finite size scaling - physnet-rz · fit each pseudo data set and plot histograms of chi-squared and...

Keith SlevinOsaka University

Finite Size Scaling

Keith Slevin. Bremen, 22nd July 2008.

Basis of FSS Method IPhysical system of size LSome coupling constant u (for simplicity consider only one coupling constant).The coupling constant is a function of some parameters W1, W2, etc. like disorder, energy etc.Some dimensionless physical quantity

( )( )1 2, ,,Q Q L u W W≡ …


Renormalization GroupRenormalization Transformation

Integrate out short wavelength fluctuationsRescale by factor b to recover same short wavelength cut-offRecover original system (hopefully) but with scaled variables

Physical quantity Q now give by

If all this makes sense then

( ) ( )1, bQ Q Lb u u R u− ′ ′= =

1 2 1 2b b b bR R R=Keith Slevin. Bremen, 22nd July 2008.

Fixed PointsFixed points of RG transform correspond to phase transitions

Linearize the RG transform near the fixed point

The derivative at the fixed point defines an exponent

( )c b cu R u=

c

c cu u

bdRu u uudu

δ δ=

′+ = + +

cu u

bdRb udu

u bα αδ δ=

′ ==


Finite Size ScalingUse the RG transform to rescale the system to some arbitrary but fixed length L0

Since L0 is fixed we may introduce a scaling function

This can be re-written in a more well known way

( )00

, LQ Q L u u ub bL

αδ δ δ′ ′= = =

( )Q uf Lαδ=

( )L uQ F ξ ξ δξ±

⎛ ⎞= ⎜ ≡⎟

⎝ ⎠Keith Slevin. Bremen, 22nd July 2008.

Critical ExponentWe usually vary one parameter at a time i.e. fix the energy and vary the disorderThe critical point i.e. the critical value of the parameter W is the solution of

The correlation length diverges at the critical point

( ) ( ) 0c WW uu uδ ==

1u νξ ξ δ να

−±≈ =


Transfer MatricesConsider a quasi-1D system of cross section L × Land length LX

Transfer matrix for bar is a product of transfer matrices for each slice

The Lyapunov exponents are the limiting values of the eigenvalues of

1

XL

XX

M M=

=∏

†1 ln2 X

X

LM ML

Ω = →∞


Lyapunov ExponentsAs 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

The output of the simulation are estimates of the LE with some precision σ as a function of system size Lenergy E and disorder W

Usually the LE vs W or E for a series of different L

( ), ,L E W Lγ γ γ≡ Γ ≡


Lyapunov Data vs System SizeData for Anderson model with box distributed random potential

8 10 12 14 16 18 201.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

Γ

size L

Gamma vs System SizeLocalised Phase

Extended Phase

Scale invariance at critical point

L Lξ

Γ ≈ →∞

2

2 2

dLW W L

L− ≈Γ ≈ → ∞

constant L≈Γ →∞


Lyapunov Data vs Disorder

15.0 15.5 16.0 16.5 17.0 17.5 18.01.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

Γ

Disorder W


FSS of Lyapunovs in 3DWe attempt to fit the data to the following form

Focus on the critical point: aim is to estimate the critical disorder Wc and critical exponent υ

( )( )f L u WαδΓ =

c

c

W Wu w wW

δ −= =

( ) 20 1 2

nnf x f f x f x f x= + + + +


Least Squares FittingThe best fit is found by minimizing the chi-squared statistic

JustificationRandom normal errors in data

Model is non-linear in some parametersIterative solution e.g. Levenberg-Marquardt

( )

( )

( 2

2)

i

ii

i

Q fχσ

⎛ ⎞−= ⎜ ⎟⎜ ⎟

⎝ ⎠∑


Reliability and PrecisionIt is extremely important to determine

the reliability of the fit the precision of all estimated quantities

Reliability of the fitGoodness of fit (GOF) is the probability that we would get a worse value of chi-squared by chanceGOF p > 0.1 (or something like that)

Cannot proceed unless GOF is OKConfidence intervals for the fitted parameters


Monte Carlo SimulationGenerate a large series of pseudo data sets

Model + random normal errorsFit each pseudo data set and plot histograms of chi-squared and each fitted parameter

Goodness of fitFrom the histogram of chi-squared

Confidence intervalsSymmetric interval centred on the best fit value for the true data set and containing 95% of the data in histogram

Alternative to MC is the Bootstrap (but no GOF)


Goodness of FitDetermine the goodness of fit by looking at the histogram of chi-squared

p=0.2


Confidence IntervalsDetermine confidence intervals from histograms of each parameter


FSS of Conductance DistributionDimensionless conductance g of a mesoscopic cube of side L

Sample-to-sample fluctuation of the conductanceFSS not applicable to g directlyApply to statistics such as

Or percentiles of the distribution of g e.g. median of g

1exp lng g g


3D Anderson Model

FSS of Lyapunovs in 2DFSS is not restricted to the critical pointe.g. Lyapunovs in 2D

Parameterise F e.g. using splinesOther fitting parameters are the localisation lengths for each energy and disorder

( ),LL F

W Eγ

ξ⎛ ⎞

= ⎜ ⎟⎜ ⎟⎝ ⎠


2D Anderson Model

4 6 8 10 12 14 160.1

1

10

100

1000γL

W

L=16, 32, 64, 128, 256, 512


2D Anderson Model (scaled)

0.01 0.1 1 10 100 10000.1

1

10

100

1000

γL

L/ξKeith Slevin. Bremen, 22nd July 2008.

Beta FunctionWe can use the results of a FSS analysis to plot the beta function

Parametric plot of beta function

( ) ( )lndQ dQQ L

d L dLf L u L u Qα αα δ δ β= = = ′ ≡


Numerical Estimate of β FunctionNumerical estimate of beta functions for Lyapunov exponents


SummaryData quality!

To do FSS you need high precision data for the largest systems you can manage

Not just for critical exponentsScaling functionsLocalisation lengthsBeta functions

Use splines etc. when needed

PrecisionYour estimates are meaningless with out systematic calculation of their precision


ReferencesRG and FSS

Goldenfeld, N. (1992). Lectures on phase transitions and the renormalization group. Cardy, J. L. (1996). Scaling and renormalization in statistical physics.

FSS and Anderson localisationSlevin, K. and T. Ohtsuki (1999). "Corrections to Scaling at the Anderson Transition." PRL 82 382.Slevin, K., P. Markos, et al. (2001). "Reconciling Conductance Fluctuations and the Scaling Theory of Localization." PRL 86 3594Slevin, K., Y. Asada, et al. (2004). "Fluctuations of the Lyapunov exponent in two-dimensional disordered systems." PRB 70 054201Asada, Y., K. Slevin, et al. (2004). "Numerical estimation of the beta function in two-dimensional systems with spin-orbit coupling." PRB 70 035115


finite size scaling - physnet-rz · fit each pseudo data set and plot histograms of chi-squared and...

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