Numerical estimates of critical exponents of the Anderson transition. Keith Slevin (Osaka University) Tomi Ohtsuki (Sophia University)

Transcription

1 Numerical estimates of critical exponents of the Anderson transition Keith Slevin (Osaka University) Tomi Ohtsuki (Sophia University)

2 Anderson Model Standard model of a disordered system H W c c c V c n n n n n nm m nm Random on-site potential [-W/2,+W/2] Nearest neighbour hoping Applied magnetic field and spin-orbit interaction can be incorporated

3 Anderson Localisation in 3D

4 Quasi-1D Bars and Strips X H X X H X 1 X H X 1 E X X 1 X 1 X M X 1 X X X X X H X 1 X 1 M X 1 1 X H X 1 E X H X X H X 1 X 1 H X 0

5 Transfer Matrices Consider a quasi-1d system of cross section L L and length L X Transfer matrix for bar or strip is a product of transfer matrices for each slice M L X M X 1 X The Lyapunov exponents are the limiting values of the eigenvalues of 1 ln 2L X M M L X

6 Lyapunov Exponents As a consequence of current conservation the LEs occur in pairs of opposite sign. It is usual (but not necessary) to focus on the smallest positive exponent L, E, W L The output of the simulation are estimates of the LE with some precision as a function of system size L energy E and disorder W

7 Gamma vs Transverse System Size Localised Phase L L Extended Phase 2 W L L ( 3) Scale invariance at critical point constant L

8 Anderson Model 3D (Box distribution) 2.6 Anderson model 3D Box L=4,6,8,10,12,14,16,18,20 precision 0.1% W

9 FSS with Corrections to Scaling I Expand the scaling function and expand the scaling variables in the control parameter 1 y L w, L w f L 1 y w L whl w F f x f x f x h x h x h x w w w w w W W W c c K. Slevin and T. Ohtsuki (1999). "Corrections to Scaling at the Anderson Transition." Physical Review Letters 82(2):

10 Least Squares Fitting The best fit is found by minimizing the chi-squared statistic Justification 2 Random normal errors in data Model is non-linear in some parameters Iterative solution e.g. Levenberg-Marquardt i ( i) f i i 2

11 Reliability and Precision It is extremely important to determine the reliability of the fit the precision of all estimated quantities Reliability of the fit Goodness of fit (GOF) is the probability that we would get a worse value of chi-squared by chance GOF p > 0.1 (or something like that) Cannot proceed unless GOF is OK Precision Confidence intervals for the fitted parameters

12 Monte Carlo Simulation Generate a large series of pseudo data sets Model + random normal errors Fit each pseudo data set and plot histograms of chisquared and each fitted parameter Goodness of fit From the histogram of chi-squared Confidence intervals 2.5%<p<97.5% interval, i.e. an interval containing 95% of the data in the histogram

13 FSS with Corrections to Scaling II Which corrections to scaling to include? If data cross at a common point then irrelevant variable not needed If critical point appears to move with system size you need an irrelevant variable Where to truncate the expansions? The minimum number of parameters that gives an acceptable goodness of fit Check that the fit is stable against the inclusion of more parameters change of the range of data included

14 Universality Classes TRS SRS orthogonal yes yes unitary no symplectic yes no TRS = time reversal symmetry Broken by magnetic field SRS = spin rotation symmetry Broken by spin-orbit interaction

15 Estimates of Exponents in 3D O(1) ±.012 Anderson ±.008 U(1) ±.009 Anderson - Hofstadter ±.026 SU(2) ±.012 Ando ±.006 Requires high precision data and careful analysis of corrections to scaling

16 Numerical Estimate of Function Numerical estimate of beta functions for Lyapunov exponents

17 Quantum Hall Effect E E 0 0 Landau Levels Broadened by disorder All states localised except for a single state at the centre of the LL Localisation length diverges at the centre of the LL Divergence described by a critical exponent Kramer, B., T. Ohtsuki, et al. (2005). "Random network models and quantum phase transitions in two dimensions." Physics Reports 417(5-6):

18 Landau Space Models Real space model with Gaussian random potential with delta function correlation Landau level basis Each level treated independently Numerical estimate of the exponent = (Huckestein 1992) Interacting electrons Time dependent Hartree-Fock (Huckestein 1999) unchanged z1

19 Critical Exponents Temperature dependence of longitudinal and Hall resistivities = Frequency dependence of longitudinal resistivity = Assuming z=1 for the dynamic critical exponent Size dependence = Values above taken from Bodo Huckestein, RMP (1995)

20 Smooth random potential and strong perpendicular magnetic field Magnetic length much less than correlation length of potential Semi-classical quantization Electrons propagate along equipotential lines Equipotential lines are closed except at the centre of the LL Tunnelling between equipotential lines at saddle points of the potential

21 Chalker-Coddington Model Nodes described by 2 2 scattering matrices; transmission coefficient t and reflection coefficient r Random phases on the links i1 i3 e 0 r t e 0 S i 2 i 4 0 e t r 0 e i1 i3 e 0 t r e 0 S i2 i4 0 e r t 0 e 1 1 t r 2x 2x e 1 e 1

22 vs x x L=8, 12, 16, 24, 32, 48, 64, 96

23 vs x x L=8, 12, 16, 24, 32, 48, 64, 96

24 vs L L

25 Corrections to scaling for Chalker- Coddington model Note the logarithm f 1 f u f u f u h u 1 h u h u 1 L x x h L x L x x x x y log L

26 FSS Analysis Details of the data set and fit Systems sizes (number of saddle points) 8, 12, 16, 24, 32, 48, 64, 96 Range of x ,0.225 Number of data 248 Number of parameters Degrees of freedom 240 Chi-squared Goodness of fit Best fit estimate of selected parameters Critical exponent [2.585,2.596] c [1.232,1.235] y [-0.70,-0.68]

27 (x=0) Centre of the Landau Level simulation data a + b L^y (p~0) a + b L^y log(l) (p~0.1) L

28 Conformal Symmetry Scaling relations for 2D systems with ATs strips 2D Position 0 of the maximum of the f() spectrum for a 2D system with c c 1 2 Results for 2D SU(2) model 0 0 =2.173±0.001 (Obuse et al 2007) (1.8340) c =1.843±0.002 (Asada et al 2004)

29 Conformal symmetry and QHE? Is this relation obeyed at the QHE transition? Obuse, H., A. R. Subramaniam, A. Furusaki, I. A. Gruzberg, A. W. W. Ludwig. Physical Review Letters, Evers, F., A. Mildenberger, and A.D. Mirlin. Physical Review Letters, author 0 c Obuse et al ( ) Evers et al ( ) Slevin et al

30 Summary Values for exponents for different universality classes are not very different High precision of raw data is needed Proper analysis requires taking account of corrections to scaling Preliminary FSS Analysis of the Chalker-Coddington model Corrections to scaling seem not to be simple power law Results seem consistent with conformal invariance