Scale-dependent correction to the dynamical conductivity of a disordered system at unitary symmetry

Anderson localization has been studied extensively for more than half a century. However, while our understanding has been greatly enhanced by calculations based on a small epsilon expansion in d = 2 + epsilon dimensions in the framework of non-linear sigma models, those results can not be safely extrapolated to d = 3. Here we calculate the leading scale-dependent correction to the frequency-dependent conductivity sigma(omega) in dimensions d <= 3. At d = 3 we find a leading correction Re{sigma(omega)} ~ |omega|, which at low frequency is much larger than the omega^2 correction deriving from the Drude law. We also determine the leading correction to the renormalization group beta-function in the metallic phase at d = 3.Comment: 5 pages, 3 figure

Family of solvable generalized random-matrix ensembles with unitary symmetry

We construct a very general family of characteristic functions describing Random Matrix Ensembles (RME) having a global unitary invariance, and containing an arbitrary, one-variable probability measure which we characterize by a `spread function'. Various choices of the spread function lead to a variety of possible generalized RMEs, which show deviations from the well-known Gaussian RME originally proposed by Wigner. We obtain the correlation functions of such generalized ensembles exactly, and show examples of how particular choices of the spread function can describe ensembles with arbitrary eigenvalue densities as well as critical ensembles with multifractality.Comment: 4 pages, to be published in Phys. Rev. E, Rapid Com

Soft matrix models and Chern-Simons partition functions

We study the properties of matrix models with soft confining potentials. Their precise mathematical characterization is that their weight function is not determined by its moments. We mainly rely on simple considerations based on orthogonal polynomials and the moment problem. In addition, some of these models are equivalent, by a simple mapping, to matrix models that appear in Chern-Simons theory. The models can be solved with q deformed orthogonal polynomials (Stieltjes-Wigert polynomials), and the deformation parameter turns out to be the usual $q$ parameter in Chern-Simons theory. In this way, we give a matrix model computation of the Chern-Simons partition function on $S^{3}$ and show that there are infinitely many matrix models with this partition function.Comment: 13 pages, 3 figure

Rotationally invariant family of L\'evy like random matrix ensembles

We introduce a family of rotationally invariant random matrix ensembles characterized by a parameter $\lambda$. While $\lambda=1$ corresponds to well-known critical ensembles, we show that $\lambda \ne 1$ describes "L\'evy like" ensembles, characterized by power law eigenvalue densities. For $\lambda > 1$ the density is bounded, as in Gaussian ensembles, but $\lambda <1$ describes ensembles characterized by densities with long tails. In particular, the model allows us to evaluate, in terms of a novel family of orthogonal polynomials, the eigenvalue correlations for L\'evy like ensembles. These correlations differ qualitatively from those in either the Gaussian or the critical ensembles.Comment: 9 pages, 5 figure

Electronic transport in strongly anisotropic disordered systems: model for the random matrix theory with non-integer beta

We study numerically an electronic transport in strongly anisotropic weakly disorderd two-dimensional systems. We find that the conductance distribution is gaussian but the conductance fluctuations increase when anisotropy becomes stronger. We interpret this result by random matrix theory with non-integer symmetry parameter beta, in accordance with recent theoretical work of K.A.Muttalib and J.R.Klauder [Phys.Rev.Lett. 82 (1999) 4272]. Analysis of the statistics of transport paramateres supports this hypothesis.Comment: 8 pages, 7 *.eps figure

New Class of Random Matrix Ensembles with Multifractal Eigenvectors

Three recently suggested random matrix ensembles (RME) are linked together by an exact mapping and plausible conjections. Since it is known that in one of these ensembles the eigenvector statistics is multifractal, we argue that all three ensembles belong to a new class of critical RME with multifractal eigenfunction statistics and a universal critical spectral statitics. The generic form of the two-level correlation function for weak and extremely strong multifractality is suggested. Applications to the spectral statistics at the Anderson transition and for certain systems on the border of chaos and integrability is discussed.Comment: 4 pages RevTeX, resubmitte

Universality of a family of Random Matrix Ensembles with logarithmic soft-confinement potentials

Recently we introduced a family of $U(N)$ invariant Random Matrix Ensembles which is characterized by a parameter $\lambda$ describing logarithmic soft-confinement potentials $V(H) \sim [\ln H]^{(1+\lambda)} \:(\lambda>0$). We showed that we can study eigenvalue correlations of these "$\lambda$-ensembles" based on the numerical construction of the corresponding orthogonal polynomials with respect to the weight function $\exp[- (\ln x)^{1+\lambda}]$. In this work, we expand our previous work and show that: i) the eigenvalue density is given by a power-law of the form $\rho(x) \propto [\ln x]^{\lambda-1}/x$ and ii) the two-level kernel has an anomalous structure, which is characteristic of the critical ensembles. We further show that the anomalous part, or the so-called "ghost-correlation peak", is controlled by the parameter $\lambda$; decreasing $\lambda$ increases the anomaly. We also identify the two-level kernel of the $\lambda$-ensembles in the semiclassical regime, which can be written in a sinh-kernel form with more general argument that reduces to that of the critical ensembles for $\lambda=1$. Finally, we discuss the universality of the $\lambda$-ensembles, which includes Wigner-Dyson universality ($\lambda \to \infty$ limit), the uncorrelated Poisson-like behavior ($\lambda \to 0$ limit), and a critical behavior for all the intermediate $\lambda$ ($0<\lambda<\infty$) in the semiclassical regime. We also comment on the implications of our results in the context of the localization-delocalization problems as well as the $N$ dependence of the two-level kernel of the fat-tail random matrices.Comment: 10 pages, 13 figure

Conductance distribution in disordered quantum wires: Crossover between the metallic and insulating regimes

We calculate the distribution of the conductance P(g) for a quasi-one-dimensional system in the metal to insulator crossover regime, based on a recent analytical method valid for all strengths of disorder. We show the evolution of P(g) as a function of the disorder parameter from a insulator to a metal. Our results agree with numerical studies reported on this problem, and with analytical results for the average and variance of g.Comment: 8 pages, 5 figures. Final version (minor changes

Level Spacing Distribution of Critical Random Matrix Ensembles

We consider unitary invariant random matrix ensembles which obey spectral statistics different from the Wigner-Dyson, including unitary ensembles with slowly (~(log x)^2) growing potentials and the finite-temperature fermi gas model. If the deformation parameters in these matrix ensembles are small, the asymptotically translational-invariant region in the spectral bulk is universally governed by a one-parameter generalization of the sine kernel. We provide an analytic expression for the distribution of the eigenvalue spacings of this universal asymptotic kernel, which is a hybrid of the Wigner-Dyson and the Poisson distributions, by determining the Fredholm determinant of the universal kernel in terms of a Painleve VI transcendental function.Comment: 5 pages, 1 figure, REVTeX; restriction on the parameter stressed, figure replaced, refs added (v2); typos (factors of pi) in (35), (36) corrected (v3); minor changes incl. title, version to appear in Phys.Rev.E (v4

Generalized Fokker-Planck Equation For Multichannel Disordered Quantum Conductors

The Dorokhov-Mello-Pereyra-Kumar (DMPK) equation, which describes the distribution of transmission eigenvalues of multichannel disordered conductors, has been enormously successful in describing a variety of detailed transport properties of mesoscopic wires. However, it is limited to the regime of quasi one dimension only. We derive a one parameter generalization of the DMPK equation, which should broaden the scope of the equation beyond the limit of quasi one dimension.Comment: 8 pages, abstract, introduction and summary rewritten for broader readership. To be published in Phys. Rev. Let