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Register a new userCritical Exponent of the Anderson Transition using Massively Parallel Supercomputing
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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.
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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.
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 propose a generalization of multifractal analysis that is applicable to the critical regime of the Anderson localization-delocalization transition. The approach reveals that the behavior of the probability distribution of wavefunction amplitudes is sufficient to characterize the transition. In combination with finite-size scaling, this formalism permits the critical parameters to be estimated without the need for conductance or other transport measurements. Applying this method to high-precision data for wavefunction statistics obtained by exact diagonalization of the three-dimensional Anderson model, we estimate the critical exponent $ u=1.58pm 0.03$.
We show that quantum wavepackets exhibit a sharp macroscopic peak as they spread in the vicinity of the critical point of the Anderson transition. The peak gives a direct access to the mutifractal properties of the wavefunctions and specifically to the multifractal dimension $d_2$. Our analysis is based on an experimentally realizable setup, the quantum kicked rotor with quasi-periodic temporal driving, an effectively 3-dimensional disordered system recently exploited to explore the physics of the Anderson transition with cold atoms.
We have estimated the critical exponent describing the divergence of the localization length at the metal-quantum spin Hall insulator transition. The critical exponent for the metal-ordinary insulator transition in quantum spin Hall systems is known to be consistent with that of topologically trivial symplectic systems. However, the precise estimation of the critical exponent for the metal-quantum spin Hall insulator transition proved to be problematic because of the existence, in this case, of edge states in the localized phase. We have overcome this difficulty by analyzing the second smallest positive Lyapunov exponent instead of the smallest positive Lyapunov exponent. We find a value for the critical exponent $ u=2.73 pm 0.02$ that is consistent with that for topologically trivial symplectic systems.
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