Why Hyperbolic Symmetry?
|
|
- Loren Peters
- 5 years ago
- Views:
Transcription
1 Why Hyperbolic Symmetry? Consider the very trivial case when N = 1 and H = h is a real Gaussian variable of unit variance. To simplify notation set E ɛ = 0 iɛ and z, w C = G(E ɛ ) 2 h = [(h iɛ)(h + iɛ)] 1 h < exp[ih(z z w w) ɛ(z z + w w)] > h dz dw = exp[ 1 2 (z z w w) 2 ɛ(z z + w w)] dz dw. Note that ɛ breaks the hyperbolic symmetry to make the integral well defined.
2 Hyperbolic sigma models and Reinforced walk Quantum Scattering with random impurities - Manhattan Pinball (Beamond, Cardy, Owczarek and Gruzberg, Ludwig, Read) Linearly Edge Reinforced Random Walk - Persi Diaconis Zirnbauer s SUSY sigma model with a hyperbolic target. Anderson like phase transition in 3D. Cycle structure created by Random Adjacent Transpositions on Λ Z d. Universality of Mean Field for macroscopic cycle lengths? (Ueltschi, Chalker, Nahum, Schramm, Toth )
3 Manhattan Pinball Quantum Network Model with random scatterers Motivation: Chalker s network model Integer Quantum Hall. Particle moves on Z 2 along streets with alternating orientations scattered by random obstructions. Equivalent to Unitary evolution with SU(2) bond disorder. Beamond, Cardy, Owczarek, and Gruzberg, Ludwig, Read
4 obstruction Figure: Manhattan Lattice
5 Theorem If p > 1/2 then all trajectories are closed with probability 1. Localization. Proof (Chalker) by percolation. Conjecture: All trajectories are closed for any p > 0 Average loop diameter e c p 2 1. Thus in 2D any randomness produces Localization Mirror model: mirrors at vertices randomly placed at ±45. Fully packed mirrors equivalent to critical percolation. There is always an Extended channel. (Kozma-Sidovaricius)
6
7 Linearly Edge Reinforced Random Walk History dependent walk W n Z d, n Z + : Walk takes nearest neighbor steps and favors edges j, k, Z d, j k = 1, it has visited in the past. Introduced by P. Diaconis while wandering the streets of Paris. He liked to return to streets he had visited in the past. Remarks: Not Markovian but is a superposition of Markov Processes Equivalent to a random walk in a correlated random environment given by statistical mechanics.
8 Definition of Reinforced Random Walk Let C jk (n) = number of times the walk has crossed edge jk up to time n and let β > 0. Prob{W n+1 = k W n = j} = 1 + C jk(n)/β N β, j k = 1. N is the normalization: N β = k (1 + C jk (n)/β), j k = 1 0 < β 1, strong reinforcement, (high temperature) β 1, weak reinforcement, (low temperature) Questions: Is ERRW localized? recurrent? transient?
9 P. Diaconis and D. Coppersmith (1986): ERRW random walk in a random environment. Environment: The rate at which an edge j, j is crossed w j,j > 0 are correlated random variables. (conductances) Distribution of w j,j > 0 is correlated, unbounded and given by an explicit statistical mechanics model. The generator for the RW is a weighted Laplacian D w s t D w s = w j,j (s j s j ) 2, s j R j j =1 Local conductance: w j,j
10 Quantum of particle on Z 3 scattered by impurities Quantum Time Evolution is analyzed via the Green s function: G(E + iɛ; j, k) = [H(ω) E iɛ] 1 (j, k). Average of G(j, k) 2 is the spin-spin correlation in SUSY Statistical mechanics. Spins s j, j Z d Λ are 4 4 super matrices in box Λ. Efetov sigma model target: U(1, 1 2)/[U(1 1) U(1 1)] Study a simpler vector version - Zirnbauer s model.
11 Zirnbauer s SUSY Hyperbolic Model - H (2 2) Sigma constraint : z 2 j x 2 j y 2 j 2η j ξ j = 1 Horospherical parametrization: s j, t j R : x = sinh t e t ( 1 2 s2 + ψψ ), y = e t s, ξ = e t ψ, η = e t ψ, S Λ = i j Λ S ij + ɛ k Λ S ij = cosh(t i t j ) (s i s j ) 2 e t i +t j + ( ψ i ψ j )(ψ i ψ j ) e t i +t j, z k z k = cosh t k + ( 1 2 s2 k + ψ k ψ k ) e t k. Local conductance: w j,j = e t j +t j.
12 Partition function and Ward identity: H (2 2) Z Λ (β, ɛ) = C exp [ βs Λ ] Λ e t j dt j ds j dψ j d ψ j By supersymmetry Z Λ (β, ɛ) 1 all β, ɛ. Ward identity: e α t 0 = e (1 α) t 0, e t 0 = 1. β ρ(e) 2 λ 2, ρ(e) = density of states, λ = disorder
13 Effective Action: Integrate out Grassmann ψ, ψ and s variables. E SUSY ({t j }) = β j j cosh(t j t j ) 1/2 log det D β,ɛ (t) D β,ɛ (t) a finite difference elliptic operator: [s ; D β,ɛ (t) s] Λ = β (j j) et j +t j (s j s j ) 2 + ɛ k Λ et k s 2 k Local conductance = e t j +t j. Saddle Point t s of Effective Action: If t s = 0, as ɛ 0 - Conduction, 3D, when β large If ɛe t s e β as ɛ 0 localization, 2D
14 Spin-Spin correlation: y 0 y k =< s 0 e t 0 s k e t k > (β, ɛ) =< e t 0+t k D β,ɛ (t) 1 (0, k) > SUSY (β, ɛ) random walk in a random environment: In 3D if the local conductance e t j +t j does not fluctuate or t j 0 Then correlation < e t 0+t k D β,ɛ (t) 1 (0, k) > SUSY (β, ɛ) ( β + ɛ) 1 (0, k) is diffusive.
15 Phase Transition in 3D Theorem (Disertori, S., Zirnbauer 10) A) For β 1 and d 3 Q Diffusion: y 0 y k ( β + ɛ) 1 (0, k), k Z 3 B) For 0 < β 1, localization: y 0 y k (β, ɛ) ɛ 1 e k /l(β)
16 Remark: The Hyperbolic sigma model (No Fermions) has No phase transition in 3D - always delocalized. Effective action is convex for all β. (Sp, Zirnbauer, Brydges) Gruzberg-Mirlin (96) Replica analysis of H (2 2) on Bethe Lattice Conjectures: A)There is a multi-fractal transition in 3D B) In 2D exponential localization holds for all β.
17 SUSY Hyperbolic sigma model is essentially equivalent to ERRW Relation made precise by Sabot and Tarres following earlier observations of Kozma, Sznitman, Gawedzki
18 Phase Transition for ERRW in 3D Theorem (Sabot-Tarres) For strong reinforcement, ERRW is recurrent and we have Localization: Prob{ W (t) W (0) R} Ce R/l(β), E W 2 (t) Const and l(β) is the localization length. Theorem (Disertori-Sabot-Tarres) For weak reinforcement, β 1, and d 3, W (t) is Transient, quasi-diffusion.
19 Ideas of Proof in 3D: Ward identities To prove diffusion must show the t field has very small fluctuation for large β: cosh p (t j t k ) (β) Const all j, k, β 1 If F (t, s, ψ, ψ) is 0Sp(2 2) invariant, then. < F > SUSY (β, ɛ) = F (0) Example: < F m > = 1 all m 1. [ F = cosh(t j t k ) + e t j +t k 1 ] 2 (s j s k ) 2 + (ψ j ψ k )(ψ j ψ k )
20 Products of Random Transpositions on Z d. Let Λ = Λ L Z d be a box of side L, P Λ (0) be the Identity permutation of its vertices at time 0. Each adjacent edge (j, j ) Λ, has an independent Poisson clock which rings at rate 1. When the clock rings, we make a transposition j j. P Λ (T ) is the product of these transpositions up to time T. Question: After time T, what is the cycle structure of P Λ (T ) for large L? In 2D are these cycles like those of the Manhattan Pinball T p 2??
21 Theorem (Caputo, Liggett, Richthammer) In Λ L, if T L 2 then P Λ (T ) uniform weight on all permutations of Λ L. Conjecture: For d 3 there is a T independent of L, such that if T > T there are macroscopic cycles of length l 1 > l 2 > l 3... where l j c j L d. Moreover for T > T as L the length distribution is Mean Field - Poisson-Dirichlet (Schramm, Ueltschi) after scaling. Thus l 1 /l 2 reaches its Mean Field distribution long before T = L 2!
22 Theorem The partition function Z Λ (β) for the spin 1/2 Quantum Heisenberg Ferromagnet is Z Λ (β) = tr e β Λ j j S j S j = E Λ (β) [ 2 #cycles ] where E Λ (β) denotes the expectation of the independent Poisson clocks up to time T = β. Macroscopic cycles correspond to Bose-Einstein condensation. Theorem In 2D there are no macroscopic cycles - Mermin-Wagner Remark: PD(2) implies that Z(β) 1 tr e β Λ j j S j S j + Λ j Λ 1 h s j = S 2 e M(β)h S 0 dµ END
Two Classical models of Quantum Dynamics
Two Classical models of Quantum Dynamics Tom Spencer Institute for Advanced Study Princeton, NJ May 1, 2018 Outline: Review of dynamics and localization for Random Schrödinger H on l 2 (Z d ) H = + λv
More informationQuasi-Diffusion in a SUSY Hyperbolic Sigma Model
Quasi-Diffusion in a SUSY Hyperbolic Sigma Model Joint work with: M. Disertori and M. Zirnbauer December 15, 2008 Outline of Talk A) Motivation: Study time evolution of quantum particle in a random environment
More informationDuality, Statistical Mechanics and Random Matrices. Bielefeld Lectures
Duality, Statistical Mechanics and Random Matrices Bielefeld Lectures Tom Spencer Institute for Advanced Study Princeton, NJ August 16, 2016 Overview Statistical mechanics motivated by Random Matrix theory
More informationClassical and Quantum Localization in two and three dimensions
Classical and Quantum Localization in two and three dimensions John Cardy University of Oxford Mathematics of Phase Transitions Warwick, November 2009 This talk is about some mathematical results on physical
More informationWhere Probability Meets Combinatorics and Statistical Mechanics
Where Probability Meets Combinatorics and Statistical Mechanics Daniel Ueltschi Department of Mathematics, University of Warwick MASDOC, 15 March 2011 Collaboration with V. Betz, N.M. Ercolani, C. Goldschmidt,
More informationKosterlitz-Thouless Transition
Heidelberg University Seminar-Lecture 5 SS 16 Menelaos Zikidis Kosterlitz-Thouless Transition 1 Motivation Back in the 70 s, the concept of a phase transition in condensed matter physics was associated
More informationIndecomposability in CFT: a pedestrian approach from lattice models
Indecomposability in CFT: a pedestrian approach from lattice models Jérôme Dubail Yale University Chapel Hill - January 27 th, 2011 Joint work with J.L. Jacobsen and H. Saleur at IPhT, Saclay and ENS Paris,
More informationdisordered topological matter time line
disordered topological matter time line disordered topological matter time line 80s quantum Hall SSH quantum Hall effect (class A) quantum Hall effect (class A) 1998 Nobel prize press release quantum Hall
More informationNematic phase of spin 1 quantum systems
Nematic phase of spin 1 quantum systems Daniel Ueltschi Department of Mathematics, University of Warwick International Conference on Applied Mathematics, Hράκλειo, 17 September 2013 D. Ueltschi (Univ.
More informationNumerical estimates of critical exponents of the Anderson transition. Keith Slevin (Osaka University) Tomi Ohtsuki (Sophia University)
Numerical estimates of critical exponents of the Anderson transition Keith Slevin (Osaka University) Tomi Ohtsuki (Sophia University) Anderson Model Standard model of a disordered system H W c c c V c
More informationNon-magnetic states. The Néel states are product states; φ N a. , E ij = 3J ij /4 2 The Néel states have higher energy (expectations; not eigenstates)
Non-magnetic states Two spins, i and j, in isolation, H ij = J ijsi S j = J ij [Si z Sj z + 1 2 (S+ i S j + S i S+ j )] For Jij>0 the ground state is the singlet; φ s ij = i j i j, E ij = 3J ij /4 2 The
More informationVarious Facets of Chalker- Coddington network model
Various Facets of Chalker- Coddington network model V. Kagalovsky Sami Shamoon College of Engineering Beer-Sheva Israel Context Integer quantum Hall effect Semiclassical picture Chalker-Coddington Coddington
More informationPERCOLATION AND COARSE CONFORMAL UNIFORMIZATION. 1. Introduction
PERCOLATION AND COARSE CONFORMAL UNIFORMIZATION ITAI BENJAMINI Abstract. We formulate conjectures regarding percolation on planar triangulations suggested by assuming (quasi) invariance under coarse conformal
More informationResurgence Structure to All Orders of Multi-bions in Deformed SUSY Quantum Mechanics
Resurgence Structure to All Orders of Multi-bions in Deformed SUSY Quantum Mechanics Toshiaki Fujimori (Keio University) based on arxiv:1607.04205, Phys.Rev. D94 (2016) arxiv:1702.00589, Phys.Rev. D95
More informationWave function multifractality at Anderson localization transitions. Alexander D. Mirlin
Wave function multifractality at Anderson localization transitions Alexander D. Mirlin Forschungszentrum Karlsruhe & Universität Karlsruhe http://www-tkm.physik.uni-karlsruhe.de/ mirlin/ F. Evers, A. Mildenberger
More informationarxiv: v2 [math.pr] 22 Aug 2017
Submitted to the Annals of Probability PHASE TRANSITION FOR THE ONCE-REINFORCED RANDOM WALK ON Z D -LIKE TREES arxiv:1604.07631v2 math.pr] 22 Aug 2017 By Daniel Kious and Vladas Sidoravicius, Courant Institute
More informationContact interactions in string theory and a reformulation of QED
Contact interactions in string theory and a reformulation of QED James Edwards QFT Seminar November 2014 Based on arxiv:1409.4948 [hep-th] and arxiv:1410.3288 [hep-th] Outline Introduction Worldline formalism
More information8.334: Statistical Mechanics II Problem Set # 4 Due: 4/9/14 Transfer Matrices & Position space renormalization
8.334: Statistical Mechanics II Problem Set # 4 Due: 4/9/14 Transfer Matrices & Position space renormalization This problem set is partly intended to introduce the transfer matrix method, which is used
More informationThe Fermion Bag Approach
The Fermion Bag Approach Anyi Li Duke University In collaboration with Shailesh Chandrasekharan 1 Motivation Monte Carlo simulation Sign problem Fermion sign problem Solutions to the sign problem Fermion
More informationarxiv: v3 [cond-mat.stat-mech] 17 Oct 2017
UNIVERSAL BEHAVIOUR OF 3D LOOP SOUP MODELS arxiv:73.953v3 [cond-mat.stat-mech] 7 Oct 27 DANIEL UELTSCHI Abstract. These notes describe several loop soup models and their universal behaviour in dimensions
More informationHyperbolic Hubbard-Stratonovich transformations
Hyperbolic Hubbard-Stratonovich transformations J. Müller-Hill 1 and M.R. Zirnbauer 1 1 Department of Physics, University of Cologne August 21, 2008 1 / 20 Outline Motivation Hubbard-Stratonovich transformation
More informationGradient interfaces with and without disorder
Gradient interfaces with and without disorder Codina Cotar University College London September 09, 2014, Toronto Outline 1 Physics motivation Example 1: Elasticity Recap-Gaussian Measure Example 2: Effective
More informationPhase transitions and critical phenomena
Phase transitions and critical phenomena Classification of phase transitions. Discontinous (st order) transitions Summary week -5 st derivatives of thermodynamic potentials jump discontinously, e.g. (
More informationFrom loop clusters and random interlacements to the Gaussian free field
From loop clusters and random interlacements to the Gaussian free field Université Paris-Sud, Orsay June 4, 014 Definition of the random walk loop soup G = (V, E) undirected connected graph. V at most
More informationLattice spin models: Crash course
Chapter 1 Lattice spin models: Crash course 1.1 Basic setup Here we will discuss the basic setup of the models to which we will direct our attention throughout this course. The basic ingredients are as
More informationNumerical Analysis of the Anderson Localization
Numerical Analysis of the Anderson Localization Peter Marko² FEI STU Bratislava FzU Praha, November 3. 23 . introduction: localized states in quantum mechanics 2. statistics and uctuations 3. metal - insulator
More informationRenormalization Group: non perturbative aspects and applications in statistical and solid state physics.
Renormalization Group: non perturbative aspects and applications in statistical and solid state physics. Bertrand Delamotte Saclay, march 3, 2009 Introduction Field theory: - infinitely many degrees of
More informationSpectrum of Holographic Wilson Loops
Spectrum of Holographic Wilson Loops Leopoldo Pando Zayas University of Michigan Continuous Advances in QCD 2011 University of Minnesota Based on arxiv:1101.5145 Alberto Faraggi and LPZ Work in Progress,
More informationAdvanced Workshop on Anderson Localization, Nonlinearity and Turbulence: a Cross-Fertilization. 23 August - 3 September, 2010
16-5 Advanced Workshop on Anderson Localization, Nonlinearity and Turbulence: a Cross-Fertilization 3 August - 3 September, 010 INTRODUCTORY Anderson Localization - Introduction Boris ALTSHULER Columbia
More informationEdge-and vertex-reinforced random walks with super-linear reinforcement on infinite graphs
Edge-and vertex-reinforced random walks with super-linear reinforcement on infinite graphs Codina Cotar University College London December 2016, Cambridge Edge-reinforced random walks (ERRW): random walks
More informationClusters and Percolation
Chapter 6 Clusters and Percolation c 2012 by W. Klein, Harvey Gould, and Jan Tobochnik 5 November 2012 6.1 Introduction In this chapter we continue our investigation of nucleation near the spinodal. We
More informationSpin Models and Gravity
Spin Models and Gravity Umut Gürsoy (CERN) 6th Regional Meeting in String Theory, Milos June 25, 2011 Spin Models and Gravity p.1 Spin systems Spin Models and Gravity p.2 Spin systems Many condensed matter
More informationIntroduction to Theory of Mesoscopic Systems
Introduction to Theory of Mesoscopic Systems Boris Altshuler Princeton University, Columbia University & NEC Laboratories America Lecture 3 Beforehand Weak Localization and Mesoscopic Fluctuations Today
More informationDensity of States for Random Band Matrices in d = 2
Density of States for Random Band Matrices in d = 2 via the supersymmetric approach Mareike Lager Institute for applied mathematics University of Bonn Joint work with Margherita Disertori ZiF Summer School
More informationStatistical Thermodynamics Solution Exercise 8 HS Solution Exercise 8
Statistical Thermodynamics Solution Exercise 8 HS 05 Solution Exercise 8 Problem : Paramagnetism - Brillouin function a According to the equation for the energy of a magnetic dipole in an external magnetic
More informationarxiv:cond-mat/ v4 [cond-mat.stat-mech] 19 Jun 2007
arxiv:cond-mat/060065v4 [cond-mat.stat-mech] 9 Jun 007 Restoration of Isotropy in the Ising Model on the Sierpiński Gasket Naoto Yajima Graduate School of Human and Environmental Studies, Kyoto University,
More informationHeidelberg University. Seminar Vortrag 1 WS2015 SUSY Quantum mechanics. QFT in dim = 0
Heidelberg University Seminar Vortrag 1 WS015 SUSY Quantum mechanics QFT in dim = 0 1 Preliminaries on QFT s: QFT is a theory that aims to bring field theories and quantum mechanics together, under the
More informationString / gauge theory duality and ferromagnetic spin chains
String / gauge theory duality and ferromagnetic spin chains M. Kruczenski Princeton Univ. In collaboration w/ Rob Myers, David Mateos, David Winters Arkady Tseytlin, Anton Ryzhov Summary Introduction mesons,,...
More informationInvestigations on the SYK Model and its Dual Space-Time
2nd Mandelstam Theoretical Physics Workshop Investigations on the SYK Model and its Dual Space-Time Kenta Suzuki A. Jevicki, KS, & J. Yoon; 1603.06246 [hep-th] A. Jevicki, & KS; 1608.07567 [hep-th] S.
More informationCFT and SLE and 2D statistical physics. Stanislav Smirnov
CFT and SLE and 2D statistical physics Stanislav Smirnov Recently much of the progress in understanding 2-dimensional critical phenomena resulted from Conformal Field Theory (last 30 years) Schramm-Loewner
More informationString/gauge theory duality and QCD
String/gauge theory duality and QCD M. Kruczenski Purdue University ASU 009 Summary Introduction String theory Gauge/string theory duality. AdS/CFT correspondence. Mesons in AdS/CFT Chiral symmetry breaking
More informationDecay of correlations in 2d quantum systems
Decay of correlations in 2d quantum systems Costanza Benassi University of Warwick Quantissima in the Serenissima II, 25th August 2017 Costanza Benassi (University of Warwick) Decay of correlations in
More informationTowards conformal invariance of 2-dim lattice models
Towards conformal invariance of 2-dim lattice models Stanislav Smirnov Université de Genève September 4, 2006 2-dim lattice models of natural phenomena: Ising, percolation, self-avoiding polymers,... Realistic
More informationThe parabolic Anderson model on Z d with time-dependent potential: Frank s works
Weierstrass Institute for Applied Analysis and Stochastics The parabolic Anderson model on Z d with time-dependent potential: Frank s works Based on Frank s works 2006 2016 jointly with Dirk Erhard (Warwick),
More information221B Lecture Notes Spontaneous Symmetry Breaking
B Lecture Notes Spontaneous Symmetry Breaking Spontaneous Symmetry Breaking Spontaneous Symmetry Breaking is an ubiquitous concept in modern physics, especially in condensed matter and particle physics.
More informationQuantum spin systems - models and computational methods
Summer School on Computational Statistical Physics August 4-11, 2010, NCCU, Taipei, Taiwan Quantum spin systems - models and computational methods Anders W. Sandvik, Boston University Lecture outline Introduction
More informationTopological protection, disorder, and interactions: Life and death at the surface of a topological superconductor
Topological protection, disorder, and interactions: Life and death at the surface of a topological superconductor Matthew S. Foster Rice University March 14 th, 2014 Collaborators: Emil Yuzbashyan (Rutgers),
More informationStatistical Mechanics
Franz Schwabl Statistical Mechanics Translated by William Brewer Second Edition With 202 Figures, 26 Tables, and 195 Problems 4u Springer Table of Contents 1. Basic Principles 1 1.1 Introduction 1 1.2
More informationHelicity fluctuation, generation of linking number and effect on resistivity
Helicity fluctuation, generation of linking number and effect on resistivity F. Spineanu 1), M. Vlad 1) 1) Association EURATOM-MEC Romania NILPRP MG-36, Magurele, Bucharest, Romania spineanu@ifin.nipne.ro
More informationPhase transitions in Hubbard Model
Phase transitions in Hubbard Model Anti-ferromagnetic and superconducting order in the Hubbard model A functional renormalization group study T.Baier, E.Bick, C.Krahl, J.Mueller, S.Friederich Phase diagram
More informationGroup theoretical methods in Machine Learning
Group theoretical methods in Machine Learning Risi Kondor Columbia University Tutorial at ICML 2007 Tiger, tiger, burning bright In the forests of the night, What immortal hand or eye Dare frame thy fearful
More informationAutomorphic Equivalence Within Gapped Phases
1 Harvard University May 18, 2011 Automorphic Equivalence Within Gapped Phases Robert Sims University of Arizona based on joint work with Sven Bachmann, Spyridon Michalakis, and Bruno Nachtergaele 2 Outline:
More informationDependent percolation: some examples and multi-scale tools
Dependent percolation: some examples and multi-scale tools Maria Eulália Vares UFRJ, Rio de Janeiro, Brasil 8th Purdue International Symposium, June 22 I. Motivation Classical Ising model (spins ±) in
More informationThe Random Matching Problem
The Random Matching Problem Enrico Maria Malatesta Universitá di Milano October 21st, 2016 Enrico Maria Malatesta (UniMi) The Random Matching Problem October 21st, 2016 1 / 15 Outline 1 Disordered Systems
More informationHIGHER SPIN CORRECTIONS TO ENTANGLEMENT ENTROPY
HIGHER SPIN CORRECTIONS TO ENTANGLEMENT ENTROPY JHEP 1406 (2014) 096, Phys.Rev. D90 (2014) 4, 041903 with Shouvik Datta ( IISc), Michael Ferlaino, S. Prem Kumar (Swansea U. ) JHEP 1504 (2015) 041 with
More informationA semiclassical ramp in SYK and in gravity
A semiclassical ramp in SYK and in gravity Douglas Stanford IAS June 25, 2018 Based on work with Phil Saad and Steve Shenker In a black hole geometry, correlators like φ(t )φ(0) seem to just decay with
More informationThe Ising model Summary of L12
The Ising model Summary of L2 Aim: Study connections between macroscopic phenomena and the underlying microscopic world for a ferromagnet. How: Study the simplest possible model of a ferromagnet containing
More informationORIGINS. E.P. Wigner, Conference on Neutron Physics by Time of Flight, November 1956
ORIGINS E.P. Wigner, Conference on Neutron Physics by Time of Flight, November 1956 P.W. Anderson, Absence of Diffusion in Certain Random Lattices ; Phys.Rev., 1958, v.109, p.1492 L.D. Landau, Fermi-Liquid
More informationMultifractality in simple systems. Eugene Bogomolny
Multifractality in simple systems Eugene Bogomolny Univ. Paris-Sud, Laboratoire de Physique Théorique et Modèles Statistiques, Orsay, France In collaboration with Yasar Atas Outlook 1 Multifractality Critical
More informationContents. Appendix A Strong limit and weak limit 35. Appendix B Glauber coherent states 37. Appendix C Generalized coherent states 41
Contents Preface 1. The structure of the space of the physical states 1 1.1 Introduction......................... 1 1.2 The space of the states of physical particles........ 2 1.3 The Weyl Heisenberg algebra
More informationAdvanced Topics in Probability
Advanced Topics in Probability Conformal Methods in 2D Statistical Mechanics Pierre Nolin Different lattices discrete models on lattices, in two dimensions: square lattice Z 2 : simplest one triangular
More informationMeron-Cluster and Nested Cluster Algorithms: Addressing the Sign Problem in Quantum Monte Carlo Simulations
Meron-Cluster and Nested Cluster Algorithms: Addressing the Sign Problem in Quantum Monte Carlo Simulations Uwe-Jens Wiese Bern University IPAM Workshop QS2009, January 26, 2009 Collaborators: B. B. Beard
More informationQuasiparticle localization in superconductors with spin-orbit scattering
PHYSICAL REVIEW B VOLUME 61, NUMBER 14 1 APRIL 2000-II Quasiparticle localization in superconductors with spin-orbit scattering T. Senthil and Matthew P. A. Fisher Institute for Theoretical Physics, University
More informationIntegrability and Finite Size Effects of (AdS 5 /CFT 4 ) β
Integrability and Finite Size Effects of (AdS 5 /CFT 4 ) β Based on arxiv:1201.2635v1 and on-going work With C. Ahn, D. Bombardelli and B-H. Lee February 21, 2012 Table of contents 1 One parameter generalization
More informationRG Limit Cycles (Part I)
RG Limit Cycles (Part I) Andy Stergiou UC San Diego based on work with Jean-François Fortin and Benjamín Grinstein Outline The physics: Background and motivation New improved SE tensor and scale invariance
More informationThe ultrametric tree of states and computation of correlation functions in spin glasses. Andrea Lucarelli
Università degli studi di Roma La Sapienza Facoltà di Scienze Matematiche, Fisiche e Naturali Scuola di Dottorato Vito Volterra Prof. Giorgio Parisi The ultrametric tree of states and computation of correlation
More informationExponential approach to equilibrium for a stochastic NLS
Exponential approach to equilibrium for a stochastic NLS CNRS and Cergy Bonn, Oct, 6, 2016 I. Stochastic NLS We start with the dispersive nonlinear Schrödinger equation (NLS): i u = u ± u p 2 u, on the
More informationPhase transitions in discrete structures
Phase transitions in discrete structures Amin Coja-Oghlan Goethe University Frankfurt Overview 1 The physics approach. [following Mézard, Montanari 09] Basics. Replica symmetry ( Belief Propagation ).
More informationMPS formulation of quasi-particle wave functions
MPS formulation of quasi-particle wave functions Eddy Ardonne Hans Hansson Jonas Kjäll Jérôme Dubail Maria Hermanns Nicolas Regnault GAQHE-Köln 2015-12-17 Outline Short review of matrix product states
More informationScale without conformal invariance
Scale without conformal invariance Andy Stergiou Department of Physics, UCSD based on arxiv:1106.2540, 1107.3840, 1110.1634, 1202.4757 with Jean-François Fortin and Benjamín Grinstein Outline The physics:
More informationI. BASICS OF STATISTICAL MECHANICS AND QUANTUM MECHANICS
I. BASICS OF STATISTICAL MECHANICS AND QUANTUM MECHANICS Marus Holzmann LPMMC, Maison de Magistère, Grenoble, and LPTMC, Jussieu, Paris marus@lptl.jussieu.fr http://www.lptl.jussieu.fr/users/marus (Dated:
More informationInterpolating between Wishart and inverse-wishart distributions
Interpolating between Wishart and inverse-wishart distributions Topological phase transitions in 1D multichannel disordered wires with a chiral symmetry Christophe Texier December 11, 2015 with Aurélien
More informationSU(3) Quantum Spin Ladders as a Regularization of the CP (2) Model at Non-Zero Density: From Classical to Quantum Simulation
SU(3) Quantum Spin Ladders as a Regularization of the CP (2) Model at Non-Zero Density: From Classical to Quantum Simulation W. Evans, U. Gerber, M. Hornung, and U.-J. Wiese arxiv:183.4767v1 [hep-lat]
More informationt Hooft Loops and S-Duality
t Hooft Loops and S-Duality Jaume Gomis KITP, Dualities in Physics and Mathematics with T. Okuda and D. Trancanelli Motivation 1) Quantum Field Theory Provide the path integral definition of all operators
More informationPAPER 51 ADVANCED QUANTUM FIELD THEORY
MATHEMATICAL TRIPOS Part III Tuesday 5 June 2007 9.00 to 2.00 PAPER 5 ADVANCED QUANTUM FIELD THEORY Attempt THREE questions. There are FOUR questions in total. The questions carry equal weight. STATIONERY
More informationFluctuations for the Ginzburg-Landau Model and Universality for SLE(4)
Fluctuations for the Ginzburg-Landau Model and Universality for SLE(4) Jason Miller Department of Mathematics, Stanford September 7, 2010 Jason Miller (Stanford Math) Fluctuations and Contours of the GL
More informationQGP, Hydrodynamics and the AdS/CFT correspondence
QGP, Hydrodynamics and the AdS/CFT correspondence Adrián Soto Stony Brook University October 25th 2010 Adrián Soto (Stony Brook University) QGP, Hydrodynamics and AdS/CFT October 25th 2010 1 / 18 Outline
More informationIntroduction to Integrability in AdS/CFT: Lecture 4
Introduction to Integrability in AdS/CFT: Lecture 4 Rafael Nepomechie University of Miami Introduction Recall: 1-loop dilatation operator for all single-trace operators in N=4 SYM is integrable 1-loop
More informationQuantum Diffusion and Delocalization for Random Band Matrices
Quantum Diffusion and Delocalization for Random Band Matrices László Erdős Ludwig-Maximilians-Universität, Munich, Germany Montreal, Mar 22, 2012 Joint with Antti Knowles (Harvard University) 1 INTRODUCTION
More informationLecture 8 Nature of ensemble: Role of symmetry, interactions and other system conditions: Part II
Lecture 8 Nature of ensemble: Role of symmetry, interactions and other system conditions: Part II We continue our discussion of symmetries and their role in matrix representation in this lecture. An example
More informationPhD in Theoretical Particle Physics Academic Year 2017/2018
July 10, 017 SISSA Entrance Examination PhD in Theoretical Particle Physics Academic Year 017/018 S olve two among the four problems presented. Problem I Consider a quantum harmonic oscillator in one spatial
More informationGaussian Free Field in beta ensembles and random surfaces. Alexei Borodin
Gaussian Free Field in beta ensembles and random surfaces Alexei Borodin Corners of random matrices and GFF spectra height function liquid region Theorem As Unscaled fluctuations Gaussian (massless) Free
More informationCluster Algorithms to Reduce Critical Slowing Down
Cluster Algorithms to Reduce Critical Slowing Down Monte Carlo simulations close to a phase transition are affected by critical slowing down. In the 2-D Ising system, the correlation length ξ becomes very
More informationPhysics 127b: Statistical Mechanics. Landau Theory of Second Order Phase Transitions. Order Parameter
Physics 127b: Statistical Mechanics Landau Theory of Second Order Phase Transitions Order Parameter Second order phase transitions occur when a new state of reduced symmetry develops continuously from
More informationIntroduction. Chapter The Purpose of Statistical Mechanics
Chapter 1 Introduction 1.1 The Purpose of Statistical Mechanics Statistical Mechanics is the mechanics developed to treat a collection of a large number of atoms or particles. Such a collection is, for
More informationDelocalization for Schrödinger operators with random Dirac masses
Delocalization for Schrödinger operators with random Dirac masses CEMPI Scientific Day Lille, 10 February 2017 Disordered Systems E.g. box containing N nuclei Motion of electron in this system High temperature
More informationFerromagnets and the classical Heisenberg model. Kay Kirkpatrick, UIUC
Ferromagnets and the classical Heisenberg model Kay Kirkpatrick, UIUC Ferromagnets and the classical Heisenberg model: asymptotics for a mean-field phase transition Kay Kirkpatrick, Urbana-Champaign June
More informationAnderson Localization from Classical Trajectories
Anderson Localization from Classical Trajectories Piet Brouwer Laboratory of Atomic and Solid State Physics Cornell University Support: NSF, Packard Foundation With: Alexander Altland (Cologne) Quantum
More informationCanonical partition functions in lattice QCD
Canonical partition functions in lattice QCD (an appetizer) christof.gattringer@uni-graz.at Julia Danzer, Christof Gattringer, Ludovit Liptak, Marina Marinkovic Canonical partition functions Grand canonical
More informationJuly 2, SISSA Entrance Examination. PhD in Theoretical Particle Physics Academic Year 2018/2019. olve two among the three problems presented.
July, 018 SISSA Entrance Examination PhD in Theoretical Particle Physics Academic Year 018/019 S olve two among the three problems presented. Problem I Consider a theory described by the Lagrangian density
More informationMagnetic ordering of local moments
Magnetic ordering Types of magnetic structure Ground state of the Heisenberg ferromagnet and antiferromagnet Spin wave High temperature susceptibility Mean field theory Magnetic ordering of local moments
More informationCritical Level Statistics and QCD Phase Transition
Critical Level Statistics and QCD Phase Transition S.M. Nishigaki Dept. Mat. Sci, Shimane Univ. quark gluon Wilsonʼs Lattice Gauge Theory = random Dirac op quenched Boltzmann weight # " tr U U U U 14 243
More informationA Tour of Spin Glasses and Their Geometry
A Tour of Spin Glasses and Their Geometry P. Le Doussal, D. Bernard, LPTENS A. Alan Middleton, Syracuse University Support from NSF, ANR IPAM: Random Shapes - Workshop I 26 March, 2007 Goals Real experiment
More informationFRACTAL DIMENSIONS FOR CERTAIN CRITICAL RANDOM MATRIX ENSEMBLES. Eugène Bogomolny
FRACTAL DIMENSIONS FOR CERTAIN CRITICAL RANDOM MATRIX ENSEMBLES Eugène Bogomolny Univ. Paris-Sud, CNRS, LPTMS, Orsay, France In collaboration with Olivier Giraud VI Brunel Workshop on PMT Brunel, 7-8 December
More informationChapter 4. RWs on Fractals and Networks.
Chapter 4. RWs on Fractals and Networks. 1. RWs on Deterministic Fractals. 2. Linear Excitation on Disordered lattice; Fracton; Spectral dimension 3. RWs on disordered lattice 4. Random Resistor Network
More informationBaby Skyrmions in AdS 3 and Extensions to (3 + 1) Dimensions
Baby Skyrmions in AdS 3 and Extensions to (3 + 1) Dimensions Durham University Work in collaboration with Matthew Elliot-Ripley June 26, 2015 Introduction to baby Skyrmions in flat space and AdS 3 Discuss
More informationMATH 564/STAT 555 Applied Stochastic Processes Homework 2, September 18, 2015 Due September 30, 2015
ID NAME SCORE MATH 56/STAT 555 Applied Stochastic Processes Homework 2, September 8, 205 Due September 30, 205 The generating function of a sequence a n n 0 is defined as As : a ns n for all s 0 for which
More informationGradient Percolation and related questions
and related questions Pierre Nolin (École Normale Supérieure & Université Paris-Sud) PhD Thesis supervised by W. Werner July 16th 2007 Introduction is a model of inhomogeneous percolation introduced by
More informationAccelerator Physics Homework #3 P470 (Problems: 1-5)
Accelerator Physics Homework #3 P470 (Problems: -5). Particle motion in the presence of magnetic field errors is (Sect. II.2) y + K(s)y = B Bρ, where y stands for either x or z. Here B = B z for x motion,
More informationThe dynamics of small particles whose size is roughly 1 µmt or. smaller, in a fluid at room temperature, is extremely erratic, and is
1 I. BROWNIAN MOTION The dynamics of small particles whose size is roughly 1 µmt or smaller, in a fluid at room temperature, is extremely erratic, and is called Brownian motion. The velocity of such particles
More information