Anderson localization: from theoretical aspects to applications

1. Anderson localization: from theoretical aspects to applications Antonio M. García-García ag3@princeton.edu http://phy-ag3.princeton.edu Princeton and ICTP Analytical approach to the 3d Anderson transition Theoretical aspects Existence of a band of metallic states in 1d Localization in Quantum Chromodynamics Applications Emilio Cuevas, Wang Jiao, James Osborn Collaborators:

2. Problem: Get analytical expressions for different quantities characterizing the metal-insulator transition in d  3 such as , level statistics. Locator expansions One parameter scaling theory Selfconsistent condition Quasiclassical approach to the Anderson transition

3. Anderson localization 50’ 70’ 70’ 80’ 90’ 00’ Perturbative locator expansion Anderson Self consistent conditions Abou Chakra, Anderson, Thouless, Vollhardt, Woelfle 1d Kotani, Pastur, Sinai, Jitomirskaya, Mott. Thouless, Wegner, Gang of four, Frolich, Spencer, Molchanov, Aizenman Scaling Dynamical localization Fishman, Grempel, Prange, Casati Weak Localization Lee Efetov, Wegner Field theory Efetov, Fyodorov,Mirlin, Klein, Zirnbauer,Kravtsov Cayley tree and rbm Computers Aoki, Schreiber, Kramer, Shapiro Experiments Aspect, Fallani, Segev

4. But my recollection is that, on the whole, the attitude was one of humoring me. 4202 citations! What if I place a particle in a random potential and wait? Tight binding model Vij nearest neighbors, I random potential Not rigorous! Small denominators Technique: Looking for inestabilities in a locator expansion Correctly predicts a metal-insulator transition in 3d and localization in 1d Interactions? Disbelief?, against the spirit of band theory

5. Perturbation theory around the insulator limit (locator expansion). No control on the approximation. It should be a good approx for d>>2. It predicts correctly localization in 1d and a transition in 3d The distribution of the self energy Si (E) is sensitive to localization. metal insulator > 0 metal = 0 insulator ~ h

6. Scaling theory of localization Phys. Rev. Lett. 42, 673 (1979), Gang of four. Based on Thouless,Wegner, scaling ideas 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 localization corrections Metal Insulator

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

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

9. Weak localization Positive correction to the resistivity of a metal at low T 1.Cooperons (Langer-Neal, maximally crossed, responsible for weak localization) and Diffusons (no localization, semiclassical) can be combined. 3. Accurate in d ~2. Self consistent condition (Wolfle-Volhardt) No control on the approximation!

10. Predictions of the self consistent theory at the transition 1. Critical exponents: Vollhardt, Wolfle,1982 2. Transition for d>2 Disagreement with numerical simulations!! Why? 3. Correct for d ~ 2

11. Why do self consistent methods fail for d = 3? 1. Always perturbative around the metallic (Vollhardt & 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.

12. Proposal: Analytical results combining the scaling theory and the self consistent condition.  and level statistics.

13. Idea 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.

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

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

16. Shapiro, Abrahams Aizenman, Warzel Cayley tree A linear number variance in the 3d case was obtained by Altshuler et al.’88 Chalker Kravtsov,Lerner

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

18. Problem: Conditions for the absence of localization in 1d Motivation Quasiperiodic potentials Nonquasiperiodic potentials Work in progress in collaboration with E Cuevas

19. Your intuition about localization Ea Random V(x) Eb 0 Ec X For any of the energies above: Will the classical motion be strongly affected by quantum effects?

20. tt t t Speckle potentials The effective 1d random potential is correlated

22. Localization/Delocalization in 1d: Exponential localization for every energy and disorder Random uncorrelated potential Bloch theorem. Absence of localization. Band theory Periodic potential In between?

23. Quasiperiodic potentials Jitomirskaya, Sinai,Harper,Aubry Similar results What it is the least smooth potential that can lead to a band of metallic states? No metallic band if V(x) is discontinuous Jitomirskaya, Aubry, Damanik Jitomirskaya, Bourgain Conjecture

24. Fourier space: Localization for >0 Delocalization in real space Long range hopping Levitov FyodorovMirlin A metallic band can exist for

25. Non quasiperiodic potentials Physics literature 1. Izrailev & Krokhin Metallic band if: Neither of them is accurate Born approximation 1a. A vanishing Lyapunov exponent does not mean metallic behavior. 1b. Higher order corrections make the Lyapunov exponent > 0 2. Not generic 2. Lyra & Moura Decaying and sparse potentials (Kunz,Simon, Soudrillard): transition but non ergodic Localization in correlated potentials: Luck, Shomerus, Efetov, Mirlin,Titov

26. Mathematical literature: Kotani, Simon, Kirsch, Minami, Damanik. Kotani’s theory of ergodic operators Non deterministic potentials No a.c. spectrum Deterministic potentials More difficult to tell Discontinuous potentials No a.c. spectrum Damanik,Stolz, Sims No a.c. spectrum

27. A band of metallic states might exist provided  > 0 and V(x) and its  derivative are bounded. Neighboring values of the potential must be correlated enough in order to avoid destructive interference. According to the scaling theory in the metallic region motion must be ballistic. How to proceed? Smoothing uncorrelated random potentials

28. Finite size scaling analysis Spectral correlations are scale invariant at the transition Thouless, Shklovski, Shapiro 93’ AGG, Cuevas Diffusive Metal Clean metal Insulator

29. Savitzsky-Golay 1. Take np values of V(n) around a given V(n0) 2. Replace V(n0) by the best fit of the np values to a polynomial of M degree 3. Repeat for all n0 Resulting potential is not continuous A band of metallic states does not exist

30. Fourier filtering 1. Fourier transform of the uncorrelated noise. 2. Remove k > kcut 3. Fourier transform back to real space Resulting potential is analytic A band of metallic states do exist

31. Gruntwald Letnikov operator Resulting potential is C-+1/2

32. A band of metallic states exists provided Is this generic?

33. Localization in systems with chiral symmetry and applications to QCD 1. Chiral phase transition in lattice QCD as a metal-insulator transition,Phys.Rev. D75 (2007) 034503, AMG, J. Osborn 2. Chiral phase transition and Anderson localization in the Instanton Liquid Model for QCD , Nucl.Phys. A770 (2006) 141-161, AMG. J. Osborn 3. Anderson transition in 3d systems with chiral symmetry, Phys. Rev. B 74, 113101 (2006), AMG, E. Cuevas 4. Long range disorder and Anderson transition in systems with chiral symmetry , AMG, K. Takahashi, Nucl.Phys. B700 (2004) 361 5. Chiral Random Matrix Model for Critical Statistics,Nucl.Phys. B586 (2000) 668-685, AMG and J. Verbaarschot

34. QCD : The Theory of the strong interactions HighEnergyg << 1 Perturbative 1. Asymptotic freedom Quark+gluons, Well understood Low Energy g ~ 1 Lattice simulations The world around us 2. Chiral symmetry breaking Massive constituent quark 3. Confinement Colorless hadrons How to extract analytical information?Instantons , Monopoles, Vortices

35. Deconfinement and chiral restoration Deconfinement:Confining potentialvanishes: Chiral Restoration:Matter becomes light: How to explain these transitions? 1. Effective, simple, model of QCD close to the phase transition (Wilczek,Pisarski,Yaffe): Universality. 2. Classical QCD solutions (t'Hooft): Instantons (chiral), Monopoles and vortices (confinement). We propose that quantum interference/tunneling plays an important role.

36. QCD at T=0, instantons and chiral symmetry breakingtHooft, Polyakov, Callan, Gross, Shuryak, Diakonov, Petrov,VanBaal Instantons:Non perturbative solutions of the classical Yang Mills equation. Tunneling between classical vacua. 1. Dirac operator has a zero mode in the field of an instanton 2. Spectral properties of the smallest eigenvalues of the Dirac operator are controled by instantons 3. Spectral properties related to chiSB. Banks-Casher relation:

37. Instanton liquid models T = 0 Multiinstanton vacuum? Non linear equations Nosuperposition Variational principles(Dyakonov), Instanton liquid model (Shuryak). Solution ILM T > 0

38. QCD vacuum as a conductor (T =0) Metal:An electron initially bounded to a single atom gets delocalized due to the overlapping with nearest neighbors QCD Vacuum:Zero modes initially bounded to an instanton get delocalized due to the overlapping with the rest of zero modes. (Diakonov and Petrov) Differences Dis.Sys: Exponential decay QCD vacuum: Power law decay

39. QCD vacuum as a disordered conductor Diakonov, Petrov, Verbaarschot, Osborn, Shuryak, Zahed,Janik Instanton positions and color orientations vary Electron Quarks Ion Instantons T = 0 TIA~ 1/R,  = 3<4 T>0 TIA~ e-R/l(T) QCD vacuum is a conductor A transition is possible Shuryak,Verbaarschot, AGG and Osborn

40. QCD Dirac operator Phys.Rev. D75 (2007) 034503 Nucl.Phys. A770 (2006) 141 with J. Osborn At the same Tc that the Chiral Phase transition undergo a metal - insulator transition A metal-insulator transition in the Dirac operator induces the QCD chiral phase transition

41. Signatures of a metal-insulator transition 1. Scale invariance of the spectral correlations. A finite size scaling analysis is then carried out to determine the transition point. 2. 3. Eigenstates are multifractals. Skolovski, Shapiro, Altshuler var Mobility edge Anderson transition

42. ILM, close to the origin, 2+1 flavors, N = 200 Metal insulator transition

43. Spectrum is scale invariant ILM with 2+1 massless flavors, We have observed a metal-insulator transition at T ~ 125 Mev

44. Localization versus chiral transition Instanton liquid model Nf=2, masless Chiral and localizzation transition occurs at the same temperature

46. Problem: To determine the importance of Anderson localization effects in deterministic (quantum chaos) systems Scaling theory in quantum chaos Metal insulator transition in quantum chaos

47. Quantum chaos studies the quantum properties of systems whose classical motion is chaotic (or not) What is quantum chaos? Bohigas-Giannoni-Schmit conjecture Classical chaos Wigner-Dyson Energy is the only integral of motion Momentum is not a good quantum number Delocalization

48. Gutzwiller-Berry-Tabor conjecture Poisson statistics (Insulator) Integrable classical motion Integrability P(s) Canonical momenta are conserved s System is localized in momentum space

49. Dynamical localization Fishman, Prange, Casati Exceptions to the BGS conjecture 1. Kicked systems Classical <p2> Quantum Dynamical localization in momentum space t 2. Harper model 3. Arithmetic billiards

50. Characterization Random Deterministic d > 2 Weak disorder Wigner-Dyson Delocalization Normal diffusion Chaotic motion Always? d = 1,2 d > 2 Strong disorder Poisson Localization Diffusion stops Integrable motion Bogomolny Altshuler, Levitov Casati, Shepelansky Critical statistics Multifractality Anomalous diffusion d > 2 Critical disorder ??????????