Single parameter scaling of 1d systems with long range correlated disorder Why correlated disorder Long standing question role of correlations in Anderson localization Potentially ID: 1045374
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1. Greg Petersen and Nancy SandlerSingle parameter scaling of 1d systems with long-range correlated disorder
2. Why correlated disorder? Long standing question: role of correlations in Anderson localization. Potentially accessible in meso and nanomaterials: disorder is or can be ‘correlated’.
3. Graphene: RIPPLED AND STRAINEDBao et al. Nature Nanotech. 2009Lau et al. Mat. Today 2012http://www.materials.manchester.ac.uk/E.E. Zumalt, Univ. of Texas at Austin
4. Multiferroics: magnetic tweedhttp://www.msm.cam.ac.uk/dmg/Research/Index.htmlN. Mathur CambridgeTheory: Porta et al PRB 2007Correlation length of disorderScaling exponent
5. BEC in Optical latticesBilly et al. Nature 2008http://www.lcf.institutoptique.fr/Groupes-de-recherche/Optique-atomique/Experiences/Transport-QuantiqueTheory: Sanchez-Palencia et al. PRL 2007.
6. Disorder correlationsQuasi-periodic real space orderRandom disorder amplitudes chosen from a discrete set of values.Specific long range correlations (spectral function) Mobility edge: Anderson transitionDiscrete number of extended statesSome (not complete!) references:Johnston and Kramer Z. Phys. B 1986 Dunlap, Wu and Phillips, PRL 1990De Moura and Lyra, PRL 1998Jitomirskaya, Ann. Math 1999Izrailev and Krokhin, PRL 1999Dominguez-Adame et al, PRL 2003Shima et al PRB 2004Kaya, EPJ B 2007Avila and Damanik, Invent. Math 2008Reviews:Evers and Mirlin, Rev. Mod. Phys. 2008Izrailev, Krokhin and Makarov, Phys. Reps. 2012This work: scale free power law correlated potential (more in Greg’s talk).
7. Outline Scaling of conductance Localization length Participation RatioG. Petersen and NS submitted.
8. How does a power law long-range disorder look like?Smoothening effect as correlations increase
9. Model and generation of potential Fast Fourier Transform Tight binding Hamiltonian:Correlation function:Spectral function:(Discrete Fourier transform)
10. Conductance Scaling I: Method Conductance from transmission function T:Green’s function*:Self-energy:Hybridization:*Recursive Green’s Function method
11. Conductance Scaling II: BETA FUNCTION?COLLAPSE!IS THIS SINGLE PARAMETER SCALING?NEGATIVE!
12. CONDUCTANCE Scaling III: Second momentSingle Parameter Scaling:ESPSShapiro, Phil. Mag. 1987Heinrichs, J.Phys.Cond Mat. 2004 (short range)
13. Conductance Scaling IV: ESPSWEAK DISORDERCORRELATIONS
14. CONDUCTANCE Scaling V: Rescaling of disorder strengthDerrida and Gardner J. Phys. France 1984Russ et al Phil. Mag. 1998Russ, PRB 2002
15. Localization length Iw/t =1Lyapunov exponent obtained from Transfer Matrix:ECRuss et al Physica A 1999Croy et al EPL 2011
16. Localization length II: EC Enhanced localizationEnhanced localization length
17. Localization length III: CRITICAL EXPONENTw/t=1
18. Participation Ratio IE/t = 0.1E/t = 1.7IS THERE ANY DIFFERENCE?
19. Participation Ratio II: Fractal exponentE/t = 0.1E/t = 1.7
20. Classical systems: Harris criterion (‘73):Consistency criterion: As the transition is approached, fluctuations should grow less than mean values.“A 2d disordered system has a continuous phase transition (2nd order) with the same critical exponentsas the pure system (no disorder) if n 1”.How does disorder affect critical exponents?
21. Weinrib and Halperin (PRB 1983): True if disorder has short-range correlations only.For a disorder potential with long-range correlations:There are two regimes:Long-range correlated disorder destabilizes the classical critical point! (=relevant perturbation => changes critical exponents)Extended Harris criterion
22. Bringing all together: ConclusionsScaling is ‘valid’ within a region determined by disorder strength that is renormalized by No Anderson transition !!!!! and D appear to follow the Extended Harris Criterion
23. SupportNSF- PIRENSF- MWN - CIAM Ohio UniversityCondensed Matter and Surface ScienceGraduate Fellowship Ohio UniversityNanoscale and Quantum PhenomenaInstitute