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2162-5 Advanced Workshop on Anderson Localization, Nonlinearity and Turbulence: a Cross-Fertilization 23 August - 3 September, 2010 INTRODUCTORY Anderson Localization - Introduction Boris ALTSHULER Columbia University, Dept. of Physics, New York NY U.S.A. Anderson localization - introduction Boris Altshuler Physics Department, Columbia University Advanced Workshop on Anderson Localization, Nonlinearity and Turbulence: a Cross-Fertilization August 23 – September 03, 2010 One--particle Localization >50 years of Anderson Localization One quantum particle Random potential (e.g., impurities) Elastic scattering q.p. Einstein (1905): r2 Dt Random walk diffusion constant always diffusion as long as the system has no memory Anderson(1958): It might be that For quantum particles r 2 t o const of D not always! Quantum interference memory 0 Einstein Relation (1905) V e DQ 2 Conductivity dn Q{ dP Density of states Diffusion Constant U I V G §I· ¨ ¸ ; © V ¹V 0 V L G A wU D 2 U wt 0 t t! x Einstein Relation (1905) V e DQ 2 Conductivity dn Q{ dP Density of states Diffusion Constant No diffusion – no conductivity Localized states – insulator Extended states - metal Metal – insulator transition Localization of single-electron wave-functions: z Disorder extended I V extended Conductance localized localized Lx Ly ½ extended ° V L ° z ° ° ® ¾ °v exp § Lz · localized ° ¨ ¸ ° ° 9 © loc ¹ ¯ ¿ Spin Diffusion Experiment Feher, G., Phys. Rev. 114, 1219 (1959); Feher, G. & Gere, E. A., Phys. Rev. 114, 1245 (1959). Light Wiersma, D.S., Bartolini, P., Lagendijk, A. & Righini R. “Localization of light in a disordered medium”, Nature 390, 671-673 (1997). Scheffold, F., Lenke, R., Tweer, R. & Maret, G. “Localization or classical diffusion of light”, Nature 398,206-270 (1999). Schwartz, T., Bartal, G., Fishman, S. & Segev, M. “Transport and Anderson localization in disordered two dimensional photonic lattices”. Nature 446, 52-55 (2007). C.M. Aegerter, M.Störzer, S.Fiebig, W. Bührer, and G. Maret : JOSA A, 24, #10, A23, (2007) Microwave Dalichaouch, R., Armstrong, J.P., Schultz, S.,Platzman, P.M. & McCall, S.L. “Microwave localization by 2-dimensional random scattering”. Nature 354, 53, (1991). Chabanov, A.A., Stoytchev, M. & Genack, A.Z. Statistical signatures of photon localization. Nature 404, 850, (2000). Pradhan, P., Sridar, S, “Correlations due to localization in quantum eigenfunctions od disordered microwave cavities”, PRL 85, (2000) Sound Weaver, R.L. Anderson localization of ultrasound. Wave Motion 12, 129-142 (1990). f = 3.04 GHz Localized State Anderson Insulator f = 7.33 GHz Extended State Anderson Metal Localization of cold atoms Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008). 87Rb Roati et al. “Anderson localization of a non-interacting Bose-Einstein condensate“. Nature 453, 895-898 (2008). • Lattice - tight binding model Anderson Model • Onsite energies j Iij -W < Hi <W uniformly distributed i Hi - random • Hopping matrix elements Iij I i and j are nearest { Iij = 0 neighbors otherwise Anderson Transition I < Ic Insulator All eigenstates are localized Localization length [ I > Ic Metal There appear states extended all over the whole system Q: Why arbitrary weak hopping I is not sufficient for the existence of the diffusion ? j i Iij Einstein (1905): Marcovian (no memory) process J diffusion ! Quantum mechanics is not marcovian There is memory in quantum propagation Why ? H2 I H1 Hamiltonian Hˆ § H1 I · ¸¸ ¨¨ © I H2 ¹ E2 E1 diagonalize § E1 ¨¨ ©0 Hˆ H 2 H1 2 I 2 0· ¸¸ E2 ¹ Hˆ § H1 I · ¸¸ ¨¨ © I H2 ¹ E2 E1 H 2 H1 2 H 2 H 1 H 2 H 1 !! I I | I H 2 H 1 I 2 What about the eigenfunctions ? I1 , H1 ; I2 , H 2 H 2 H 1 !! I \ 1, 2 \ 1 , E1 ; \ 2 , E2 H 2 H 1 I § I · ¸¸M 2,1 M1, 2 O¨¨ \ | M r M H H 1 , 2 1 , 2 2 ,1 2 1 ¹ © Off-resonance Eigenfunctions are close to the original onsite wave functions Resonance In both eigenstates the probability is equally shared between the sites Anderson insulator Few isolated resonances Anderson metal There are many resonances and they overlap Typically each site is in Transition: resonance with some other one Anderson Transition I > Ic I < Ic localized and extended never coexist! all states are localized extended DoS - mobility edges (one particle) DoS Condition for Localization: I< energy mismatch # of n.neighbors energy mismatch H i H j typ W # of nearest neighbors 2d A bit more precise: Logarithm is due to the resonances, which are not nearest neighbors Condition for Localization: Q:Is it correct ? A1:Is exact on the Cayley tree Ic W , K ln K K is the branching number K 2 Anderson Model on a Cayley tree Condition for Localization: Q:Is it correct ? A1:Is exact on the Cayley tree Ic W , K ln K K is the branching number K a good approximation at high dimensions. Is Is qualitatively correct for d t 3 2 Anderson’s recipe: 1. take descrete spectrum EP of H0 2. Add an infinitesimal Im part iK to EP insulator 3. Evaluate Im6 P 1 2 4 1) N o f limits 2) K o0 4. take limit K o 0 but only after N o f 5. “What we really need to know is the probability distribution of Im6, not its average…” ! metal Probability Distribution of *=Im 6 K is an infinitesimal width (Im metal insulator Look for: V part of the self-energy due to a coupling with a bath) of one-electron eigenstates Condition for Localization: Q:Is it correct ? A1:Is exact on the Cayley tree Ic W , K ln K K is the branching number K 2 a good approximation at high dimensions. Is Is qualitatively correct for d t 3 A2: For low dimensions – NO. I c f for d 1, 2 All states are localized. Reason – loop trajectories 1D Localization Gertsenshtein & Vasil'ev, Exactly solved: 1959 all states are localized Conjectured: Mott & Twose, 1961 ... Condition for Localization: Q:Is it correct ? For low dimensions – NO. I f for d 1, 2 A2:All states are localized. Reason – loop trajectories c M & & ³ pdr Phase accumulated when traveling along the loop M1 M 2 O The particle can go around the loop in two directions Memory! Einstein: there is no diffusion at too short scales – there is memory, i.e., the process is not marcovian. Due to the localization effects diffusion description fails at large scales. Quantum interference Memory M1 M 2 O Large scales are important diffusion constant depends on the system size Einstein relation for the conductivity V V Conductance G d 2 VL dn Q{ dP e DQ 2 2 G g L for a cubic sample of the size L g L hD L2 d 1QL e d Dh QL 2 h L Thouless energy mean level spacing Dimensionless Thouless conductance Scaling theory of Localization (Abrahams, Anderson, Licciardello and Ramakrishnan 1979) 2 Dimensionless Thouless g = Gh/e g = ET / G conductance L = 2L = 4L = 8L .... without quantum corrections ET vL -2 ET ET ET ET G1 G1 G1 G1 g g g g G1vL -d d log g E g d log L d log g E g d log L Universal, i.e., material independent E – function is Limits: g !! 1 g v Ld 2 g 1 g v e L [ E g But It depends on the global symmetries, e.g., it is different with and without T-invariance (in orthogonal and unitary ensembles) §1· d 2 O ¨ ¸ ©g¹ E g | log g 0 !0 d!2 ?? d 2 0 d2 1 E(g) E - function d log g d log L unstable fixed point 3D gc | 1 -1 E g g 2D 1D Metal – insulator transition in 3D All states are localized for d=1,2 RG approach Effective Field Theory of Localization – Nonlinear V - model 1 E(g) E - function d log g d log L unstable fixed point 3D gc | 1 -1 cd E g d 2 g cd ? r ? E g g 2D 1D g L V cl Ld 2 cd d 2 dz2 § L· c2 log ¨ ¸ d ©l¹ 2 vF dt c ³W Dtcd 2 t P t d 1 Gg v F L | log g D DW v F 2 1 } SQ g Q D! Gg E (g) Gg g | P tmax 2 v F L log D l L log S l 2 2 Sg Universal !!! For d=1,2 all states are localized. M & & ³ pdr Phase accumulated when traveling along the loop M1 M 2 O The particle can go around the loop in two directions Memory! Weak Localization: The localization length 9 can be large Inelastic processes lead to dephasing, which is characterized by the dephasing length L M If 9 !! LM , then only small corrections to a conventional metallic behavior R.A. Chentsov “On the variation of electrical resistivity of tellurium in magnetic field at low temperatures”, Zh. Exp. Theor. Fiz. v.18, 375-385, (1948). Magnetoresistance ) O No magnetic field M M O With magnetic field H MM S)) Length Scales 1/2 Magnetic length LH = (hc/eH) 1/2 Dephasing length LM = (D WM) G g H § LH fd ¨ ¨ LM © · ¸¸ ¹ Universal functions Magnetoresistance measurements allow to study inelastic collisions of electrons with phonons and other electrons Weak Localization Negative Magnetoresistance Chentsov (1949) Aharonov-Bohm effect Theory Experiment B.A., Aronov & Spivak (1981) Sharvin & Sharvin (1981) Temperature dependence of the conductivity one-electron picture Chemical potential DoS DoS V T o 0 ! 0 V T v e E c H F T DoS V T 0 T Temperature dependence of the conductivity one-electron picture Assume that all the states are localized; e.g. d = 1,2 DoS V T 0 T Inelastic processes transitions between localized states E energy mismatch D T 0 V 0 (any mechanism) Phonon-assisted hopping !Z E D !Z HD H E V 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 Z= 0 will give the same exponential Spectral statistics and Localization Spectral statistics RANDOM MATRIX THEORY ensemble of Hermitian matrices with random matrix element N uN N o ED - spectrum (set of eigenvalues) G 1 { ED 1 ED - mean level spacing, determines the density of states ...... - ensemble averaging s{ ED 1 ED P s G1 - spacing between nearest neighbors - distribution function of nearest neighbors spacing between Spectral Rigidity P s 0 0 Level repulsion P s 1 v s E E 1,2,4 P(s) Wigner-Dyson; GOE Poisson Gaussian Orthogonal Ensemble Orthogonal E P( s ) v s E Unitary E Simplectic E Poisson – completely uncorrelated levels • Lattice - tight binding model Anderson Model • Onsite energies j i Hi - random • Hopping matrix elements Iij Iij -W < Hi <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 ? Anderson Transition Strong disorder Weak disorder I < Ic I > Ic Insulator Metal All eigenstates are localized Localization length [ There appear states extended all over the whole system The eigenstates, which are localized at different places will not repel each other Any two extended eigenstates repel each other Poisson spectral statistics Wigner – Dyson spectral statistics Anderson Localization and Spectral Statistics 0 Ic I Localized states Insulator Extended states Metal Poisson spectral statistics Wigner-Dyson spectral statistics Extended Level repulsion, anticrossings, Wigner-Dyson spectral statistics states: Localized Poisson spectral statistics states: Invariant (basis independent) definition In general: Localization in the space of quantum numbers. KAM tori localized states. Glossary Classical Quantum Integrable Integrable H0 & H0 I Hˆ 0 ¦P EP P P, KAM Localized Ergodic – distributed all over the energy shell Chaotic Extended ? P & I Many--Body Localization BA, Gefen, Kamenev & Levitov, 1997 Basko, Aleiner & BA, 2005. . . Example: Random Ising model in the perpendicular field Will not discuss today in detail Hˆ N N i 1 i 1 z z z x x ˆ ˆ ˆ ˆ ˆ ˆ B V J V V I V { H I V ¦ i i ¦ ij i j ¦ i 0 ¦ i i 1 & N iz j Random Ising model in a parallel field V i - Pauli matrices, V z i i 1, 2,..., N ; N !! 1 Perpendicular field 1 r 2 V z i Without perpendicular field all commute with the Hamiltonian, i.e. they are integrals of motion Hˆ N N i 1 i 1 z z z x x ˆ ˆ ˆ ˆ ˆ ˆ B V J V V I V { H I V ¦ i i ¦ ij i j ¦ i 0 ¦ i i 1 iz j Random Ising model in a parallel field & N V i - Pauli matrices i 1, 2,..., N ; N !! 1 Perpendicular field Withoutz perpendicular field all V i commute with the Hamiltonian, i.e. they are integrals of motion Anderson Model on ^V z ` determines i N-dimensional cube x ˆ ˆ ˆ V V V H ^V ` 0 i onsite energy hoping between nearest neighbors a site Hˆ N N N i 1 i 1 z z z x x ˆ ˆ ˆ ˆ ˆ ˆ { B V J V V I V H I V ¦ i i ¦ ij i j ¦ i 0 ¦ i i 1 iz j Anderson Model on N-dimensional cube Usually: Here: # of dimensions d o const # of dimensions system linear size Lof system linear size d N of L 1 6-dimensional cube 9-dimensional cube Hˆ N N N i 1 i 1 z z z x x ˆ ˆ ˆ ˆ ˆ ˆ { B V J V V I V H I V ¦ i i ¦ ij i j ¦ i 0 ¦ i i 1 iz j Anderson Model on N-dimensional cube Localization: •No relaxation •No equipartition •No temperature •No thermodynamics Glass ??