This chapter describes the progress made during the past three decades in the finite size scaling analysis of the critical phenomena of the Anderson transition. The scaling theory of localization and the Anderson model of localization are briefly sketched. The finite size scaling method is described. Recent results for the critical exponents of the different symmetry classes are summarised. The importance of corrections to scaling are emphasised. A comparison with experiment is made, and a direction for future work is suggested.This chapter describes the progress made during the past three decades in the finite size scaling analysis of the critical phenomena of the Anderson transition. The scaling theory of localization and the Anderson model of localization are briefly sketched. The finite size scaling method is described. Recent results for the critical exponents of the different symmetry classes are summarised. The importance of corrections to scaling are emphasised. A comparison with experiment is made, and a direction for future work is suggested.

arXiv:1004.0285International Journal of Modern Physics BMay 20, 201031 citations

Finite-Size Scaling and Critical Exponents. A New Approach and Its Application to Anderson Localisation

Etienne Hofstetter, Michael Schreiber

Based on finite-size scaling ideas a new approach to analyse the singularities associated with second-order phase transitions is presented. Under certain requirements, the method allows to derive an upper limit for the critical exponent from finite-size data that describe the system far away from the transition. By extrapolation the critical exponent ν is determined. As an example, the approach is applied to the metal-insulator transition in disordered systems. The result ν = 1.35 ± 0.15 is independent of the employed random distribution. With field theoretical arguments we discuss and generalize this result. Our results are consistent with the assumption that one-parameter scaling and universality are valid in the 3D Anderson model.Based on finite-size scaling ideas a new approach to analyse the singularities associated with second-order phase transitions is presented. Under certain requirements, the method allows to derive an upper limit for the critical exponent from finite-size data that describe the system far away from the transition. By extrapolation the critical exponent ν is determined. As an example, the approach is applied to the metal-insulator transition in disordered systems. The result ν = 1.35 ± 0.15 is independent of the employed random distribution. With field theoretical arguments we discuss and generalize this result. Our results are consistent with the assumption that one-parameter scaling and universality are valid in the 3D Anderson model.

Europhysics Letters (EPL)Mar 20, 199341 citations

On the Anderson transition on the Bethe lattice

Giorgio Parisi A, Saverio Pascazio, Francesca Pietracaprina +2 more

We study the Anderson model on the Bethe lattice by working directly with real energies $E$ . We show that the criterion for the stability of the populations leads to the same results as other, more traditional methods based on the imaginary part of the self energies for identifying the localized and delocalized phases. We present new numerical and analytical results, including an accurate finite-size scaling of the transition point, returning a critical length that diverges with exponent $\nu=1/2$ in the delocalized region of the transition, as well as a concise proof of the asymptotic formula for the critical disorder, as given in [P.W. Anderson 1958]. We discuss how the forward approximation used in analytic treatments of MBL fits into this scenario and how one can interpolate between it and the correct asymptotic analysis.We study the Anderson model on the Bethe lattice by working directly with real energies $E$. We show that the criterion for the stability of the populations leads to the same results as other, more traditional methods based on the imaginary part of the self energies for identifying the localized and delocalized phases. We present new numerical and analytical results, including an accurate finite-size scaling of the transition point, returning a critical length that diverges with exponent $\nu=1/2$ in the delocalized region of the transition, as well as a concise proof of the asymptotic formula for the critical disorder, as given in [P.W. Anderson 1958]. We discuss how the forward approximation used in analytic treatments of MBL fits into this scenario and how one can interpolate between it and the correct asymptotic analysis.

arXiv:1812.03531Dec 9, 20181 citation

A MICROSCOPIC THEORY OF FINITE-SIZE SCALING

Christian Borgs

In this paper, I give an overview over a recently developed rigorous theory of finite-size scaling near first order transitions. Leaving out details of the mathematical proofs, the main emphasis is put on the underlying physical ideas and the discussion of the validity of the results for regions which are, in the sense of mathematical rigour, not covered by the original papers. I present both the finite-size scaling for cubic systems and for long cylinders, discussing also a recent controversy which stems from our work on the finite-size scaling of the mass gap in long cylinders.In this paper, I give an overview over a recently developed rigorous theory of finite-size scaling near first order transitions. Leaving out details of the mathematical proofs, the main emphasis is put on the underlying physical ideas and the discussion of the validity of the results for regions which are, in the sense of mathematical rigour, not covered by the original papers. I present both the finite-size scaling for cubic systems and for long cylinders, discussing also a recent controversy which stems from our work on the finite-size scaling of the mass gap in long cylinders.

International Journal of Modern Physics COct 1, 19926 citations

Multifractal finite-size scaling at the Anderson transition in the unitary symmetry class

Jakob Lindinger, Alberto Rodríguez

We use multifractal finite-size scaling to perform a high-precision numerical study of the critical properties of the Anderson localization-delocalization transition in the unitary symmetry class, considering the Anderson model including a random magnetic flux. We demonstrate the scale invariance of the distribution of wave-function intensities at the critical point and study its behavior across the transition. Our analysis, involving more than $4\ifmmode\times\else\texttimes\fi{}{10}^{6}$ independently generated wave functions of system sizes up to ${L}^{3}={150}^{3}$ , yields accurate estimates for the critical exponent of the localization length, $\ensuremath{\nu}=1.446(1.440,1.452)$, the critical value of the disorder strength, and the multifractal exponents.We use multifractal finite-size scaling to perform a high-precision numerical study of the critical properties of the Anderson localization-delocalization transition in the unitary symmetry class, considering the Anderson model including a random magnetic flux. We demonstrate the scale invariance of the distribution of wave-function intensities at the critical point and study its behavior across the transition. Our analysis, involving more than $4\ifmmode\times\else\texttimes\fi{}{10}^{6}$ independently generated wave functions of system sizes up to ${L}^{3}={150}^{3}$, yields accurate estimates for the critical exponent of the localization length, $\ensuremath{\nu}=1.446(1.440,1.452)$, the critical value of the disorder strength, and the multifractal exponents.

arXiv:1707.05701Physical review. B./Physical review. BOct 5, 201723 citations

Monte Carlo studies of phase transitions and critical phenomena

Ibragimkhan K. Kamilov, Akai Kurbanovich Murtazaev, Kh. K. Aliev

The current state of Monte Carlo work on phase transitions and critical phenomena is reviewed. Both classical and quantum Monte Carlo results are discussed with emphasis on statistical lattice-model studies involving the highly accurate calculation of critical exponents. It is shown that finite-size scaling is an effective method whereby not only simple lattice models but also those of complex practical magnetic materials can be treated. For models allowing crossover phenomena, it is shown that both finite-size scaling theory and conventional power-law functions must be used in carrying out a Monte Carlo analysis.The current state of Monte Carlo work on phase transitions and critical phenomena is reviewed. Both classical and quantum Monte Carlo results are discussed with emphasis on statistical lattice-model studies involving the highly accurate calculation of critical exponents. It is shown that finite-size scaling is an effective method whereby not only simple lattice models but also those of complex practical magnetic materials can be treated. For models allowing crossover phenomena, it is shown that both finite-size scaling theory and conventional power-law functions must be used in carrying out a Monte Carlo analysis.

Physics-UspekhiJul 31, 199965 citations

We study the Anderson model on the Bethe lattice by working directly with real energies

E

. We show that the criterion for the stability of the populations leads to the same results as other, more traditional methods based on the imaginary part of the self energies for identifying the localized and delocalized phases. We present new numerical and analytical results, including an accurate finite-size scaling of the transition point, returning a critical length that diverges with exponent

\nu=1/2

in the delocalized region of the transition, as well as a concise proof of the asymptotic formula for the critical disorder, as given in [P.W. Anderson 1958]. We discuss how the forward approximation used in analytic treatments of MBL fits into this scenario and how one can interpolate between it and the correct asymptotic analysis.

{L}^{3}={150}^{3}

, yields accurate estimates for the critical exponent of the localization length, $\ensuremath{\nu}=1.446(1.440,1.452)$, the critical value of the disorder strength, and the multifractal exponents.