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A semiclassical, quantitative approach to the Anderson transition

A semiclassical, quantitative approach to the Anderson transition. Antonio M. Garc í a-Garc í a ag3@princeton.edu Princeton University

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A semiclassical, quantitative approach to the Anderson transition

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  1. 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.  The upper critical dimension is infinity. Level Statistics at the transition: AGG, Emilio Cuevas, Phys. Rev. B 75, 174203 (2007), AGG arXiv:0709.1292

  2. Main Goals: 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? Transition Insulator Metal Abs. Continuous Singular Pure point spectrum Multifractal Wigner-Dyson statistics Critical statistics Poisson statistics

  3. 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 r Metal Insulator Transition Why? E

  4. Approaches to localization: 1. Numerical simulations Currently reliable 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. 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+e (Wegner, Hikami, Efetov) 2. Analytical Strong disorder/localization. NOquantitative theory but: Some of the main results of the field are already included in the original paper by Anderson 1957!!

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

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

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

  8. 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.

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

  10. Correctly predicts a transition for d>2 Predictions of the self consistent theory 1. Critical exponents: Vollhardt, Wolfle d = 4 Upper critical dimension! 2. Critical disorder: Wc3d ~ 14 Anderson, Abou Chacra, Thouless Kroha, Wolfle, Kotov, Sadovskii 3. Critical conductance: also B. Shapiro and E. Abrahams 1980

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

  12. SECOND PART What we did: 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 2. Analytical results combining the scaling theory and the self consistent condition, AGG, arXiv:0709.1292 Critical exponents, critical disorder, level statistics.

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

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

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

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

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

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

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

  20. Cayley tree Aizenman, Warzel

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

  22. 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. CONCLUSIONS What is next? 1. Experimental verification. 2. Anderson transition in correlated potential

  23. 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.

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

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