Martin Janßen - Academia.edu (original) (raw)

Papers by Martin Janßen

Generated Dynamics of Markov and Quantum Processes, 2016

Eprint Arxiv Cond Mat 9710297, Oct 1, 1997

$Research paper thumbnail of Multifractality and scaling in disordered mesoscopic systems$

Zeitschrift Fur Physik B Condensed Matter, 1991

Phys Rev B, 1998

We introduce a network model to describe two-dimensional disordered electron systems with spin-or... more We introduce a network model to describe two-dimensional disordered electron systems with spin-orbit 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 nu=2.51pm0.18\nu=2.51\pm 0.18nu=2.51pm0.18. Based on a conformal mapping we also calculate the scaling exponent of the typical local density of states alpha_0=2.174pm0.003\alpha_0=2.174 \pm 0.003alpha_0=2.174pm0.003.

Zeitschrift Fur Physik B Condensed Matter, 1987

Physica B Condensed Matter, 1999

We study hierarchical network models for the integer quantum Hall eect. The renormalization¯ow of... more We study hierarchical network models for the integer quantum Hall eect. The renormalization¯ow of the conductance distribution, the critical exponent of the localization length and the multifractal exponents of critical eigenstates are determined. The hierarchical models display the qualitative physics responsible for the quantum Hall eect and allow, in principle, for an accurate determination of critical exponents.

$Research paper thumbnail of Multifractal analysis of disorder induced metal-insulator transitions$

Physica a, 1992

We give a physical picture for the muhifractal nature of mesoscopic observables at the disorder i... more We give a physical picture for the muhifractal nature of mesoscopic observables at the disorder induced metal-insulator transition (MIT) and present their scaling laws in terms of normalized observables. We calculate the f(a) spectrum of the Thouless numbers at the MIT point of the lowest Landau band of a quantum Hall system and thereby establish an alternative method to finite size scaling for calculating critical exponents. We conclude that the MIT of a quantum Hall system shows universal one-parameter scaling. * Work performed within the research program of the Sonderforschungsbereich 341, Kohr-Aachen-Jiilich.

Physical Review B, 1998

We introduce a network model to describe two-dimensional disordered electron systems with spinorb... more 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.

$Research paper thumbnail of Multifractal Analysis of Broadly-Distributed Observables at Criticality$

International Journal of Modern Physics B, 1994

The multifractal analysis of disorder-induced localization-delocalization transitions is reviewed... more The multifractal analysis of disorder-induced localization-delocalization transitions is reviewed. Scaling properties of this transition are generic for multi parameter coherent systems which show broadly-distributed observables at criticality. The multifractal analysis of local measures is extended to more general observables including scaling variables such as the conductance in the localization problem. The relation of multifractal dimensions to critical exponents such as the order parameter exponent β and the correlation length exponent ν is investigated, We discuss a number of scaling relations between spectra of critical exponents, showing that all of the critical exponents necessary to characterize the critical phenomenon can be obtained within the generalized multifractal analysis. Furthermore we show how bounds for the correlation length exponent ν are obtained by the typical order parameter exponent α0 and make contact between the multifractal analysis and the finite size ...

World Scientific Lecture Notes in Physics, 2001

... The investigation reveals the multifractal properties of electronic states and opens the poss... more ... The investigation reveals the multifractal properties of electronic states and opens the possibility to consider the local density of states as an appropriate order parameter field for the LD ... The typical value, as given by the geometric mean, is the global order parameter. ...

Physical Review B, 1997

We study a number of hierarchical network models related to the Chalker-Coddington model of quant... more We study a number of hierarchical network models related to the Chalker-Coddington model of quantum percolation. Our aim is to describe the physics of the quantum Hall transition. The hierarchical network models are constructed by combining series and parallel composition of quantum resistors. The localization-delocalization transition occurring in these models is treated by real space renormalization techniques. Essentially, the localization-delocalization transition is due to a competition between two one-dimensional localization mechanisms.

Physical review letters, Jan 27, 1996

We present two novel approaches to establish the local density of states as an order parameter fi... more We present two novel approaches to establish the local density of states as an order parameter field for the Anderson transition problem. We first demonstrate for 2D quantum Hall systems the validity of conformal scaling relations which are characteristic of order parameter fields. Second we show the equivalence between the critical statistics of eigenvectors of the Hamiltonian and of the transfer matrix, respectively. Based on this equivalence we obtain the order parameter exponent α0 ≈ 3.4 for 3D quantum Hall systems.

Philosophical Magazine B, 1998

We study an ensemble of random matrices (the Rosenzweig-Porter model) which, in contrast to the s... more We study an ensemble of random matrices (the Rosenzweig-Porter model) which, in contrast to the standard Gaussian ensemble, is not invariant under changes of basis. We show that a rather complete understanding of its level correlations can be obtained within the standard framework of diagrammatic perturbation theory. The structure of the perturbation expansion allows for an interpretation of the level structure on simple physical grounds, an aspect that is missing in the exact analysis (T.

Zeitschrift für Physik B Condensed Matter, 1997

We consider a disordered two-dimensional system of independent lattice electrons in a perpendicul... more We consider a disordered two-dimensional system of independent lattice electrons in a perpendicular magnetic field with rigid confinement in one direction and generalized periodic boundary conditions (GPBC) in the other direction. The objects investigated numerically are the orbits in the plane spanned by the energy eigenvalues and the corresponding center of mass coordinate in the confined direction, parameterized by the phase characterizing the GPBC. The Kubo Hall conductivity is expressed in terms of the winding numbers of these orbits. For vanishing disorder the spectrum of the system consists of Harper bands with energy levels corresponding to the edge states within the band gaps. Disorder leads to broadening of the bands. For sufficiently large systems localized states occur in the band tails. We find that within the mobility gaps of bulk states the Diophantine equation determines the value of the Hall conductivity as known for systems with torus geometry (PBCs in both directions). Within the spectral bands of extended states the Hall conductivity fluctuates strongly. For sufficiently large systems the generic behavior of localization-delocalization transitions characteristic for the quantum Hall effect are recovered.

Physical Review E, 2000

We study the statistics of eigenvectors in correlated random band matrix models. These models are... more We study the statistics of eigenvectors in correlated random band matrix models. These models are characterized by two parameters, the band width B(N) of a Hermitian N × N matrix and the correlation parameter C(N) describing correlations of matrix elements along diagonal lines. The correlated band matrices show a much richer phenomenology than models without correlation as soon as the correlation parameter scales sufficiently fast with matrix size. In particular, for B(N) ∼ √ N and C(N) ∼ √ N , the model shows a localization-delocalization transition of the quantum Hall type.

Physical Review B, 1999

On the basis of the Chalker-Coddington network model, a numerical and analytical study is made of... more On the basis of the Chalker-Coddington network model, a numerical and analytical study is made of the statistics of point-contact conductances for systems in the integer quantum Hall regime. In the Hall plateau region the point-contact conductances reflect strong localization of the electrons, while near the plateau transition they exhibit strong mesoscopic fluctuations. By mapping the network model on a supersymmetric vertex model with GL(2|2) symmetry, and postulating a twopoint correlator in keeping with the rules of conformal field theory, we derive an explicit expression for the distribution of conductances at criticality. There is only one free parameter, the power law exponent of the typical conductance. Its value is computed numerically to be Xt = 0.640±0.009. The predicted conductance distribution agrees well with the numerical data. For large distances between the two contacts, the distribution can be described by a multifractal spectrum solely determined by Xt. Our results demonstrate that multifractality can show up in appropriate transport experiments.

Solid State Communications, 1991

Physics Reports, 1998

This review is intended to give a pedagogical and unified view on the subject of the statistics a... more This review is intended to give a pedagogical and unified view on the subject of the statistics and scaling of physical quantities in disordered electron systems at very low temperatures. Quantum coherence at low temperatures and randomness of microscopic details can cause large fluctuations of physical quantities. In such mesoscopic systems a localization-delocalization transition can occur which forms a critical phenomenon. Accordingly, a one-parameter scaling theory was formulated stressing the role of conductance as the (oneparameter) scaling variable. The localized and delocalized phases are separated by a critical point determined by a critical value of conductance. However, the notion of an order parameter was not fully clarified in this theory. The one-parameter scaling theory has been questioned once it was noticed that physical quantities are broadly distributed and that average values are not characteristic for the distributions. Based on presently available analytical and numerical results we focus here on the description of the total distribution functions and their flow with increasing system size. Still, one-parameter scaling theory does work in terms of typical values of the local density of states and of the conductance which serve as order parameter and scaling variable of the localization-delocalization transition, respectively. Below a certain length scale, ξ c , related to the value of the typical conductance, local quantities are multifractally distributed. This multifractal behavior becomes universal on approaching the localization-delocalization transition with ξ c playing the role of a correlation length.

Physical Review Letters, 1998

Physical Review Letters, 1999

Scattering theoretical network models for general coherent wave mechanical systems with quenched ... more 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.