particles mix. At time zero, chambers of oxygen and nitrogen are separated by a wall. When the wall is removed, the gasses mix thoroughly as time tends to infinity. In contrast think of the rotation of a sphere. All points move along latitudes, and ergodicity fails due to existence of invariant equatorial bands. Ergodicity is stable if it persists under perturbation of the dynamical system. In this paper we ask: \How common are ergodicity and stable ergodicity?" and we propose an answer along the lines of the Boltzmann hypothesis { \very." There are two competing forces that govern ergodicity { hyperbolicity and the Kolmogorov-Arnold-Moser (KAM) phenomenon. The former promotes ergodicity and the latter impedes it. One of the striking applications of KAM theory and its more recent variants is the existence of open sets of volume preserving dynamical systems, each of which possesses a positive measure set of invariant tori and hence fails to be ergodic. Stable ergodicity fails dramatically for these systems. But does the lack of ergodicity persist if the system is weakly coupled to another? That is, what happens if you have a KAM system or one of its perturbations that refuses to be ergodic, due to these positive measure sets of invariant tori, but somewhere in the universe there is a hyperbolic or partially hyperbolic system weakly coupled to it? Does the lack of egrodicity persist? The answer is \no," at least under reasonable conditions on the hyperbolic factor.
  • Invariant Measures and Arithmetic Unique Ergodicity

    Invariant Measures and Arithmetic Unique Ergodicity

    Annals of Mathematics, 163 (2006), 165–219 Invariant measures and arithmetic quantum unique ergodicity By Elon Lindenstrauss* Appendix with D. Rudolph Abstract We classify measures on the locally homogeneous space Γ\ SL(2, R) × L which are invariant and have positive entropy under the diagonal subgroup of SL(2, R) and recurrent under L. This classification can be used to show arithmetic quantum unique ergodicity for compact arithmetic surfaces, and a similar but slightly weaker result for the finite volume case. Other applications are also presented. In the appendix, joint with D. Rudolph, we present a maximal ergodic theorem, related to a theorem of Hurewicz, which is used in theproofofthe main result. 1. Introduction We recall that the group L is S-algebraic if it is a finite product of algebraic groups over R, C,orQp, where S stands for the set of fields that appear in this product. An S-algebraic homogeneous space is the quotient of an S-algebraic group by a compact subgroup. Let L be an S-algebraic group, K a compact subgroup of L, G = SL(2, R) × L and Γ a discrete subgroup of G (for example, Γ can be a lattice of G), and consider the quotient X =Γ\G/K. The diagonal subgroup et 0 A = : t ∈ R ⊂ SL(2, R) 0 e−t acts on X by right translation. In this paper we wish to study probablilty measures µ on X invariant under this action. Without further restrictions, one does not expect any meaningful classi- fication of such measures. For example, one may take L = SL(2, Qp), K = *The author acknowledges support of NSF grant DMS-0196124.
  • Weak Ergodicity Breaking from Quantum Many-Body Scars

    Weak Ergodicity Breaking from Quantum Many-Body Scars

    This is a repository copy of Weak ergodicity breaking from quantum many-body scars. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/130860/ Version: Accepted Version Article: Turner, CJ orcid.org/0000-0003-2500-3438, Michailidis, AA, Abanin, DA et al. (2 more authors) (2018) Weak ergodicity breaking from quantum many-body scars. Nature Physics, 14 (7). pp. 745-749. ISSN 1745-2473 https://doi.org/10.1038/s41567-018-0137-5 © 2018 Macmillan Publishers Limited. This is an author produced version of a paper accepted for publication in Nature Physics. Uploaded in accordance with the publisher's self-archiving policy. Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ Quantum many-body scars C. J. Turner1, A. A. Michailidis1, D. A. Abanin2, M. Serbyn3, and Z. Papi´c1 1School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom 2Department of Theoretical Physics, University of Geneva, 24 quai Ernest-Ansermet, 1211 Geneva, Switzerland and 3IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria (Dated: November 16, 2017) Certain wave functions of non-interacting quantum chaotic systems can exhibit “scars” in the fabric of their real-space density profile.
  • Lecture Notes Or in Any Other Reasonable Lecture Notes

    Lecture Notes Or in Any Other Reasonable Lecture Notes

    Ergodicity Economics Ole Peters and Alexander Adamou 2018/06/30 at 12:11:59 1 Contents 1 Tossing a coin4 1.1 The game...............................5 1.1.1 Averaging over many trials.................6 1.1.2 Averaging over time.....................7 1.1.3 Expectation value......................9 1.1.4 Time average......................... 14 1.1.5 Ergodic observables..................... 19 1.2 Rates................................. 23 1.3 Brownian motion........................... 24 1.4 Geometric Brownian motion..................... 28 1.4.1 Itˆocalculus.......................... 31 2 Decision theory 33 2.1 Models and science fiction...................... 34 2.2 Gambles................................ 34 2.3 Repetition and wealth evolution................... 36 2.4 Growth rates............................. 39 2.5 The decision axiom.......................... 41 2.6 The expected-wealth and expected-utility paradigms....... 45 2.7 General dynamics........................... 48 2.7.1 Why discuss general dynamics?............... 48 2.7.2 Technical setup........................ 50 2.7.3 Dynamic from a utility function.............. 54 2.7.4 Utility function from a dynamic.............. 56 2.8 The St Petersburg paradox..................... 59 2.9 The insurance puzzle......................... 65 2.9.1 Solution in the time paradigm............... 69 2.9.2 The classical solution of the insurance puzzle....... 71 3 Populations 72 3.1 Every man for himself........................ 73 3.1.1 Log-normal distribution................... 73 3.1.2 Two growth rates....................... 74 3.1.3 Measuring inequality..................... 75 3.1.4 Wealth condensation..................... 76 3.1.5 Rescaled wealth........................ 77 3.1.6 u-normal distributions and Jensen’s inequality...... 78 3.1.7 power-law resemblance.................... 79 3.2 Finite populations.......................... 80 3.2.1 Sums of log-normals....................
  • Recent Progress on the Quantum Unique Ergodicity Conjecture

    Recent Progress on the Quantum Unique Ergodicity Conjecture

    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 48, Number 2, April 2011, Pages 211–228 S 0273-0979(2011)01323-4 Article electronically published on January 10, 2011 RECENT PROGRESS ON THE QUANTUM UNIQUE ERGODICITY CONJECTURE PETER SARNAK Abstract. We report on some recent striking advances on the quantum unique ergodicity, or QUE conjecture, concerning the distribution of large frequency eigenfunctions of the Laplacian on a negatively curved manifold. The account falls naturally into two categories. The first concerns the general conjecture where the tools are more or less limited to microlocal analysis and the dynamics of the geodesic flow. The second is concerned with arithmetic such manifolds where tools from number theory and ergodic theory of flows on homogeneous spaces can be combined with the general methods to resolve the basic conjecture as well as its holomorphic analogue. Our main emphasis is on the second category, especially where QUE has been proven. This note is not meant to be a survey of these topics, and the discussion is not chronological. Our aim is to expose these recent developments after introducing the necessary backround which places them in their proper context. 2. The general QUE conjecture In Figure 1 the domains ΩE, ΩS,andΩB are an ellipse,astadium and a Barnett billiard table, respectively. Superimposed on these are the densities of a consecutive sequence of high frequency eigenfunctions (states, modes) of the Laplacian. That is, they are solutions to ⎧ ⎪ ⎪ φj + λj φj = 0 in Ω , ⎪ ⎨⎪ φ = 0 (Dirichlet boundary conditions), (0) ∂Ω ⎪ ⎪ ⎪ 2 ⎩ |φj | dxdy =1. Ω ∂2 ∂2 ≤ ··· Here = divgrad = ∂x2 + ∂y2 , λ1 <λ2 λ3 are the eigenvalues, and the eigenfunctions are normalized to have unit L2-norm.
  • Quantum Many-Body Scars Or Non-Ergodic Quantum Dynamics in Highly Excited States of a Kinematically Constrained Rydberg Chain

    Quantum Many-Body Scars Or Non-Ergodic Quantum Dynamics in Highly Excited States of a Kinematically Constrained Rydberg Chain

    Quantum many-body scars or Non-ergodic Quantum Dynamics in Highly Excited States of a Kinematically Constrained Rydberg Chain Christopher J. Turner1, A. A. Michailidis1, D. A. Abanin2, M. Serbyn3, Z. Papi´c1 1School of Physics and Astronomy, University of Leeds 2Department of Theoretical Physics, University of Geneva 3IST Austria 15th December 2017 Lancaster, NQM2 arXiv:1711.03528 Outline What is a quantum scar? 1.0 L = 28 An experimental phenomena L = 32 0.8 2 i | ) t ( 0.6 2 Z | 2 0.4 4 Z Exact L Why is it happening? | h 2 0.2 FSA i | ψ 2 0.0 | n 0 10 20 30 t | h 0 What else is going on? 2 L 2 i | ψ 1 | n | h 0 0 10 20 30 n Quantum scars I First discussed by Heller 1984 in quantum stadium billiards. I Here, classically unstable periodic orbits of the stadium billiards (right) scarring a wavefunction (left). I One might expect unstable classical period orbits to be lost in the transition to quantum mechanics as the particle becomes \blurred". I This model is quantum ergodic but not quantum unique ergodic1. Think eigenstate thermalisation for all eigenstates vs. almost all eigenstates. 1Hassell 2010. ARTICLE doi:10.1038/nature24622 Probing many-body dynamics on a 51-atom quantum simulator Hannes Bernien1, Sylvain Schwartz1,2, Alexander Keesling1, Harry Levine1, Ahmed Omran1, Hannes Pichler1,3, Soonwon Choi1, Alexander S. Zibrov1, Manuel Endres4, Markus Greiner1, Vladan Vuletić2 & Mikhail D. Lukin1 Controllable, coherent many-body systems can provide insights into the fundamental properties of quantum matter, enable the realization of new quantum phases and could ultimately lead to computational systems that outperform existing computers based2 on classical approaches.
  • Geometry and Dynamics in the Fractional Quantum Hall Effect: from Graviton Oscillation to Quantum Many-Body Scars

    Geometry and Dynamics in the Fractional Quantum Hall Effect: from Graviton Oscillation to Quantum Many-Body Scars

    Geometry and dynamics in the fractional quantum Hall effect: from graviton oscillation to quantum many-body scars Zlatko Papic Anyons in Quantum Many-Body Systems, MPI PKS, Dresden, 21/01/2019 Historical timeline of quantum Hall effects Leinaas/Myrheim Moore and Read statistics in 2D Laughlin's theory introduce non-Abelian statistics of FQHE Thouless, [Moore, Read, Nucl. Phys. B. 360, 362 (1991)] Kosterlitz, Haldane Topological order IQHE/FQHE in graphene Integer QHE [K. Novoselov, P. Kim, E. Andrei] [Klitzing, Dorda, Pepper, X-G Wen PRL 45, 494 (1980)] 2005- 1982 mid 1980s 2003 2009 2012- 1977 1980 1983 1989 1991 2005- 2006 Majorana Fractional QHE Composite fermions zero modes (> 100 observed Jain [Delft, Princeton, states so far) TKNN Weizmann, [Tsui, Stormer, Gossard, Topological insulators Copenhagen...] PRL 48, 1559 (1982)] [Haldane; Kane, Mele; Bernevig, Hughes, Zhang; ...] FCIs, SPTs,... Numerical work Anyons (Haldane & Rezayi) confirms Kitaev's simpler models of Wilczek Laughlin's theory topological physics in lattice/spin systems Historical timeline of quantum Hall effects Leinaas/Myrheim Moore and Read statistics in 2D Laughlin's theory introduce non-Abelian statistics of FQHE Thouless, [Moore, Read, Nucl. Phys. B. 360, 362 (1991)] Kosterlitz, Haldane Topological order IQHE/FQHE in graphene Integer QHE [K. Novoselov, P. Kim, E. Andrei] [Klitzing, Dorda, Pepper, X-G Wen PRL 45, 494 (1980)] Q: FQHE is a mature subject (nearing 402005- years). mid 1980s 2009 2012- What1982 is there left to understand about2003 FQHE? 1977 1980 1983 1989 1991 2005- 2006 Majorana Fractional QHE Composite fermions zero modes (> 100 observed Jain [Delft, Princeton, states so far) TKNN Weizmann, [Tsui, Stormer, Gossard, Topological insulators Copenhagen...] PRL 48, 1559 (1982)] [Haldane; Kane, Mele; Bernevig, Hughes, Zhang; ...] FCIs, SPTs,..