We introduce a network model to describe two-dimensional disordered electron systems with spinorbit scattering. The network model is defined by a discrete unitary time evolution operator. We establish by numerical transfer matrix calculations that the model exhibits a localization-delocalization transition. We determine the corresponding phase diagram in the parameter space of disorder scattering strength and spin-orbit scattering strength. Near the critical point we determine by statistical analysis a one-parameter scaling function and the critical exponent of the localization length to be ν = 2.51 ± 0.18. Based on a conformal mapping we also calculate the scaling exponent of the typical local density of states α0 = 2.174 ± 0.003.

Systems with bond disorder are defined through lattice Hamiltonians that are of pure nearest neighbour hopping type, i.e. do not contain on-site contributions. They stand representative for the general family of disordered systems with chiral symmetries. Application of the Dorokhov-Mello-Pereyra-Kumar transfer matrix technique [P. W. Brouwer et al., Phys. Rev. Lett 81, 862(1998); Phys. Rev. Lett. 84, 1913Lett. 84, (2000] has shown that both spectral and transport properties of quasi one-dimensional systems belonging to this category are highly unusual. Most notably, regimes with absence of exponential Anderson localization are observed, the single particle density of states exhibits singular structure in the vicinity of the band centre, and the manifestation of these phenomena depends in an apparently topological manner on the even-or oddness of the channel number. In this paper we re-consider the problem from the complementary perspective of the non-linear σ-model.Relying on the standard analogy between one-dimensional statistical field theories and zero-dimensional quantum mechanics, we will relate the problem to the behaviour of a quantum point particle subject to an Aharonov-Bohm flux.We will build on this analogy to re-derive earlier DMPK results, identify a new class of even/odd staggering phenomena (now dependent on the total number of sites in the system) and trace back the anomalous behaviour of the bond disordered system to a simple physical mechanism, viz. the flux periodicity of the quantum Aharonov-Bohm system. We will also touch upon connections to the low energy physics of other lattice systems, notably disordered chiral systems in 0 and 2 dimensions and antiferromagnetic spin chains.

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...

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Networks of random quantum scatterers @S-matricesA form paradigmatic models for the propagation of coherent w aves in random media. S-matrix network models cover universal localization-delocalization properties and have some advantages over more traditional Hamiltonian models. In particularD a straightforward implementation of real space renormalization techniques is possible. Starting from a nite elementary cell of the S-matrix networkD hierarchical network models can be constructed by recursion. The localization-delocalization properties are contained in the ow of the forward scattering strength @9conductance9A under increasing system size. With the aid of 9small scale9 n umerics qualitative aspects of the localization-delocalization properties of S-matrix network models can be worked out.