Anderson Localization Looking Forward
|
|
- Edward Ray
- 5 years ago
- Views:
Transcription
1 Anderson Localization Looking Forward Boris Altshuler Physics Department, Columbia University Collaborations: Also Igor Aleiner Denis Basko, Gora Shlyapnikov, Vincent Michal, Vladimir Kravtsov, Lecture2 September, 9, 2015
2 Outline 1. Introduction 2. Anderson Model; Anderson Metal and Anderson Insulator 3. Localization beyond the real space. Integrability and chaos. 4. Spectral Statistics and Localization 5. Many-Body Localization. 6. Many-Body Localization of the interacting fermions. 7. Many-Body localization of weakly interacting bosons. 8. Many-Body Localization and Ergodicity
3 extended f = 3.04 GHz f = 7.33 GHz Anderson Model W, I localized Localized j I ij i E Ĥ x E x I 2 Finite motion 0 DoS x I 1
4 Vˆ, Matrix element of the perturbation I 0 I,..., I 1 I d Anderson Model! AL hops are local one can distinguish near and far KAM perturbation is smooth enough
5 Pradhan & Sridar, PRL, 2000 Sinai billiard Square billiard Disordered localized 0 W Localized momentum space extended Disordered extended Localized real space I 0
6 Classical Integrable H0 H0 I ; I t 0 Perturbation Glossary Quantum Integrable E Perturbation V; I t 0 Vˆ V, KAM Localized Ergodic (chaotic) Extended? ˆ 0 H,, I
7 Question: What is the reason to speak about localization if we in? general do not know the space in which the system is localized Need an invariant (basis independent) criterion of the localization
8 Part 4. Spectral Statistics and Localization
9 RANDOM MATRIX THEORY N N E s 1 P... s E 1 E 1 E 1 Spectral Rigidity ensemble of Hermitian matrices with random matrix element E - spectrum (set of eigenvalues) - mean level spacing - ensemble averaging - spacing between nearest neighbors - distribution function of nearest neighbors spacing between P s 0 0 Spectral statistics N Level repulsion P s 1 s 1, 2, 4
10 Wigner-Dyson; GOE Poisson Gaussian Orthogonal Ensemble Unitary 2 Orthogonal 1 Simplectic 4 Poisson completely uncorrelated levels
11 RANDOM MATRICES N N matrices with random matrix elements. N Dyson Ensembles Matrix elements real complex 2 2 matrices Ensemble orthogonal unitary simplectic realization T-inv potential broken T-invariance (e.g., by magnetic field) T-inv, but with spinorbital coupling
12 Hˆ Reason for H H H H P s 0 0: when s E E H H H small small small Recall the Wigner von Neumann noncrossing rule 1. The assumption is that the matrix elements are statistically independent. Therefore probability of two levels to be degenerate vanishes. 2. If H 12 is real (orthogonal ensemble), then for s to be small two statistically independent variables ((H 22 - H 11 ) and H 12 ) should be small and thus P( s) s 1
13 Hˆ H H H H p H H p H 2 2 P E E d H H dh E E H H H Distribution function of the diagonal matrix elements Distribution function of the off-diagonal matrix elements Distribution function of the spacing P s
14 Hˆ Reason for H H H H P s 0 0: when s E E H H H small small small 1. The assumption is that the matrix elements are statistically independent. Therefore probability of two levels to be degenerate vanishes. 2. If H 12 is real (orthogonal ensemble), then for s to be small two statistically independent variables ((H 22 - H 11 ) and H 12 ) should be small and thus P( s) s 1 3. Complex H 12 (unitary ensemble) both Re(H 12 ) and Im(H 12 ) are statistically independent three independent 2 random variables should be small P( s) s 2
15 Anderson Model Lattice - tight binding model Onsite energies i - random j i Hopping matrix elements I ij Hˆ i i aˆ i aˆ i I aˆ i, jn. n. i aˆ j I ij -W < i <W uniformly distributed Is there much in common between Random Matrices and Hamiltonians with random potential? Q: What are the spectral statistics? of a finite size Anderson model
16 Strong disorder Anderson Transition Weak disorder I < I c Insulator All eigenstates are localized Localization length x The eigenstates, which are localized at different places will not repel each other I > I c Metal There appear states extended all over the whole system Any two extended eigenstates repel each other Poisson spectral statistics Wigner Dyson spectral statistics
17 Zharekeschev & Kramer. Exact diagonalization of the Anderson model Disorder W
18 Anderson transition in terms of pure level statistics P(s)
19 Part 5. Quantum Chaos, Integrability and Localization
20 ATOMS NUCLEI Wigner: Main goal is to classify the eigenstates in terms of the quantum numbers For the nuclear excitations this program does not work Study spectral statistics of a particular quantum system a given nucleus Spectra: {E } Random Matrices Ensemble Ensemble averaging Atomic Nuclei Spectral averaging (over ) Particular quantum system Nevertheless Statistics of the nuclear spectra are almost exactly the same as the Random Matrix Statistics
21 P(s) Particular nucleus 166 Er N. Bohr, Nature 137 (1936) 344. P(s) s Spectra of several nuclei combined (after spacing) rescaling by the mean level
22 Q: Why the random matrix? theory (RMT) works so well for nuclear spectra Original answer: These are systems with a large number of degrees of freedom, and therefore the complexity is high Later it became clear that there exist very simple systems with as many as 2 degrees of freedom (d=2), which demonstrate RMT - like spectral statistics
23 Classical Dynamical Systems with d degrees of freedom Integrable The variables can be separated [ d one-dimensional Systems problems [d integrals of motion Chaotic Systems Examples Rectangular and circular billiard, Kepler problem,..., 1d Hubbard model and other exactly solvable models,.. The variables can not be separated [ there is only one integral of motion - energy B Sinai billiard Stadium Kepler problem in magnetic field
24 0 Bohigas Giannoni Schmit conjecture Chaotic classical analog Wigner- Dyson spectral statistics No quantum numbers except energy
25 Classical Integrable? Quantum Poisson Chaotic? Wigner- Dyson
26 Square billiard Integrable All chaotic systems resemble each other. Sinai billiard Chaotic Disordered localized All integrable systems are integrable in their own way Disordered extended
27 Sinai billiard Pradhan &Sridar, PRL, 2000 Square billiard Localized momentum space extended Disordered extended Disordered localized Localized real space
28 Spectral statistics Extended states: Localized states: Level repulsion, avoided crossings, Wigner-Dyson spectral statistics (random matrices) Poisson spectral statistics No level repulsion probability density P(s) nearest neighbor s spacing s Invariant (basis independent) definition
29 Example: Stadium Localization in the angular momentum space. R a a R - parameter
30 Localization and diffusion in the angular momentum space a 0 R 0 Integrable circular billiard Angular momentum is the integral of motion Chaotic stadium 0; 1 Diffusion in the angular momentum space 5 D 2
31 a 0 R 0 Integrable circular billiard Angular momentum is the integral of motion 0; 1 Diffusion in the angular momentum space 5 D 2 Chaotic stadium Localization and diffusion in the angular momentum space Poisson 0.01 g=0.012 Wigner-Dyson 0.1 g=4
32 Part 6. Many-Body Localization a) Spin systems; Quantum Computer
33 Example: Random Ising model in the perpendicular field Will not discuss today in detail N N N z z z x ˆ ˆ ˆ ˆ ˆ x ˆ i i ij i j i 0 i i1 i j i1 i1 Hˆ B J I H I i Random Ising model in a parallel field - Pauli matrices, z i i 1,2,..., N; N Perpendicular field Without perpendicular field all z i commute with the Hamiltonian, i.e. they are integrals of motion
34 i N N N z z z x ˆ ˆ ˆ ˆ ˆ x ˆ i i ij i j i 0 i i1 i j i1 i1 Hˆ B J I H I - Pauli matrices i 1,2,..., N; N 1 Random Ising model in a parallel field Perpendicular field Without perpendicular field all commute with the Hamiltonian, i.e. they are integrals of motion z i z i determines a site of an N-dimensional hypercube
35 i H 0 i N N N z z z x ˆ ˆ ˆ ˆ ˆ x ˆ i i ij i j i 0 i i1 i j i1 i1 Hˆ B J I H I - Pauli matrices i 1,2,..., N; N 1 Anderson Model on N-dimensional cube onsite energy Random Ising model in a parallel field z i perp. field x Perpendicular field Without perpendicular field all commute with the Hamiltonian, i.e. they are integrals of motion z i determines a site ˆ ˆ ˆ hoping between nearest neighbors
36 N N N z z z x ˆ ˆ ˆ ˆ ˆ x ˆ i i ij i j i 0 i i1 i j i1 i1 Hˆ B J I H I Anderson Model on N-dimensional cube Usually: # of dimensions d const Here: # of dimensions d N system linear size L system linear size L 1
37 6-dimensional cube 9-dimensional cube
38 Part 6. Many-Body Localization b) Interacting particles
39 one particle, one level per site, onsite disorder nearest neighbor hoping Hamiltonian: Hˆ Hˆ Vˆ 0 Conventional Anderson Model many (N) particles no } interaction. Individual energies k, k 1,... N are conserved Basis: i, Hˆ Vˆ 0 i I i, jn. n. i i i i labels sites N conservation laws integrable system j i Ĥ E E n k k labels one-particle eigenstates; n - occupation numbers; n
40 many (N) particles no } interaction. Individual energies k, k 1,... N are conserved N conservation laws integrable system Ĥ E E n k k labels one-particle eigenstates; n - occupation numbers; n Role of the interaction: Transitions between the ideal gas states Basis: Hamiltonian: Hˆ Hˆ ˆ ˆ 0 V, H0 E Interaction: Vˆ Localization in Fock space I,, nn.?
41 Disorder + Weak Interaction Basis: Hamiltonian: Hˆ Hˆ Vˆ, Hˆ 0 0 E Interaction: Vˆ,, nn.? I Anderson model Q: What is the lattice?
42 Part 6. Many-Body Localization c) Statistical mechanics
43 Main postulate of the Gibbs StatMechequipartition (microcanonical distribution): In the equilibrium all states with the same energy are realized with the same probability. Without interaction between particles the equilibrium would never be reached each one-particle energy is conserved. Common believe: Even weak interaction should drive the system to the equilibrium. It might be not true for many-body localized states!!!
44 Localization in Fock space What does it mean? No two-body operator can cause transitions between many-body states that are close in energy. No dissipation due to the external field ideal insulator (glass) The concept of the equilibrium looses its meaning no relaxation to a thermal state. No entropy production Temperature Energy
45 Time-reversal symmetry = T-invariance Equations of motion are invariant under t t For each classical trajectory there is another trajectory, which is its inversion in time Violation of the T-invariance e.g. by magnetic field H
46 Statistical mechanics Irreversibility - arrow of time t t x r r t r t Has nothing to do with the violation of the T-invariance Has everything to do with the delocalization Extended states irreversible dynamics Localized states dynamics is to some extent reversible The same is true for many body systems t
47 Heat Theorem Nerns, Einstein, Planck, Polany, [Nernst is] not open to reason, because he is not enough of a logician ( der Vernunft nich zuga nglich (zu wenig Logiker)). Einstein, letter to Ehrenfest Is it possible to reach zero temperature? Is it possible to reach thermal equilibrium close to the ground state?
48 Part 6. Many-Body Localization d) Interacting fermions; phononless transport
49 Temperature dependence of the conductivity one-electron picture Mobility edge Chemical potential T 0 0 DoS DoS DoS T E T e c F T 0 T there are extended states II c all states are localized II c
50 Temperature dependence of the conductivity one-electron picture Assume that all the states are localized; e.g. d = 1,2 DoS T 0 T
51 Inelastic processes transitions between localized states energy mismatch T 0 0 (any mechanism)
52 Phonon-assisted hopping w w T 0 0 Variable Range Hopping N.F. Mott (1968) Mechanism-dependent prefactor Optimized phase volume Any bath with a continuous spectrum of delocalized excitations down to w= 0 will give the same exponential
53 Common belief: Given: Anderson Insulator weak e-e interactions? Can hopping conductivity exist without phonons 1. All one-electron states are localized 2. Electrons interact with each other 3. The system is closed (no phonons) 4. Temperature is low but finite Phonon assisted hopping transport Find: DC conductivity (T,w=0) (zero or finite?)
54 Q: Can e-h pairs lead to phonon-less variable range hopping in the same way as phonons do? A#1: Sure 1. Recall phonon-less AC conductivity: Sir N.F. Mott (1970) 2. Fluctuation Dissipation Theorem: there should be Johnson-Nyquist noise 3. Use this noise as a bath instead of phonons 4. Self-consistency (whatever it means)
55 Q: Can e-h pairs lead to phonon-less variable range hopping in the same way as phonons do? A#1: Sure A#2: No way (L. Fleishman. P.W. Anderson (1980)) Except maybe Coulomb interaction in 3D is contributed by rare resonances R d R g a matrix element vanishes b 0
56 No phonons??? No transport T Problem: If the localization length exceeds L, then metal. } At In a metal e e interaction leads to a finite L high enough temperatures conductivity should be finite even without phonons
57 Q: Can e-h pairs lead to phonon-less variable range hopping in the same way as phonons do? A#1: Sure A#2: No way (L. Fleishman. P.W. Anderson (1980)) A#3: Finite temperature Metal-Insulator Transition (Basko, Aleiner, BA (2006)) insulator 0 metal Drude
58 Finite temperature Metal-Insulator Transition Many body wave functions are localized Many body in functional space localization! insulator metal Drude Interaction strength 0 d 1 spacing Localization Definitions: Insulator not 0 d dt 0 Metal not 0 d dt 0
59 Many body Anderson-like Model many particles, several levels per site, onsite disorder local interaction Hamiltonian: H Hˆ Vˆ Vˆ Vˆ Ĥ0 E.., n 1,.., n 1,.., i, j n. n. Vˆ 2,, I U i j Vˆ.., n 1,.., n 1,.., n 1,.., n 1,.. i i i i 1 i n i Basis: labels sites 0,1 Vˆ 2 n i labels levels occupation numbers I U
60 Stability of the insulating phase: NO spontaneous generation of broadening ( ) 0 is always a solution i linear stability analysis ( x ) 2 2 ( x ) ( x ) 2 After n iterations of the equations of the Self Consistent Born Approximation P n ( ) 3 2 const T ln 1 n first then ( ) < 1 insulator is stable!
61 Physics of the transition: cascades Conventional wisdom: For phonon assisted hopping one phonon one electron hop It is maybe correct at low temperatures, but the higher the temperature the easier it becomes to create e-h pairs. Therefore with increasing the temperature the typical number of pairs created n c (i.e. the number of hops) increases. Thus phonons create cascades of hops. Typical size of the cascade Localization length
62 Physics of the transition: cascades Conventional wisdom: For phonon assisted hopping one phonon one electron hop It is maybe correct at low temperatures, but the higher the temperature the easier it becomes to create e-h pairs. Therefore with increasing the temperature the typical number of pairs created n c (i.e. the number of hops) increases. Thus phonons create cascades of hops. T T. At some temperature c c This is the critical temperature. Above T c one phonon creates infinitely many pairs, i.e., phonons are not needed for charge transport. T n
Introduction 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 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 informationTheory of Mesoscopic Systems
Theory of Mesoscopic Systems Boris Altshuler Princeton University, Columbia University & NEC Laboratories America Lecture 2 08 June 2006 Brownian Motion - Diffusion Einstein-Sutherland Relation for electric
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 informationDisordered Quantum Systems
Disordered Quantum Systems Boris Altshuler Physics Department, Columbia University and NEC Laboratories America Collaboration: Igor Aleiner, Columbia University Part 1: Introduction Part 2: BCS + disorder
More informationIntroduction to Mesoscopics. Boris Altshuler Princeton University, NEC Laboratories America,
Not Yet Introduction to Mesoscopics Boris Altshuler Princeton University, NEC Laboratories America, ORIGINS E.P. Wigner, Conference on Neutron Physics by Time of Flight, November 1956 P.W. Anderson, Absence
More informationChapter 29. Quantum Chaos
Chapter 29 Quantum Chaos What happens to a Hamiltonian system that for classical mechanics is chaotic when we include a nonzero h? There is no problem in principle to answering this question: given a classical
More informationEmergence of chaotic scattering in ultracold lanthanides.
Emergence of chaotic scattering in ultracold lanthanides. Phys. Rev. X 5, 041029 arxiv preprint 1506.05221 A. Frisch, S. Baier, K. Aikawa, L. Chomaz, M. J. Mark, F. Ferlaino in collaboration with : Dy
More informationIntroduction to Theory of Mesoscopic Systems
Introduction to Theory of Mesoscopic Systems Boris Altshuler Princeton University, Columbia University & NEC Laboratories America Lecture 5 Beforehand Yesterday Today Anderson Localization, Mesoscopic
More informationUniversality for random matrices and log-gases
Universality for random matrices and log-gases László Erdős IST, Austria Ludwig-Maximilians-Universität, Munich, Germany Encounters Between Discrete and Continuous Mathematics Eötvös Loránd University,
More informationThe Transition to Chaos
Linda E. Reichl The Transition to Chaos Conservative Classical Systems and Quantum Manifestations Second Edition With 180 Illustrations v I.,,-,,t,...,* ', Springer Dedication Acknowledgements v vii 1
More information(De)-localization in mean-field quantum glasses
QMATH13, Georgia Tech, Atlanta October 10, 2016 (De)-localization in mean-field quantum glasses Chris R. Laumann (Boston U) Chris Baldwin (UW/BU) Arijeet Pal (Oxford) Antonello Scardicchio (ICTP) CRL,
More informationPurely electronic transport in dirty boson insulators
Purely electronic transport in dirty boson insulators Markus Müller Ann. Phys. (Berlin) 18, 849 (2009). Discussions with M. Feigel man, M.P.A. Fisher, L. Ioffe, V. Kravtsov, Abdus Salam International Center
More informationIs Quantum Mechanics Chaotic? Steven Anlage
Is Quantum Mechanics Chaotic? Steven Anlage Physics 40 0.5 Simple Chaos 1-Dimensional Iterated Maps The Logistic Map: x = 4 x (1 x ) n+ 1 μ n n Parameter: μ Initial condition: 0 = 0.5 μ 0.8 x 0 = 0.100
More informationList of Comprehensive Exams Topics
List of Comprehensive Exams Topics Mechanics 1. Basic Mechanics Newton s laws and conservation laws, the virial theorem 2. The Lagrangian and Hamiltonian Formalism The Lagrange formalism and the principle
More informationMany-Body Localization. Geoffrey Ji
Many-Body Localization Geoffrey Ji Outline Aside: Quantum thermalization; ETH Single-particle (Anderson) localization Many-body localization Some phenomenology (l-bit model) Numerics & Experiments Thermalization
More informationQuantum Chaos as a Practical Tool in Many-Body Physics
Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky NSCL/ Michigan State University Supported by NSF Statistical Nuclear Physics SNP2008 Athens, Ohio July 8, 2008 THANKS B. Alex
More informationQuantum Billiards. Martin Sieber (Bristol) Postgraduate Research Conference: Mathematical Billiard and their Applications
Quantum Billiards Martin Sieber (Bristol) Postgraduate Research Conference: Mathematical Billiard and their Applications University of Bristol, June 21-24 2010 Most pictures are courtesy of Arnd Bäcker
More informationarxiv:cond-mat/ v1 [cond-mat.stat-mech] 13 Mar 2003
arxiv:cond-mat/0303262v1 [cond-mat.stat-mech] 13 Mar 2003 Quantum fluctuations and random matrix theory Maciej M. Duras Institute of Physics, Cracow University of Technology, ulica Podchor ażych 1, PL-30084
More informationQuantum Chaos as a Practical Tool in Many-Body Physics ESQGP Shuryak fest
Quantum Chaos as a Practical Tool in Many-Body Physics ESQGP Shuryak fest Vladimir Zelevinsky NSCL/ Michigan State University Stony Brook October 3, 2008 Budker Institute of Nuclear Physics, Novosibirsk
More informationThe Mott Metal-Insulator Transition
Florian Gebhard The Mott Metal-Insulator Transition Models and Methods With 38 Figures Springer 1. Metal Insulator Transitions 1 1.1 Classification of Metals and Insulators 2 1.1.1 Definition of Metal
More informationOutline for Fundamentals of Statistical Physics Leo P. Kadanoff
Outline for Fundamentals of Statistical Physics Leo P. Kadanoff text: Statistical Physics, Statics, Dynamics, Renormalization Leo Kadanoff I also referred often to Wikipedia and found it accurate and helpful.
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 informationRegular & Chaotic. collective modes in nuclei. Pavel Cejnar. ipnp.troja.mff.cuni.cz
Pavel Cejnar Regular & Chaotic collective modes in nuclei Institute of Particle and Nuclear Physics Faculty of Mathematics and Physics Charles University, Prague, Czech Republic cejnar @ ipnp.troja.mff.cuni.cz
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 informationQUANTUM MECHANICS. Franz Schwabl. Translated by Ronald Kates. ff Springer
Franz Schwabl QUANTUM MECHANICS Translated by Ronald Kates Second Revised Edition With 122Figures, 16Tables, Numerous Worked Examples, and 126 Problems ff Springer Contents 1. Historical and Experimental
More informationLecture 6: Fluctuation-Dissipation theorem and introduction to systems of interest
Lecture 6: Fluctuation-Dissipation theorem and introduction to systems of interest In the last lecture, we have discussed how one can describe the response of a well-equilibriated macroscopic system to
More informationSpectral Fluctuations in A=32 Nuclei Using the Framework of the Nuclear Shell Model
American Journal of Physics and Applications 2017; 5(): 5-40 http://www.sciencepublishinggroup.com/j/ajpa doi: 10.11648/j.ajpa.2017050.11 ISSN: 20-4286 (Print); ISSN: 20-408 (Online) Spectral Fluctuations
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 informationRecent results in quantum chaos and its applications to nuclei and particles
Recent results in quantum chaos and its applications to nuclei and particles J. M. G. Gómez, L. Muñoz, J. Retamosa Universidad Complutense de Madrid R. A. Molina, A. Relaño Instituto de Estructura de la
More informationQUANTUM CHAOS IN NUCLEAR PHYSICS
QUANTUM CHAOS IN NUCLEAR PHYSICS Investigation of quantum chaos in nuclear physics is strongly hampered by the absence of even the definition of quantum chaos, not to mention the numerical criterion of
More informationin-medium pair wave functions the Cooper pair wave function the superconducting order parameter anomalous averages of the field operators
(by A. A. Shanenko) in-medium wave functions in-medium pair-wave functions and spatial pair particle correlations momentum condensation and ODLRO (off-diagonal long range order) U(1) symmetry breaking
More informationTitleQuantum Chaos in Generic Systems.
TitleQuantum Chaos in Generic Systems Author(s) Robnik, Marko Citation 物性研究 (2004), 82(5): 662-665 Issue Date 2004-08-20 URL http://hdl.handle.net/2433/97885 Right Type Departmental Bulletin Paper Textversion
More informationLecture 5. Hartree-Fock Theory. WS2010/11: Introduction to Nuclear and Particle Physics
Lecture 5 Hartree-Fock Theory WS2010/11: Introduction to Nuclear and Particle Physics Particle-number representation: General formalism The simplest starting point for a many-body state is a system of
More informationA new type of PT-symmetric random matrix ensembles
A new type of PT-symmetric random matrix ensembles Eva-Maria Graefe Department of Mathematics, Imperial College London, UK joint work with Steve Mudute-Ndumbe and Matthew Taylor Department of Mathematics,
More informationChemistry 483 Lecture Topics Fall 2009
Chemistry 483 Lecture Topics Fall 2009 Text PHYSICAL CHEMISTRY A Molecular Approach McQuarrie and Simon A. Background (M&S,Chapter 1) Blackbody Radiation Photoelectric effect DeBroglie Wavelength Atomic
More informationEmergent Fluctuation Theorem for Pure Quantum States
Emergent Fluctuation Theorem for Pure Quantum States Takahiro Sagawa Department of Applied Physics, The University of Tokyo 16 June 2016, YITP, Kyoto YKIS2016: Quantum Matter, Spacetime and Information
More informationTopics for the Qualifying Examination
Topics for the Qualifying Examination Quantum Mechanics I and II 1. Quantum kinematics and dynamics 1.1 Postulates of Quantum Mechanics. 1.2 Configuration space vs. Hilbert space, wave function vs. state
More informationRandom Matrix: From Wigner to Quantum Chaos
Random Matrix: From Wigner to Quantum Chaos Horng-Tzer Yau Harvard University Joint work with P. Bourgade, L. Erdős, B. Schlein and J. Yin 1 Perhaps I am now too courageous when I try to guess the distribution
More informationQuantum Physics II (8.05) Fall 2002 Outline
Quantum Physics II (8.05) Fall 2002 Outline 1. General structure of quantum mechanics. 8.04 was based primarily on wave mechanics. We review that foundation with the intent to build a more formal basis
More informationQuantum Origins of Irreversibility
Quantum Origins of Irreversibility Presented to the S. Daniel Abraham Honors Program in Partial Fulfillment of the Requirements for Completion of the Program Stern College for Women Yeshiva University
More informationSpin-Orbit Interactions in Semiconductor Nanostructures
Spin-Orbit Interactions in Semiconductor Nanostructures Branislav K. Nikolić Department of Physics and Astronomy, University of Delaware, U.S.A. http://www.physics.udel.edu/~bnikolic Spin-Orbit Hamiltonians
More informationIntroduction to Heisenberg model. Javier Junquera
Introduction to Heisenberg model Javier Junquera Most important reference followed in this lecture Magnetism in Condensed Matter Physics Stephen Blundell Oxford Master Series in Condensed Matter Physics
More informationd 3 r d 3 vf( r, v) = N (2) = CV C = n where n N/V is the total number of molecules per unit volume. Hence e βmv2 /2 d 3 rd 3 v (5)
LECTURE 12 Maxwell Velocity Distribution Suppose we have a dilute gas of molecules, each with mass m. If the gas is dilute enough, we can ignore the interactions between the molecules and the energy will
More informationPreface Introduction to the electron liquid
Table of Preface page xvii 1 Introduction to the electron liquid 1 1.1 A tale of many electrons 1 1.2 Where the electrons roam: physical realizations of the electron liquid 5 1.2.1 Three dimensions 5 1.2.2
More information(# = %(& )(* +,(- Closed system, well-defined energy (or e.g. E± E/2): Microcanonical ensemble
Recall from before: Internal energy (or Entropy): &, *, - (# = %(& )(* +,(- Closed system, well-defined energy (or e.g. E± E/2): Microcanonical ensemble & = /01Ω maximized Ω: fundamental statistical quantity
More informationMagnets, 1D quantum system, and quantum Phase transitions
134 Phys620.nb 10 Magnets, 1D quantum system, and quantum Phase transitions In 1D, fermions can be mapped into bosons, and vice versa. 10.1. magnetization and frustrated magnets (in any dimensions) Consider
More informationLuigi Paolasini
Luigi Paolasini paolasini@esrf.fr LECTURE 7: Magnetic excitations - Phase transitions and the Landau mean-field theory. - Heisenberg and Ising models. - Magnetic excitations. External parameter, as for
More informationNewton s Method and Localization
Newton s Method and Localization Workshop on Analytical Aspects of Mathematical Physics John Imbrie May 30, 2013 Overview Diagonalizing the Hamiltonian is a goal in quantum theory. I would like to discuss
More informationTheory of Mesoscopic Systems
Theory of Mesoscopic Systems Boris Altshuler Princeton University, Columbia University & NEC Laboratories America Lecture 1 01 June 2006 Meso: : intermediate between micro and macro Main Topics: Disordered
More informationIntroduction to Theory of Mesoscopic Systems. Boris Altshuler Columbia University & NEC Laboratories America
Introduction to Theory of Mesoscopic Systems Boris Altshuler Columbia University & NEC Laboratories America Introduction Einstein s Miraculous Year - 1905 Six papers: 1. The light-quantum and the photoelectric
More informationLecture #13 1. Incorporating a vector potential into the Hamiltonian 2. Spin postulates 3. Description of spin states 4. Identical particles in
Lecture #3. Incorporating a vector potential into the Hamiltonian. Spin postulates 3. Description of spin states 4. Identical particles in classical and QM 5. Exchange degeneracy - the fundamental problem
More informationEhud Altman. Weizmann Institute and Visiting Miller Prof. UC Berkeley
Emergent Phenomena And Universality In Quantum Systems Far From Thermal Equilibrium Ehud Altman Weizmann Institute and Visiting Miller Prof. UC Berkeley A typical experiment in traditional Condensed Matter
More informationUniversality of local spectral statistics of random matrices
Universality of local spectral statistics of random matrices László Erdős Ludwig-Maximilians-Universität, Munich, Germany CRM, Montreal, Mar 19, 2012 Joint with P. Bourgade, B. Schlein, H.T. Yau, and J.
More informationStudents are required to pass a minimum of 15 AU of PAP courses including the following courses:
School of Physical and Mathematical Sciences Division of Physics and Applied Physics Minor in Physics Curriculum - Minor in Physics Requirements for the Minor: Students are required to pass a minimum of
More informationSpins and spin-orbit coupling in semiconductors, metals, and nanostructures
B. Halperin Spin lecture 1 Spins and spin-orbit coupling in semiconductors, metals, and nanostructures Behavior of non-equilibrium spin populations. Spin relaxation and spin transport. How does one produce
More informationSTATISTICAL PHYSICS. Statics, Dynamics and Renormalization. Leo P Kadanoff. Departments of Physics & Mathematics University of Chicago
STATISTICAL PHYSICS Statics, Dynamics and Renormalization Leo P Kadanoff Departments of Physics & Mathematics University of Chicago \o * World Scientific Singapore»New Jersey London»HongKong Contents Introduction
More informationarxiv: v2 [cond-mat.stat-mech] 15 Apr 2016
Quantum Chaos and Thermalization in Isolated Systems of Interacting Particles F Borgonovi a, F M Izrailev b,d, L F Santos c, V G Zelevinsky d, arxiv:160201874v2 [cond-matstat-mech] 15 Apr 2016 a Dipartimento
More informationExperimental evidence of wave chaos signature in a microwave cavity with several singular perturbations
Chaotic Modeling and Simulation (CMSIM) 2: 205-214, 2018 Experimental evidence of wave chaos signature in a microwave cavity with several singular perturbations El M. Ganapolskii, Zoya E. Eremenko O.Ya.
More informationLECTURE 3 WORM ALGORITHM FOR QUANTUM STATISTICAL MODELS
LECTURE 3 WORM ALGORITHM FOR QUANTUM STATISTICAL MODELS LECTURE 3 WORM ALGORITHM FOR QUANTUM STATISTICAL MODELS Path-integral for lattice bosons*: oriented closed loops, of course LECTURE 3 WORM ALGORITHM
More informationElectronic structure of correlated electron systems. Lecture 2
Electronic structure of correlated electron systems Lecture 2 Band Structure approach vs atomic Band structure Delocalized Bloch states Fill up states with electrons starting from the lowest energy No
More informationDisordered topological insulators with time-reversal symmetry: Z 2 invariants
Keio Topo. Science (2016/11/18) Disordered topological insulators with time-reversal symmetry: Z 2 invariants Hosho Katsura Department of Physics, UTokyo Collaborators: Yutaka Akagi (UTokyo) Tohru Koma
More informationChaotic Scattering of Microwaves in Billiards: Induced Time-Reversal Symmetry Breaking and Fluctuations in GOE and GUE Systems 2008
Chaotic Scattering of Microwaves in Billiards: Induced Time-Reversal Symmetry Breaking and Fluctuations in GOE and GUE Systems 2008 Quantum billiards and microwave resonators as a model of the compound
More informationHopping transport in disordered solids
Hopping transport in disordered solids Dominique Spehner Institut Fourier, Grenoble, France Workshop on Quantum Transport, Lexington, March 17, 2006 p. 1 Outline of the talk Hopping transport Models for
More informationPotentially useful reading Sakurai and Napolitano, sections 1.5 (Rotation), Schumacher & Westmoreland chapter 2
Problem Set 2: Interferometers & Random Matrices Graduate Quantum I Physics 6572 James Sethna Due Friday Sept. 5 Last correction at August 28, 2014, 8:32 pm Potentially useful reading Sakurai and Napolitano,
More informationLecture 6. Fermion Pairing. WS2010/11: Introduction to Nuclear and Particle Physics
Lecture 6 Fermion Pairing WS2010/11: Introduction to Nuclear and Particle Physics Experimental indications for Cooper-Pairing Solid state physics: Pairing of electrons near the Fermi surface with antiparallel
More informationQuantum Physics II (8.05) Fall 2002 Assignment 12 and Study Aid
Quantum Physics II (8.05) Fall 2002 Assignment 12 and Study Aid Announcement This handout includes 9 problems. The first 5 are the problem set due. The last 4 cover material from the final few lectures
More informationThe Gutzwiller Density Functional Theory
The Gutzwiller Density Functional Theory Jörg Bünemann, BTU Cottbus I) Introduction 1. Model for an H 2 -molecule 2. Transition metals and their compounds II) Gutzwiller variational theory 1. Gutzwiller
More informationSpin- and heat pumps from approximately integrable spin-chains Achim Rosch, Cologne
Spin- and heat pumps from approximately integrable spin-chains Achim Rosch, Cologne Zala Lenarčič, Florian Lange, Achim Rosch University of Cologne theory of weakly driven quantum system role of approximate
More informationThe Oxford Solid State Basics
The Oxford Solid State Basics Steven H. Simon University of Oxford OXFORD UNIVERSITY PRESS Contents 1 About Condensed Matter Physics 1 1.1 What Is Condensed Matter Physics 1 1.2 Why Do We Study Condensed
More informationarxiv: v1 [cond-mat.str-el] 7 Oct 2017
Universal Spectral Correlations in the Chaotic Wave Function, and the Development of Quantum Chaos Xiao Chen 1, and Andreas W.W. Ludwig 2 1 Kavli Institute for Theoretical Physics, arxiv:1710.02686v1 [cond-mat.str-el]
More informationA. F. J. Levi 1 EE539: Engineering Quantum Mechanics. Fall 2017.
A. F. J. Levi 1 Engineering Quantum Mechanics. Fall 2017. TTh 9.00 a.m. 10.50 a.m., VHE 210. Web site: http://alevi.usc.edu Web site: http://classes.usc.edu/term-20173/classes/ee EE539: Abstract and Prerequisites
More informationGreen's Function in. Condensed Matter Physics. Wang Huaiyu. Alpha Science International Ltd. SCIENCE PRESS 2 Beijing \S7 Oxford, U.K.
Green's Function in Condensed Matter Physics Wang Huaiyu SCIENCE PRESS 2 Beijing \S7 Oxford, U.K. Alpha Science International Ltd. CONTENTS Part I Green's Functions in Mathematical Physics Chapter 1 Time-Independent
More information(Quantum) chaos theory and statistical physics far from equilibrium:
(Quantum) chaos theory and statistical physics far from equilibrium: Introducing the group for Non-equilibrium quantum and statistical physics Department of physics, Faculty of mathematics and physics,
More informationNPTEL
NPTEL Syllabus Nonequilibrium Statistical Mechanics - Video course COURSE OUTLINE Thermal fluctuations, Langevin dynamics, Brownian motion and diffusion, Fokker-Planck equations, linear response theory,
More informationNUMERICAL METHODS FOR QUANTUM IMPURITY MODELS
NUMERICAL METHODS FOR QUANTUM IMPURITY MODELS http://www.staff.science.uu.nl/~mitch003/nrg.html March 2015 Anrew Mitchell Utrecht University Quantum impurity problems Part 1: Quantum impurity problems
More informationQuantum Mechanics: Fundamentals
Kurt Gottfried Tung-Mow Yan Quantum Mechanics: Fundamentals Second Edition With 75 Figures Springer Preface vii Fundamental Concepts 1 1.1 Complementarity and Uncertainty 1 (a) Complementarity 2 (b) The
More informationPreface. Preface to the Third Edition. Preface to the Second Edition. Preface to the First Edition. 1 Introduction 1
xi Contents Preface Preface to the Third Edition Preface to the Second Edition Preface to the First Edition v vii viii ix 1 Introduction 1 I GENERAL THEORY OF OPEN QUANTUM SYSTEMS 5 Diverse limited approaches:
More informationNON-EQUILIBRIUM DYNAMICS IN
NON-EQUILIBRIUM DYNAMICS IN ISOLATED QUANTUM SYSTEMS Masud Haque Maynooth University Dept. Mathematical Physics Maynooth, Ireland Max-Planck Institute for Physics of Complex Systems (MPI-PKS) Dresden,
More informationThe Postulates of Quantum Mechanics Common operators in QM: Potential Energy. Often depends on position operator: Kinetic Energy 1-D case: 3-D case
The Postulates of Quantum Mechanics Common operators in QM: Potential Energy Often depends on position operator: Kinetic Energy 1-D case: 3-D case Time Total energy = Hamiltonian To find out about the
More informationThe Hubbard model in cold atoms and in the high-tc cuprates
The Hubbard model in cold atoms and in the high-tc cuprates Daniel E. Sheehy Aspen, June 2009 Sheehy@LSU.EDU What are the key outstanding problems from condensed matter physics which ultracold atoms and
More informationLecture 14 The Free Electron Gas: Density of States
Lecture 4 The Free Electron Gas: Density of States Today:. Spin.. Fermionic nature of electrons. 3. Understanding the properties of metals: the free electron model and the role of Pauli s exclusion principle.
More informationBlack holes and random matrices
Black holes and random matrices Stephen Shenker Stanford University Kadanoff Symposium Stephen Shenker (Stanford University) Black holes and random matrices Kadanoff Symposium 1 / 18 Black holes and chaos
More informationSemiclassical Electron Transport
Semiclassical Electron Transport Branislav K. Niolić Department of Physics and Astronomy, University of Delaware, U.S.A. PHYS 64: Introduction to Solid State Physics http://www.physics.udel.edu/~bniolic/teaching/phys64/phys64.html
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 informationQuantum Physics in the Nanoworld
Hans Lüth Quantum Physics in the Nanoworld Schrödinger's Cat and the Dwarfs 4) Springer Contents 1 Introduction 1 1.1 General and Historical Remarks 1 1.2 Importance for Science and Technology 3 1.3 Philosophical
More informationTopological insulator with time-reversal symmetry
Phys620.nb 101 7 Topological insulator with time-reversal symmetry Q: Can we get a topological insulator that preserves the time-reversal symmetry? A: Yes, with the help of the spin degree of freedom.
More informationQuantum chaos. University of Ljubljana Faculty of Mathematics and Physics. Seminar - 1st year, 2nd cycle degree. Author: Ana Flack
University of Ljubljana Faculty of Mathematics and Physics Seminar - 1st year, 2nd cycle degree Quantum chaos Author: Ana Flack Mentor: izred. prof. dr. Marko šnidari Ljubljana, May 2018 Abstract The basic
More informationLecture 19: Building Atoms and Molecules
Lecture 19: Building Atoms and Molecules +e r n = 3 n = 2 n = 1 +e +e r y even Lecture 19, p 1 Today Nuclear Magnetic Resonance Using RF photons to drive transitions between nuclear spin orientations in
More informationNon-equilibrium Dynamics of One-dimensional Many-body Quantum Systems. Jonathan Karp
Non-equilibrium Dynamics of One-dimensional Many-body Quantum Systems Thesis Submitted in Partial Fulfillment of the Requirements of the Jay and Jeanie Schottenstein Honors Program Yeshiva College Yeshiva
More informationTrapped ion spin-boson quantum simulators: Non-equilibrium physics and thermalization. Diego Porras
Trapped ion spin-boson quantum simulators: Non-equilibrium physics and thermalization Diego Porras Outline Spin-boson trapped ion quantum simulators o Rabi Lattice/Jahn-Teller models o Gauge symmetries
More informationP. W. Atkins and R. S. Friedman. Molecular Quantum Mechanics THIRD EDITION
P. W. Atkins and R. S. Friedman Molecular Quantum Mechanics THIRD EDITION Oxford New York Tokyo OXFORD UNIVERSITY PRESS 1997 Introduction and orientation 1 Black-body radiation 1 Heat capacities 2 The
More informationApplication of the Lanczos Algorithm to Anderson Localization
Application of the Lanczos Algorithm to Anderson Localization Adam Anderson The University of Chicago UW REU 2009 Advisor: David Thouless Effect of Impurities in Materials Naively, one might expect that
More informationAshvin Vishwanath UC Berkeley
TOPOLOGY + LOCALIZATION: QUANTUM COHERENCE IN HOT MATTER Ashvin Vishwanath UC Berkeley arxiv:1307.4092 (to appear in Nature Comm.) Thanks to David Huse for inspiring discussions Yasaman Bahri (Berkeley)
More informationPotential energy, from Coulomb's law. Potential is spherically symmetric. Therefore, solutions must have form
Lecture 6 Page 1 Atoms L6.P1 Review of hydrogen atom Heavy proton (put at the origin), charge e and much lighter electron, charge -e. Potential energy, from Coulomb's law Potential is spherically symmetric.
More information3 Symmetry Protected Topological Phase
Physics 3b Lecture 16 Caltech, 05/30/18 3 Symmetry Protected Topological Phase 3.1 Breakdown of noninteracting SPT phases with interaction Building on our previous discussion of the Majorana chain and
More informationThermalization in Quantum Systems
Thermalization in Quantum Systems Jonas Larson Stockholm University and Universität zu Köln Dresden 18/4-2014 Motivation Long time evolution of closed quantum systems not fully understood. Cold atom system
More informationLS coupling. 2 2 n + H s o + H h f + H B. (1) 2m
LS coupling 1 The big picture We start from the Hamiltonian of an atomic system: H = [ ] 2 2 n Ze2 1 + 1 e 2 1 + H s o + H h f + H B. (1) 2m n e 4πɛ 0 r n 2 4πɛ 0 r nm n,m Here n runs pver the electrons,
More informationMetropolis, 2D Ising model
Metropolis, 2D Ising model You can get visual understanding from the java applets available, like: http://physics.ucsc.edu/~peter/ising/ising.html Average value of spin is magnetization. Abs of this as
More informationErgodicity of quantum eigenfunctions in classically chaotic systems
Ergodicity of quantum eigenfunctions in classically chaotic systems Mar 1, 24 Alex Barnett barnett@cims.nyu.edu Courant Institute work in collaboration with Peter Sarnak, Courant/Princeton p.1 Classical
More information