Long Range Spatial Correlations in One-

Dimensional Anderson Models

Greg M. PetersenNancy Sandler

Ohio UniversityDepartment of Physics and Astronomy

1D Anderson Transition?

Greg M. Petersen

Evidence For

Dunlap, Wu, and Phillips, PRL (1990)

Moura and Lyra, PRL (1998)

Evidence Against

Kotani and Simon, Commun. Math. Phys (1987)

García-García and Cuevas, PRB (2009)

Cain et al. EPL (2011)

Abrahams et al. PRL (1979)Johnston and Kramer Z Phys. B (1986)

E/t

The Model

α=.1

α=.5

α=1

Greg M. Petersen

Generation Method: 1. Find spectral density 2. Generate {V(k)} from Gaussian with variance S(k) 3. Apply conditions V(k) = V*(-k) 4. Take inverse FT to get {Є

i}

Recursive Green's Function Method

Greg M. Petersen

Klimeck http://nanohub.org/resources/165 (2004)

Lead LeadConductor

Also get DOS

Verification of Single Parameter Scaling

Greg M. Petersen

Slope

All Localized

Transfer Matrix Method

Greg M. Petersen

Less Localized More Localized

Crossover Energy

Analysis of the Crossing Energy

Greg M. Petersen

More Localized

Less Localized

Participation Ratio

Greg M. Petersen

- Wavefunctions are characterized by fractal exponents.

Fractal Exponent D of IPR

Greg M. Petersen

E=0.1

E=1.3

E=2.5

Character of eigenstates changes for alpha less than 1.

Exam

Greg M. Petersen

Cain et al. EPL (2011) – no transition

Petersen, Sandler (2012)- no transition

Moura and Lyra, PRL (1998)- transition

Conclusions

- All states localize

- Single parameter scaling is verified

Thank you for your attention!

- Found more and less localized regions

Greg M. Petersen

- Determined dependence of W/t on crossing energy- Calculated the fractal dimension D by IPR

- D is conditional dependent on alpha