Correlation Functions in Disordered Systems

{Recently, we found that the correlation between the eigenvalues of random hermitean matrices exhibits universal behavior. Here we study this universal behavior and develop a diagrammatic approach which enables us to extend our previous work to the case in which the random matrix evolves in time or varies as some external parameters vary. We compute the current-current correlation function, discuss various generalizations, and compare our work with the work of other authors. We study the distribution of eigenvalues of Hamiltonians consisting of a sum of a deterministic term and a random term. The correlation between the eigenvalues when the deterministic term is varied is calculated.}Comment: 19 pages, figures not included (available on request), Tex, NSF-ITP-93-12

Anomalously large conductance fluctuations in weakly disordered graphene

We have studied numerically the mesoscopic fluctuations of the conductance of a graphene strip (width W large compared to length L), in an ensemble of samples with different realizations of the random electrostatic potential landscape. For strong disorder (potential fluctuations comparable to the hopping energy), the variance of the conductance approaches the value predicted by the Altshuler-Lee-Stone theory of universal conductance fluctuations. For weaker disorder the variance is greatly enhanced if the potential is smooth on the scale of the atomic separation. There is no enhancement if the potential varies on the atomic scale, indicating that the absence of backscattering on the honeycomb lattice is at the origin of the anomalously large fluctuations.Comment: 5 pages, 8 figure

Statistical translation invariance protects a topological insulator from interactions

We investigate the effect of interactions on the stability of a disordered, two-dimensional topological insulator realized as an array of nanowires or chains of magnetic atoms on a superconducting substrate. The Majorana zero-energy modes present at the ends of the wires overlap, forming a dispersive edge mode with thermal conductance determined by the central charge $c$ of the low-energy effective field theory of the edge. We show numerically that, in the presence of disorder, the $c=1/2$ Majorana edge mode remains delocalized up to extremely strong attractive interactions, while repulsive interactions drive a transition to a $c=3/2$ edge phase localized by disorder. The absence of localization for strong attractive interactions is explained by a self-duality symmetry of the statistical ensemble of disorder configurations and of the edge interactions, originating from translation invariance on the length scale of the underlying mesoscopic array.Comment: 5+2 pages, 8 figure

Manipulation of photon statistics of highly degenerate chaotic radiation

Highly degenerate chaotic radiation has a Gaussian density matrix and a large occupation number of modes $f$. If it is passed through a weakly transmitting barrier, its counting statistics is close to Poissonian. We show that a second identical barrier, in series with the first, drastically modifies the statistics. The variance of the photocount is increased above the mean by a factor $f$ times a numerical coefficient. The photocount distribution reaches a limiting form with a Gaussian body and highly asymmetric tails. These are general consequences of the combination of weak transmission and multiple scattering.Comment: 4 pages, 2 figure

Medium/high field magnetoconductance in chaotic quantum dots

The magnetoconductance G in chaotic quantum dots at medium/high magnetic fluxes Phi is calculated by means of a tight binding Hamiltonian on a square lattice. Chaotic dots are simulated by introducing diagonal disorder on surface sites of L x L clusters. It is shown that when the ratio W/L is sufficiently large, W being the leads width, G increases steadily showing a maximum at a flux Phi_max ~ W. Bulk disordered ballistic cavities (with an amount of impurities proportional to L) does not show this effect. On the other hand, for magnetic fluxes larger than that for which the cyclotron radius is of the order of L/2, the average magnetoconductance inceases almost linearly with the flux with a slope proportional to W^2, shows a maximum and then decreases stepwise. These results closely follow a theory proposed by Beenakker and van Houten to explain the magnetoconductance of two point contacts in series.Comment: RevTeX including six postscript figure

Classical limit of transport in quantum kicked maps

We investigate the behavior of weak localization, conductance fluctuations, and shot noise of a chaotic scatterer in the semiclassical limit. Time resolved numerical results, obtained by truncating the time-evolution of a kicked quantum map after a certain number of iterations, are compared to semiclassical theory. Considering how the appearance of quantum effects is delayed as a function of the Ehrenfest time gives a new method to compare theory and numerical simulations. We find that both weak localization and shot noise agree with semiclassical theory, which predicts exponential suppression with increasing Ehrenfest time. However, conductance fluctuations exhibit different behavior, with only a slight dependence on the Ehrenfest time.Comment: 17 pages, 13 figures. Final versio

Metallic phase of the quantum Hall effect in four-dimensional space

We study the phase diagram of the quantum Hall effect in four-dimensional (4D) space. Unlike in 2D, in 4D there exists a metallic as well as an insulating phase, depending on the disorder strength. The critical exponent $\nu\approx 1.2$ of the diverging localization length at the quantum Hall insulator-to-metal transition differs from the semiclassical value $\nu=1$ of 4D Anderson transitions in the presence of time-reversal symmetry. Our numerical analysis is based on a mapping of the 4D Hamiltonian onto a 1D dynamical system, providing a route towards the experimental realization of the 4D quantum Hall effect.Comment: 4+epsilon pages, 3 figure

Universal correlations for deterministic plus random Hamiltonians

We consider the (smoothed) average correlation between the density of energy levels of a disordered system, in which the Hamiltonian is equal to the sum of a deterministic H0 and of a random potential $\varphi$. Remarkably, this correlation function may be explicitly determined in the limit of large matrices, for any unperturbed H0 and for a class of probability distribution P$(\varphi)$ of the random potential. We find a compact representation of the correlation function. From this representation one obtains readily the short distance behavior, which has been conjectured in various contexts to be universal. Indeed we find that it is totally independent of both H0 and P($\varphi$).Comment: 26P, (+5 figures not included

Correlations between eigenvalues of large random matrices with independent entries

We derive the connected correlation functions for eigenvalues of large Hermitian random matrices with independently distributed elements using both a diagrammatic and a renormalization group (RG) inspired approach. With the diagrammatic method we obtain a general form for the one, two and three-point connected Green function for this class of ensembles when matrix elements are identically distributed, and then discuss the derivation of higher order functions by the same approach. Using the RG approach we re-derive the one and two-point Green functions and show they are unchanged by choosing certain ensembles with non-identically distributed elements. Throughout, we compare the Green functions we obtain to those from the class of ensembles with unitary invariant distributions and discuss universality in both ensemble classes.Comment: 23 pages, RevTex, hard figures available from [email protected]

Conductance Fluctuations in a Disordered Double-Barrier Junction

We consider the effect of disorder on coherent tunneling through two barriers in series, in the regime of overlapping transmission resonances. We present analytical calculations (using random-matrix theory) and numerical simulations (on a lattice) to show that strong mode-mixing in the inter-barrier region induces mesoscopic fluctuations in the conductance $G$ of universal magnitude $e^2/h$ for a symmetric junction. For an asymmetric junction, the root-mean-square fluctuations depend on the ratio $\nu$ of the two tunnel resistances according to ${rms} G = (4e^2/h)\beta^{-1/2} \nu(1+\nu)^{-2}$, where $\beta = 1 (2)$ in the presence (absence) of time-reversal symmetry.Comment: 12 pages, REVTeX-3.0, 2 figures, submitted to Physical Review