Modeling Disordered Quantum Systems With Dynamical Networks

Abstract

It is the purpose of the present article to show that so-called network models, originally designed to describe static properties of disordered electronic systems, can be easily generalized to quantum-dynamical models, which then allow for an investigation of dynamical and spectral aspects. This concept is exemplified by the Chalker-Coddington model for the quantum Hall effect and a three-dimensional generalization of it. We simulate phase coherent diffusion of wave packets and consider spatial and spectral correlations of network eigenstates as well as the distribution of (quasi-)energy levels. Apart from that, it is demonstrated how network models can be used to determine two-point conductances. Our numerical calculations for the three-dimensional model at the Metal-Insulator transition point delivers, among others, an anomalous diffusion exponent of η = 3 − D 2 = 1.7 ± 0.1. The methods presented here in detail have been used partially in earlier work. 577 Int. J. Mod. Phys. C 1999.10:577-606. Downloaded from www.worldscientific.com by FLINDERS UNIVERSITY LIBRARY on 02/08/15. For personal use only. Int. J. Mod. Phys. C 1999.10:577-606. Downloaded from www.worldscientific.com by FLINDERS UNIVERSITY LIBRARY on 02/08/15. For personal use only. c Equivalently, instead of the generalized dimensionDq, exponents τq ≡ (q −1)Dq or the Legendretransformed f (α) of the function τq are often used to describe a multifractal density. 34 Int. J. Mod. Phys. C 1999.10:577-606. Downloaded from www.worldscientific.com by FLINDERS UNIVERSITY LIBRARY on 02/08/15. For personal use only.

Multifractality of wave functions at the quantum Hall transition revisited

“…The best estimate for ␣ 0 from previous numerical work is 2.26Ϯ0.01. 16 The value of ␣ Ϫ obtained in Ref. 16 …”

“…20 In order to obtain the wavefunction we translate the lattice dynamics into a unitary time evolution operator U which describes the wave packet propagation on the network in discrete time steps. 16 The desired critical wavefunctions are the eigenfunctions of U, which are found by numerical diagonalization.…”

“…For comparison, the biggest realization of the network model reported in the literature that we are aware of is smaller by a factor of five in linear dimensions. 16 Since an accurate extrapolation to the thermodynamic limit was of primary importance for our work, it was crucial that we could observe the finite size corrections over almost two orders of magnitude in L.…”

Phase rotation symmetry and the topology of oriented scattering networks

“…When time-reversal symmetry is preserved, as it is in most photonic systems, it is fair to use similar one-way oriented networks to describe one of the two "spin" copies of the system, provided that certain spin-flip processes can be neglected [7,[15][16][17]. Due to this particular resurgence of an effective time, a fruitful analogy between scattering networks and Floquet dynamics was envisioned [8,18,19], an important consequence of which is the discovery of anomalous chiral edge states in such systems, while there is remarkably no external periodic driving as it would be in a Floquet system. The efficiency of this approach motivated two recent microwave experiments that probed the existence of these anomalous topological edge states [15,16].…”

Multifractality at the Quantum Hall Transition: Beyond the Parabolic Paradigm

“…We determine the lattice time evolution operator U for the Chalker-Coddington network model [11,12] with periodic boundary conditions and N ¼ 2L d nodes. For each realization of disorder, eight eigenstates ðrÞ with eigenvalues closest to unity are found with a standard sparse matrix package [13][14][15] from exact diagonalization of U.…”