Localization in Nonchiral Network Models for Two-Dimensional Disordered Wave Mechanical Systems

Abstract

Scattering theoretical network models for general coherent wave mechanical systems with quenched disorder are investigated. We focus on universality classes for two dimensional systems with no preferred orientation: Systems of spinless waves undergoing scattering events with broken or unbroken time reversal symmetry and systems of spin 1/2 waves with time reversal symmetric scattering. The phase diagram in the parameter space of scattering strengths is determined. The model breaking time reversal symmetry contains the critical point of quantum Hall systems but, like the model with unbroken time reversal symmetry, only one attractive fixed point, namely that of strong localization. Multifractal exponents and quasi-one-dimensional localization lengths are calculated numerically and found to be related by conformal invariance. Furthermore, they agree quantitatively with theoretical predictions. For non-vanishing spin scattering strength the spin 1/2 systems show localization-delocalization transitions. Large intensity fluctuations and spatial localization are fascinating universal features in any coherent wave mechanical system subject to quenched disorder. A modeling that covers essential symmetries and characteristic length scales, but does not rely on particular dispersion relations and specific details is highly desirable in various fields of theoretical physics, e.g. in optics, mesoscopic electronics, and quantum chaos (see e.g. In contrast to the CC-model here we deal with systems in the absence of fields that would introduce a handedness (chirality). However we do allow for the breaking of time reversal symmetry. In particular, we address three classes of non-chiral (NC) 2D NWMs with respect to their quantum localization properties. The first model describes time reversal symmetric scattering and is denoted as O2NC-model, where 'O' stands for 'orthogonal' (a corresponding Hamiltonian can be diagonalized by orthogonal matrices). Second, a similar model with broken time reversal symmetry ('U' for 'unitary') is introduced denoted as U2NC-model. It describes e.g. the motion of mesoscopic (spinless) electrons in the presence of random magnetic fields with zero mean and disorder potentials. The third model deals with time reversal symmetric scattering also in spin degrees of freedom. It is denoted as S2NC-model ('S' for 'symplectic') and a detailed analysis for this model is given in [10]. Here we focus on the U/O2NC-model, discuss their construction and the phase diagram in detail, and present a quantitative check of analytical results for quasi-1D localization lengths, multifractal exponents and conformal invariance in the weak localization regime [9].Quite generally, a NWM can be constructed as follows. Take a regular network of N sites and N bonds. Each bond α carries propagating wave modes (n + α incoming modes and n − α outgoing modes) represented by complex amplitudes, ψ n ± α . On the sites unitary S-matrices map incoming to outgoing amplitudes. The elements of each S-matrix are (in general) random quantit...

Renormalization group approach to energy level statistics at the integer quantum Hall transition

“…Without magnetic field, the propagation of electron waves along each link of the network is allowed in both directions. In two dimensions the transmission coefficient of the network is zero for all parameters of the scattering matrix at the nodes, 53 illustrating complete localization of electronic states. On the other hand, the twochannel network model with inter-channel mixing, that models spin-orbit interaction, exhibits a localizationdelocalization transition 54 that is also in accord with the scaling theory of localization.…”

Integer quantum Hall transition in the presence of a long-range-correlated quenched disorder

“…[27][28][29][30][31][32][33][34][35][36][37][38] The CC model is a strong magnetic field (chiral) limit of a general network model, first introduced by Shapiro 39 and later utilized for the study of localization-delocalization transitions within different universality classes. [40][41][42][43][44][45] In addition to describing the QH transition, the CC model applies to a much broader class of critical phenomena since the correspondence between the CC model and thermodynamic, field-theory and Dirac-fermions models 46-54 was demonstrated.…”

Unitary-stochastic matrix ensembles and spectral statistics

“…Quantum networks have been used with great success to model quantum phenomena observed in disordered metals and mesoscopic systems (Shapiro 1982, Chalker andCoddington 1988); typical behaviour found in extended, diffusive systems such as localisation -delocalisation transitions (Freche et al 1999), multifractal properties of wavefunctions at the transition point Metzler 1995, Huckestein andKlesse 1999), transport properties (Pascaud and Montambaux 1999, Huckestein et al 2000) and the statistical properties of quantum spectra (Klesse and Metzler 1997) have been studied on graphs in the limit of infinite network size. Recently, Kottos andSmilansky (1997, 1999) proposed to study quantum spectra of non-diffusive graphs with only relatively few vertices or nodes.…”