Anomalously localized electronic states in three-dimensional disordered samples Philipp Cain and Michael Schreiber Theory of disordered systems, Institute of Physics, TU Chemnitz CompPhys07 November 30 2007
Motivation Anomalously localized states Anomalously localized states (ALS) ALS originate from statistical fluctuations of the disorder potential. Even for weak average disorder specific local disorder configurations lead to states that are localized in a finite region. ALS in the metallic regime can be neglected for infinite system because of their point-like spectra. [V. Uski et al., PRB 62, R7699 (2000)] ALS at the transition regime can lead to deviations in the critical properties even for infinite system size [H. Obuse and K. Yakubo, PRB 69, 125301 (2004)] Quantitative definition of ALS required. P. Cain et al (TU Chemnitz) ALS in 3D disordered samples CompPhys07 Nov2007 2 / 19
Anderson Model of Localization Anderson Model of Localization Anderson Model of Localization Non-interacting electrons in a disordered energy landscape [P. W. Anderson, Phys. Rev. 109, 1492 (1958)] H = N ɛ i i i + t i,j i j. <i,j> i=1 Random onsite energies ɛ i [ W 2, W ] 2 describe potential disorder. Nearest-neighbor hopping integrals t i,j = 1 set energy scale and boundary conditions (here periodic). Disorder induced transition from metal (delocalization) to insulator (localization) at disorder strength W c = 16.5. P. Cain et al (TU Chemnitz) ALS in 3D disordered samples CompPhys07 Nov2007 3 / 19
Anderson Model of Localization Anderson Model of Localization Density of States Mobility edge E c separates extended states in the band center from localized states in the tails. When approaching the transition (W W c ) all states become localized (E c 0). 0.08 0.06 W=10 W=W c =16.5 W=25 ρ(e) 0.04 0.02 E c 0-20 -10 0 10 20 E P. Cain et al (TU Chemnitz) ALS in 3D disordered samples CompPhys07 Nov2007 4 / 19
Anderson Model of Localization Eigenvalues and Eigenfunctions Numerical methods Efficient diagonalization of large sparse matrices. Full spectrum very expensive (time,memory). Therefore only selected Eigenvalues and Eigenfunctions. For a long time Cullum/Willoughby implementation of the Lanczos algorithm state of the art (2000: 14 critical wave functions for N = 111 3 took 27 days on a XP1000 667 Mhz). Only recently more efficient Jacobi-Davidson method with multilevel incomplete LU preconditioning [JADAMILU] (2007: 20 critical wave functions for N = 120 3 take 1 hour on a Opteron 8216 2.4 Ghz). Huge memory requirements. system size N 40 3 80 3 120 3 memory in byte 500K 3.9M 13.2M # of wave functions 1600 1000 1040 P. Cain et al (TU Chemnitz) ALS in 3D disordered samples CompPhys07 Nov2007 5 / 19
Identifying ALS Investigating the wave function Distribution of wave function amplitudes 1 µ 1000 1/2 mean µ mean 1000 µ mean 1000 3/2 µ mean f(µ) 0.8 0.6 0.4 f(µ) = µ j <µ µ j with µ j = Ψ j 2. 0.2 ALS MS 0-7 -6-5 -4-3 -2-1 0 log 10 (µ) 54 from the 1040 critical states for N = 120 3. P. Cain et al (TU Chemnitz) ALS in 3D disordered samples CompPhys07 Nov2007 6 / 19
Identifying ALS ALS vs. MS Investigating the wave function µj > µmean ALS MS P. Cain et al (TU Chemnitz) ALS in 3D disordered samples CompPhys07 Nov2007 7 / 19
Identifying ALS Investigating the wave function ALS vs. MS µ j > 1000µ mean ALS MS P. Cain et al (TU Chemnitz) ALS in 3D disordered samples CompPhys07 Nov2007 8 / 19
Identifying ALS Investigating the wave function ALS vs. MS µ j > 1000µ mean ALS MS P. Cain et al (TU Chemnitz) ALS in 3D disordered samples CompPhys07 Nov2007 9 / 19
Identifying ALS Investigating the wave function ALS vs. MS µ j > 1000 3 µ mean ALS # of µ i > ALS MS µ mean 17103 96622 1000 µmean 1442 7810 1000 µ mean 111 142 1000 3 µmean 7 0 P. Cain et al (TU Chemnitz) ALS in 3D disordered samples CompPhys07 Nov2007 10 / 19
Identifying ALS Multifractal correlations Multifractality Divergence of the correlation length at the transition implies scale invariant critical wave function multifractal (MF) properties. Consider a local box measure for box width L b and define µ i (L b ) = j box(i) Ψ j 2 Z q (L b ) = i µq i (L b) For a MF wave function (MS) one observes a power law with mass exponent τ(q). Z q (L b ) L τ(q) b P. Cain et al (TU Chemnitz) ALS in 3D disordered samples CompPhys07 Nov2007 11 / 19
Identifying ALS Multifractal correlations MF correlations I Study general MF correlation function M q (L b, r, L) µ q i (L b)µ q i+r (L b) L where. L is the average over all pairs of boxes with fixed distance r within system of width L. Employing scaling behavior it follows [M. Janssen, Int. J. Mod. Phys. B 8, 943 (1994)] M q (L b, r, L) L x(q) b L y(q) r z(q). P. Cain et al (TU Chemnitz) ALS in 3D disordered samples CompPhys07 Nov2007 12 / 19
Identifying ALS Multifractal correlations MF correlations II Restrict choice to special cases for fixed system width L where: [H. Obuse and K. Yakubo, PRB 69, 125301 (2004)] r = 0: L b = 1: M q (L b ) M q (L b, r = 0, L) L x(q) b M q (r) M q (L b = 1, r, L) r z(q) L d+τ(2q) b Influence of ALS should result in observable deviations from scaling. Variances Γ of the correlation exponents obtained from the power-law fit gives quantitative measure for each wave function Γ = Γ z + αγ τ where the α compensates the different mean values of Γ z and Γ τ. Suitable limit Γ separates ALS (Γ > Γ ) from MS (Γ Γ ). P. Cain et al (TU Chemnitz) ALS in 3D disordered samples CompPhys07 Nov2007 13 / 19
Identifying ALS Multifractal correlations Multifractal correlation functions M 2 (L b ) and M 2 (r) -12-14 Γ=0.64 0-17 Γ=0.08 0 log 10 M 2 (r) -16-18 -2 log 10 M 2 (L b ) -4 log 10 M 2 (r) -18-19 -5-10 log 10 M 2 (L b ) -20-6 -22 0.5 1 1.5 2 log 10 r, log 10 L b -8-20 0.5 1 1.5 2 log 10 r, log 10 L b -15 ALS MS P. Cain et al (TU Chemnitz) ALS in 3D disordered samples CompPhys07 Nov2007 14 / 19
Identifying ALS Distribution of the variance Γ Multifractal correlations 5 P(Γ) 4 3 2 Γ=0.15 Γ=0.20 Γ=0.25 120 100 80 60 40 30 20 1 0 0 0.2 0.4 0.6 0.8 Γ [H. Obuse and K. Yakubo, PRB 69, 125301 (2004)] P. Cain et al (TU Chemnitz) ALS in 3D disordered samples CompPhys07 Nov2007 15 / 19
Influence of ALS MF exponent z Computation of z(q) Investigate scaling relation [M. Janssen, Int. J. Mod. Phys. B 8, 943 (1994)] z(q)=d+2τ(q) τ(2q) derived from asymptotic behavior (vanishing r = L b vs. large r = L) of M q (r, L b, L) µ q i (L b)µ q i+r (L b) L Lx(q) b L y(q) r z(q). Compare correlation exponent z(q) calculated from M q (r) r z(q) with z(q) obtained from the mass exponent τ(q) as computed from M q (L b ) L 2τ(q) b. P. Cain et al (TU Chemnitz) ALS in 3D disordered samples CompPhys07 Nov2007 16 / 19
Comparison of z(q) Influence of ALS MF exponent z 3 z(q) 2 1 z Q (q) z R (q) all states Γ< 0.5 Γ< 0.4 Γ< 0.3 0 0 1 2 3 q [H. Obuse and K. Yakubo, J. Phys. Soc. Japan, 73, 2164 (2004)] Significant deviations between different methods for q > 2. For q > 2.5 value exceeds theoretical upper limit z(q) < 3 even for smallest Γ. P. Cain et al (TU Chemnitz) ALS in 3D disordered samples CompPhys07 Nov2007 17 / 19
Influence of ALS Correlation Dimension D 2 Fluctuation of the Correlation Dimension D 2 Definition: Z 2 (L b ) L D 2 b thus D 2 = τ(2) 0.7 2.0 width P(D 2 ) 0.6 0.5 0.4 all states Γ< 0.5 Γ< 0.4 Γ< 0.3 0.01 0.03 0.05 L -1 20 30 40 60 80 100 120 1.5 1.0 P(D 2 ) 0 0.5 1 1.5 2 D 2 0.0 ALS lead to finite width for L 0.5 [H. Obuse and K. Yakubo, PRB 69, 125301 (2004)] P. Cain et al (TU Chemnitz) ALS in 3D disordered samples CompPhys07 Nov2007 18 / 19
Conclusion Conclusion ALS affect critical properties at the delocalization-localization transition in the 3D Anderson model of localization. Influence of ALS less pronounced as for the 2D SU(2) model. For better statistics are more critical states and more disorder realizations required. higher numerical effort for 3D in comparison to 2D. P. Cain et al (TU Chemnitz) ALS in 3D disordered samples CompPhys07 Nov2007 19 / 19