Finite-size scaling and multifractality at the Anderson transition for the three Wigner-Dyson symmetry classes in three dimensions

Abstract

The disorder induced metal-insulator transition is investigated in a three-dimensional simple cubic lattice and compared for the presence and absence of time-reversal and spin-rotational symmetry, i.e. in the three conventional symmetry classes. Large scale numerical simulations have been performed on systems with linear sizes up to L = 100 in order to obtain eigenstates at the band center, E = 0. The multifractal dimensions, exponents Dq and αq, have been determined in the range of −1 ≤ q ≤ 2. The finite-size scaling of the generalized multifractal exponents provide the critical exponents for the different symmetry classes in accordance with values known from the literature based on high precision transfer matrix techniques. The multifractal exponents of the different symmetry classes provide further characterization of the Anderson transition, which was missing from the literature so far.

Deconfinement, chiral transition and localisation in a QCD-like model

“…Here "unitary" refers to the symmetry class of the model in the Random Matrix Theory (RMT) classification of random matrix ensembles [42], which is shared by the staggered Dirac operator [35]. A study of the multifractal properties of the eigenmodes at criticality [43] confirmed the result for the critical exponent, and also showed that the multifractal exponents of the critical eigenmodes in QCD are compatible with those of the 3D unitary Anderson model [44]. This provided further evidence that the delocalisation transitions in the two models belong to the same universality class.…”

Anderson transition and multifractals in the spectrum of the Dirac operator of quantum chromodynamics at high temperature

“…Our methods and notations are essentially the same as in Ref. 23, to which we refer the reader for a more detailed discussion. There is however one important difference, concerning the way in which the transition is approached.…”

“…In recent high-precision calculations [22][23][24] , MFSS has been successfully employed to determine the MFEs of critical eigenfunctions, as well as to obtain a more precise estimate of the critical disorder and of the critical exponents, for Anderson models in different symmetry classes. In this work we want to perform a similar MFSS analysis to study the Anderson localizationdelocalization transition in the spectrum of the Dirac operator in QCD.…”

“…A second irrelevant term, proportional to L −y , is expected to be less important and will be neglected in the analysis 22,23 . Fits to the numerical data are performed by expanding G r and G ir in the variables L 1 ν and 1 ν up to order n r and n ir , respectively.…”

Multifractality in Fock Space of the Ground State of the Bose-Hubbard Hamiltonian

“…Such a generalised multifractal analysis has been used to perform high-precision studies of the critical properties of the transition in the non-interacting case [17][18][19][20][21][22]. This formalism is also potentially applicable in the presence of interactions, and work along this line is currently being pursued [23,24].…”