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Register a new userDimensional Dependence of Critical Exponent of the Anderson Transition in the Orthogonal Universality Class
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We report improved numerical estimates of the critical exponent of the Anderson transition in Andersons model of localization in $d=4$ and $d=5$ dimensions. We also report a new Borel-Pade analysis of existing $epsilon$ expansion results that incorporates the asymptotic behaviour for $dto infty$ and gives better agreement with available numerical results.
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We report a careful finite size scaling study of the metal insulator transition in Andersons model of localisation. We focus on the estimation of the critical exponent $ u$ that describes the divergence of the localisation length. We verify the universality of this critical exponent for three different distributions of the random potential: box, normal and Cauchy. Our results for the critical exponent are consistent with the measured values obtained in experiments on the dynamical localisation transition in the quantum kicked rotor realised in a cold atomic gas.
To date the most precise estimations of the critical exponent for the Anderson transition have been made using the transfer matrix method. This method involves the simulation of extremely long quasi one-dimensional systems. The method is inherently serial and is not well suited to modern massively parallel supercomputers. The obvious alternative is to simulate a large ensemble of hypercubic systems and average. While this permits taking full advantage of both OpenMP and MPI on massively parallel supercomputers, a straight forward implementation results in data that does not scale. We show that this problem can be avoided by generating random sets of orthogonal starting vectors with an appropriate stationary probability distribution. We have applied this method to the Anderson transition in the three-dimensional orthogonal universality class and been able to increase the largest $Ltimes L$ cross section simulated from $L=24$ (New J. Physics, 16, 015012 (2014)) to $L=64$ here. This permits an estimation of the critical exponent with improved precision and without the necessity of introducing an irrelevant scaling variable. In addition, this approach is better suited to simulations with correlated random potentials such as is needed in quantum Hall or cold atom systems.
We study the box-measure correlation function of quantum states at the Anderson transition point with taking care of anomalously localized states (ALS). By eliminating ALS from the ensemble of critical wavefunctions, we confirm, for the first time, the scaling relation z(q)=d+2tau(q)-tau(2q) for a wide range of q, where q is the order of box-measure moments and z(q) and tau(q) are the correlation and the mass exponents, respectively. The influence of ALS to the calculation of z(q) is also discussed.
We analyze the scaling behavior of the higher Lyapunov exponents at the Anderson transition. We estimate the critical exponent and verify its universality and that of the critical conductance distribution for box, Gaussian and Lorentzian distributions of the random potential.
We use large-scale Monte Carlo simulations to test the Weinrib-Halperin criterion that predicts new universality classes in the presence of sufficiently slowly decaying power-law-correlated quenched disorder. While new universality classes are reasonably well established, the predicted exponents are controversial. We propose a method of growing such correlated disorder using the three-dimensional Ising model as benchmark systems both for generating disorder and studying the resulting phase transition. Critical equilibrium configurations of a disorder-free system are used to define the two-value distributed random bonds with a small power-law exponent given by the pure Ising exponent. Finite-size scaling analysis shows a new universality class with a single phase transition, but the critical exponents $ u_d=1.13(5), eta_d=0.48(3)$ differ significantly from theoretical predictions. We find that depending on details of the disorder generation, disorder-averaged quantities can develop peaks at two temperatures for finite sizes. Finally, a layer model with the two values of bonds spatially separated to halves of the system genuinely has multiple phase transitions and thermodynamic properties can be flexibly tuned by adjusting the model parameters.
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