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Register a new userBorel-Pade re-summation of the $beta$-functions describing Anderson localisation in the Wigner-Dyson symmetry classes
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We describe a Borel-Pade re-summation of the $beta$-function in the three Wigner-Dyson symmetry classes. Using this approximate $beta$-function we discuss the dimensional dependence of the critical exponent and compare with numerical estimates. We also estimate the lower critical dimension of the symplectic symmetry class.
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Products of random matrix products of $mathrm{SL}(2,mathbb{R})$, corresponding to transfer matrices for the one-dimensional Schrodinger equation with a random potential $V$, are studied. I consider both the case where the potential has a finite second moment $langle V^2rangle<infty$ and the case where its distribution presents a power law tail $p(V)sim|V|^{-1-alpha}$ for $0<alpha<2$. I study the generalized Lyapunov exponent of the random matrix product (i.e. the cumulant generating function of the logarithm of the wave function). In the high energy/weak disorder limit, it is shown to be given by a universal formula controlled by a unique scale (single parameter scaling). For $langle V^2rangle<infty$, one recovers Gaussian fluctuations with the variance equal to the mean value: $gamma_2simeqgamma_1$. For $langle V^2rangle=infty$, one finds $gamma_2simeq(2/alpha),gamma_1$ and non Gaussian large deviations, related to the universal limiting behaviour of the conductance distribution $W(g)sim g^{-1+alpha/2}$ for $gto0$.
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 wavefunction intensities at the critical point and study its behavior across the transition. Our analysis, involving more than $4times10^6$ independently generated wavefunctions of system sizes up to $L^3=150^3$, yields accurate estimates for the critical exponent of the localization length, $ u=1.446 (1.440,1.452)$, the critical value of the disorder strength and the multifractal exponents.
We study the multifractal analysis (MFA) of electronic wavefunctions at the localisation-delocalisation transition in the 3D Anderson model for very large system sizes up to $240^3$. The singularity spectrum $f(alpha)$ is numerically obtained using the textsl{ensemble average} of the scaling law for the generalized inverse participation ratios $P_q$, employing box-size and system-size scaling. The validity of a recently reported symmetry law [Phys. Rev. Lett. 97, 046803 (2006)] for the multifractal spectrum is carefully analysed at the metal-insulator transition (MIT). The results are compared to those obtained using different approaches, in particular the typical average of the scaling law. System-size scaling with ensemble average appears as the most adequate method to carry out the numerical MFA. Some conjectures about the true shape of $f(alpha)$ in the thermodynamic limit are also made.
The multifractality of the critical eigenstate at the metal to insulator transition (MIT) in the three-dimensional Anderson model of localization is characterized by its associated singularity spectrum f(alpha). Recent works in 1D and 2D critical systems have suggested an exact symmetry relation in f(alpha). Here we show the validity of the symmetry at the Anderson MIT with high numerical accuracy and for very large system sizes. We discuss the necessary statistical analysis that supports this conclusion. We have obtained the f(alpha) from the box- and system-size scaling of the typical average of the generalized inverse participation ratios. We show that the best symmetry in f(alpha) for typical averaging is achieved by system-size scaling, following a strategy that emphasizes using larger system sizes even if this necessitates fewer disorder realizations.
This is a course on Random Matrix Theory which includes traditional as well as advanced topics presented with an extensive use of classical logarithmic plasma analogy and that of the quantum systems of one-dimensional interacting fermions with inverse square interaction (Calogero-Sutherland model). Certain non-invariant random matrix ensembles are also considered with the emphasis on the eigenfunction statistics in them. The course can also be viewed as introduction to theory of localization where the (non-invariant) random matrix ensembles play a role of the toy models to illustrate functional methods based on super-vector/super-matrix representations.
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