Anderson localization on a simplex

We derive a field-theoretical representation for the moments of the eigenstates in the generalized Anderson model. The representation is exact and can be used for the Anderson model with generic non-random hopping elements in any dimensions. We apply this method to the simplex model, for which the hopping amplitude between any two lattice sites is the same, and find that the eigenstates are localized at any strength of disorder. Our analytical predictions are in excellent agreement with the results of numerical simulations.Comment: 18 page

Virial expansion of the non-linear sigma model in the strong coupling limit

We develop a perturbative approach to study the supersymmetric non-linear sigma model characterized by a generic coupling matrix in the strong coupling limit. The method allows us to calculate explicitly the moments of the eigenfunctions and the two-level correlation function in the lowest order of the perturbative expansion. We find that the obtained expressions are equivalent to the results derived before for the corresponding random matrix ensembles. Such an equivalence is elucidated and generalized to all orders of the perturbative expansion by mapping the sigma model onto the field theory describing the almost diagonal random matrices.Comment: 17 page

Fingerprints of classical diffusion in open 2D mesoscopic systems in the metallic regime

We investigate the distribution of the resonance widths ${\cal P}(\Gamma)$ and Wigner delay times ${\cal P}(\tau_W)$ for scattering from two-dimensional systems in the diffusive regime. We obtain the forms of these distributions (log-normal for large $\tau_W$ and small $\Gamma$, and power law in the opposite case) for different symmetry classes and show that they are determined by the underlying diffusive classical dynamics. Our theoretical arguments are supported by extensive numerical calculations.Comment: 7 pages, 3 figure

Scattering approach to Anderson localization

We develop a novel approach to the Anderson localization problem in a d-dimensional disordered sample of dimension L×Md−1. Attaching a perfect lead with the cross section Md−1 to one side of the sample, we derive evolution equations for the scattering matrix and the Wigner-Smith time delay matrix as a function of L. Using them one obtains the Fokker-Planck equation for the distribution of the proper delay times and the evolution equation for their density at weak disorder. The latter can be mapped onto a nonlinear partial differential equation of the Burgers type, for which a complete analytical solution for arbitrary L is constructed. Analyzing the solution for a cubic sample with M=L in the limit L→∞, we find that for d2 to the metallic fixed point, and provide explicit results for the density of the delay times in these two limits

Dynamical scaling for critical states: is Chalker's ansatz valid for strong fractality?

The dynamical scaling for statistics of critical multifractal eigenstates proposed by Chalker is analytically verified for the critical random matrix ensemble in the limit of strong multifractality controlled by the small parameter $b\ll 1$. The power law behavior of the quantum return probability $P_{N}(\tau)$ as a function of the matrix size $N$ or time $\tau$ is confirmed in the limits $\tau/N\rightarrow\infty$ and $N/\tau\rightarrow\infty$, respectively, and it is shown that the exponents characterizing these power laws are equal to each other up to the order $b^{2}$. The corresponding analytical expression for the fractal dimension $d_{2}$ is found.Comment: 4 pages, 1 figur

Entanglement entropy in Fermi gases and Anderson's orthogonality catastrophe

We study the ground-state entanglement entropy of a finite subsystem of size L of an infinite system of noninteracting fermions scattered by a potential of finite range a. We derive a general relation between the scattering matrix and the overlap matrix and use it to prove that for a one-dimensional symmetric potential the von Neumann entropy, the Rényi entropies, and the full counting statistics are robust against potential scattering, provided that L/a≫1. The results of numerical calculations support the validity of this conclusion for a generic potential

Level number variance and spectral compressibility in a critical two-dimensional random matrix model

We study level number variance in a two-dimensional random matrix model characterized by a power-law decay of the matrix elements. The amplitude of the decay is controlled by the parameter b. We find analytically that at small values of b the level number variance behaves linearly, with the compressibility chi between 0 and 1, which is typical for critical systems. For large values of b, we derive that chi=0, as one would normally expect in the metallic phase. Using numerical simulations we determine the critical value of b at which the transition between these two phases occurs.Comment: 6 page

Reentrance effect in a graphene n-p-n junction coupled to a superconductor

We study the interplay of Klein tunneling (= interband tunneling) between n-doped and p-doped regions in graphene and Andreev reflection (= electron-hole conversion) at a superconducting electrode. The tunneling conductance of an n-p-n junction initially increases upon lowering the temperature, while the coherence time of the electron-hole pairs is still less than their lifetime, but then drops back again when the coherence time exceeds the lifetime. This reentrance effect, known from diffusive conductors and ballistic quantum dots, provides a method to detect phase coherent Klein tunneling of electron-hole pairs.Comment: 4 pages, 3 figure

How spin-orbit interaction can cause electronic shot noise

The shot noise in the electrical current through a ballistic chaotic quantum dot with N-channel point contacts is suppressed for N --> infinity, because of the transition from stochastic scattering of quantum wave packets to deterministic dynamics of classical trajectories. The dynamics of the electron spin remains quantum mechanical in this transition, and can affect the electrical current via spin-orbit interaction. We explain how the role of the channel number N in determining the shot noise is taken over by the ratio l_{so}/lambda_F of spin precession length l_{so} and Fermi wave length lambda_F, and present computer simulations in a two-dimensional billiard geometry (Lyapunov exponent alpha, mean dwell time tau_{dwell}, point contact width W) to demonstrate the scaling (lambda_F/l_{so})^{1/alpha tau_{dwell}} of the shot noise in the regime lambda_F << l_{so} << W.Comment: 4 pages, 3 figure

Localization-delocalization transition in the quasi-one-dimensional ladder chain with correlated disorder

The generalization of the dimer model on a two-leg ladder is defined and investigated both, analytically and numerically. For the closed system we calculate the Landauer resistance analytically and found the presence of the point of delocalization at the band center which is confirmed by the numerical calculations of the Lyapunov exponent. We calculate also analytically the localization length index and present the numerical investigations of the density of states (DOS). For the open counterpart of this model the distribution of the Wigner delay times is calculated numerically. It is shown how the localization-delocalization transition manifest itself in the behavior of the distribution.Comment: 9 pages, 10 figures, Revte