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A semiclassical, quantitative approach to the Anderson transition Antonio M. García-García ag3@princeton.edu Princeton University We study analytically and numerically the metal-insulator transition in a d dimensional, non interacting (short range) disordered system by combining the self-consistent theory of localization with the one parameter scaling theory. We study analytically and numerically the metal-insulator transition in a d dimensional, non interacting (short range) disordered system by combining the self-consistent theory of localization with the one parameter scaling theory. The upper critical dimension is infinity. Level Statistics at the transition: AGG, Emilio Cuevas, Phys. Rev. B 75, 174203 (2007), AGG arXiv:0709.1292

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1. For a given disorder, E and d, how is the quantum dynamics? Metal or insulator like? 2. When does a metal insulator transition occurs?. How is it described? Metal Abs. Continuous Abs. Continuous Wigner-Dyson statistics Insulator Pure point spectrum Poisson statistics Main Goals: Transition Singular Multifractal Critical statistics

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What we (physicists) know believe: d = 1 An insulator for any disorder d =2 An insulator for any disorder d > 2 Disorder strong enough: Insulator Disorder weak enough: Mobility edge Why? Metal Insulator Transition E

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1. Self-consistent theory from the insulator side, valid only for d >>2. No interference. Abou-Chacra, Anderson. 1. Self-consistent theory from the insulator side, valid only for d >>2. No interference. Abou-Chacra, Anderson. 2. Self-consistent theory from the metallic side, valid only for d ~ 2. No tunneling. Vollhardt and Wolfle. 2. Self-consistent theory from the metallic side, valid only for d ~ 2. No tunneling. Vollhardt and Wolfle. 3. One parameter scaling theory. (1980) Correct (?) but qualitative 3. One parameter scaling theory. Anderson et al.(1980) Correct (?) but qualitative. Weak disorder/localization. Perturbation theory. Well understood. Relevant in the transition in d = 2+ (Wegner, Hikami, Efetov) 1. Numerical simulations Some of the main results of the field are already included in the original paper by Anderson 1957!! 2. Analytical Approaches to localization : Currently reliable Strong disorder/localization. NO quantitative theory but:

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Energy Scales 1. Mean level spacing: 2. Thouless energy: t T (L) is the travel time to cross a box of size L Dimensionless Thouless conductance Diffusive motion without quantum corrections Metal Insulator

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Scaling theory of localization The change in the conductance with the system size only depends on the conductance itself g Weak localization

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Predictions of the scaling theory at the transition 1. Diffusion becomes anomalous 2. Diffusion coefficient become size and momentum dependent 3. g=g c is scale invariant therefore level statistics are scale invariant as well Imry, Slevin

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1.Cooperons (Langer-Neal, maximally crossed, responsible for weak localization) and Diffusons (no localization, semiclassical) can be combined. 2. Perturbation theory around the metallic limit. 3. No control on the approximation.

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No control on the approximation. Exact for a Cayley tree. It should be a good approx for d>>2. = 0 metal insulator > 0 metal insulator The distribution of the self energy S i (E) is sensitive to localization. Perturbation theory around the insulator limit (locator expansion).

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Predictions of the self consistent theory 1. Critical exponents: 2. Critical disorder: 3. Critical conductance: d = 4 Upper critical dimension! Correctly predicts a transition for d>2 also B. Shapiro and E. Abrahams 1980 W c 3d ~ 14 Kroha, Wolfle, Kotov, Sadovskii Anderson, Abou Chacra, Thouless Vollhardt, Wolfle

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Numerical results at the transition 1. Scale invariance of the spectral correlations. 2. Intermediate level statistics 2. Intermediate level statistics 3. Critical exponents 3. Critical exponents 4. Critical disorder 4. Critical disorder 5. Anomalous diffusion Finite scale analysis, Shapiro, et al. 93 Disagreement with the selfconsistent theory ! Insulator Metal ? Schreiber, Grussbach Mirlin, Evers, Cuevas, Schreiber, Slevin Agreement scaling theory var

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SECOND PART What we did: Phys. Rev. B 75, 174203 (2007), 1. Numerical results for the Anderson transition in d=4,5,6, AGG and E. Cuevas, Phys. Rev. B 75, 174203 (2007), Critical exponents, critical disorder, level statistics AGG, arXiv:0709.1292 2. Analytical results combining the scaling theory and the self consistent condition, AGG, arXiv:0709.1292 Critical exponents, critical disorder, level statistics.

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Numerical Results: Anderson model cubic lattice, d=[3,6] Metal Insulator

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Critical exponents and Critical Disorder Cayley tree Upper critical dimension is infinity W c /ln(W c /2) Self consistent theory OK but

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Level Statistics A(d)-1 ln(P(s))

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Analytical results 1. Always perturbative around the metallic (Wolhardt & Wolfle) or the insulator state (Anderson, Abou Chacra, Thouless). A new basis for localization is needed A new basis for localization is needed 2. Anomalous diffusion at the transition (predicted by the scaling theory) is not taken into account. Why do self consistent methods fail for d 3?

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2. Right at the transition the quantum dynamics is well described by a process of anomalous diffusion. with no further localization corrections. Idea! (AGG arXiv:0709.1292) Solve the self consistent equation assuming that the diffusion coefficient is renormalized as predicted by the scaling theory Assumptions: 1. All the quantum corrections missing in the self consistent treatment are included by just renormalizing the coefficient of diffusion following the scaling theory.

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Technical details: Critical exponents The critical exponent ν, can be obtained by solving the above equation for with D (ω) = 0. 2

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Level Statistics: Starting point: Anomalous diffusion predicted by the scaling theory Semiclassically, only “diffusons” Two levels correlation function

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Cayley tree Aizenman, Warzel

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Comparison with numerical results 1. Critical exponents: Excellent 2, Level statistics: Good (problem with g c ) 3. Critical disorder: Not better than before

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CONCLUSIONS 1.We obtain analytical results at the transition by combining the scaling theory with the self consistent in d>3. 2. The upper critical dimension is infinity 3. Analytical results on the level statistics agree with numerical simulations. What is next? 1. Experimental verification. 2. Anderson transition in correlated potential

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Experiments: Our findings may be used to test experimentally the Anderson transition by using ultracold atoms techniques. One places a dilute sample of ultracold Na/Cs in a periodic step-like standing wave which is pulsed in time to approximate a delta function then the atom momentum distribution is measured. The classical singularity cannot be reproduced in the lab. However (AGG, W Jiao, PRA 2006) an approximate singularity will still show typical features of a metal insulator transition.

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Spectral signatures of a metal (Wigner-Dyson): 1. Level Repulsion 2. Spectral Rigidity Spectral Signatures of an insulator: (Poisson) s P(s)

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