Borel–Padé Re-summation of the β-functions Describing Anderson Localisation in the Wigner–Dyson Symmetry Classes

Abstract

We describe a Borel-Padé re-summation of the β-function in the three Wigner-Dyson symmetry classes. Using this approximate β-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.

Conformal Bootstrap Analysis for Localization: Symplectic Case

“…At D = 8, we have no such phenomena, and this suggests that D = 8 may not be the upper critical point of the localization. The upper critical dimension D = ∞ has been suggested for the localization [14,23], and our present work does not exclude this possibility.…”

“…In this paper, assuming we don't know the fusion rule for the operator product expansions, we simply use the approximation by the use of small numbers of determinants, which involve a limited number of the scale dimensions. We have obtained the values of the critical exponent ν which roughly agrees with the numerical estimation by the finite size scaling method, and also we obtain the lower critical dimension D c = 1.25 and the upper critical dimension as D c = ∞, which are consistent with the numerical result of finite size scaling [13,14]. For the case of polymer [4], the analysis of small size determinants provide a rather accurate value of the critical exponent ν.…”

“…Our obtained result from bootstrap is ν = 2.5 − 2.75 in the two dimensional case, and ν = 1.01 for three dimensional case, which are obtained 4 × 4 determinants. The numerical result by the finite size scaling for ν in the two dimensions is 2.75, and the lower critical dimension is estimated as D c = 1.44 − 1.55 [14]. The difference between of the finite size scaling results and the bootstrap method will be expected to be small by the increasing size of the determinants in taking more scaling dimensions ∆.…”

“…The strong spin-orbit case belongs to the symplectic case [8]. The numerical analysis of the localization has precise predictions of the critical exponents due to the finite scaling [11,12,13], and the Borel-Pad'e re-summation for the results of β-function has been proposed to obtain the exponents [14]. We specify the model, which satisfies the conditions of (3) in below, and we call it as a localization model, in which the density of state does not show the any singular behavior.…”

Drawing Phase Diagrams of Random Quantum Systems by Deep Learning the Wave Functions

“…The critical exponent ν is known to be 1/2 in the limit of infinite dimensions, [287][288][289] and the Borel-Padé approximation 290) successfully interpolates the exponents in low to high dimensions. 291,292) The Anderson transition in four dimensions can be studied, for example, using the quantum kicked rotor with amplitude modulation realized in atomic matter waves. 293) For human beings, the wave functions in 4D space are difficult to imagine and analyze, since our eyes and brains are already trained to observe 2D and 3D images.…”