Long Range Spatial Correlations in One- Dimensional Anderson Models Greg M. Petersen Nancy Sandler Ohio University Department 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 (2004) Lead Conductor Also get DOS

Verification of Single Parameter Scaling Greg M. Petersen Slope All Localized

Transfer Matrix Method Greg M. Petersen Less LocalizedMore 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