Renormalization group analysis of the Anderson model on random regular graphs.

We present a renormalization group (RG) analysis of the problem of Anderson localization on a random regular graph (RRG) which generalizes the RG of Abrahams, Anderson, Licciardello, and Ramakrishnan to infinite-dimensional graphs. The RG equations necessarily involve two parameters (one being the changing connectivity of subtrees), but we show that the one-parameter scaling hypothesis is recovered for sufficiently large system sizes for both eigenstates and spectrum observables. We also explain the nonmonotonic behavior of dynamical and spectral quantities as a function of the system size for values of disorder close to the transition, by identifying two terms in the beta function of the running fractal dimension of different signs and functional dependence.

Duality in Power-Law Localization in Disordered One-Dimensional Systems.

The transport of excitations between pinned particles in many physical systems may be mapped to single-particle models with power-law hopping, 1/r^{a}. For randomly spaced particles, these models present an effective peculiar disorder that leads to surprising localization properties. We show that in one-dimensional systems almost all eigenstates (except for a few states close to the ground state) are power-law localized for any value of a>0.

Nonergodic Phases in Strongly Disordered Random Regular Graphs.

We combine numerical diagonalization with semianalytical calculations to prove the existence of the intermediate nonergodic but delocalized phase in the Anderson model on disordered hierarchical lattices. We suggest a new generalized population dynamics that is able to detect the violation of ergodicity of the delocalized states within the Abou-Chakra, Anderson, and Thouless recursive scheme. This result is supplemented by statistics of random wave functions extracted from exact diagonalization of the Anderson model on ensemble of disordered random regular graphs (RRG) of N sites with the connectivity K=2.

Multifractality of random eigenfunctions and generalization of Jarzynski equality.

Systems driven out of equilibrium experience large fluctuations of the dissipated work. The same is true for wavefunction amplitudes in disordered systems close to the Anderson localization transition. In both cases, the probability distribution function is given by the large-deviation ansatz.

Anderson localization on the Bethe lattice: nonergodicity of extended States.

Statistical analysis of the eigenfunctions of the Anderson tight-binding model with on-site disorder on regular random graphs strongly suggests that the extended states are multifractal at any finite disorder. The spectrum of fractal dimensions f(α) defined in Eq. (3) remains positive for α noticeably far from 1 even when the disorder is several times weaker than the one which leads to the Anderson localization; i.

Magnetic field-induced giant enhancement of electron-phonon energy transfer in strongly disordered conductors.

Relaxation of soft modes (e.g., charge density in gated semiconductor heterostructures, spin density in the presence of magnetic field) slowed down by disorder may lead to giant enhancement of energy transfer (cooling power) between overheated electrons and phonons at low bath temperature.

Lévy flights and multifractality in quantum critical diffusion and in classical random walks on fractals.

We employ the method of virial expansion to compute the retarded density correlation function (generalized diffusion propagator) in the critical random matrix ensemble in the limit of strong multifractality. We find that the long-range nature of the Hamiltonian is a common root of both multifractality and Lévy flights, which show up in the power-law intermediate- and long-distance behaviors, respectively, of the density correlation function. We review certain models of classical random walks on fractals and show the similarity of the density correlation function in them to that for the quantum problem described by the random critical long-range Hamiltonians.

Horizon in random matrix theory, the Hawking radiation, and flow of cold atoms.

We propose a Gaussian scalar field theory in a curved 2D metric with an event horizon as the low-energy effective theory for a weakly confined, invariant random matrix ensemble (RME). The presence of an event horizon naturally generates a bath of Hawking radiation, which introduces a finite temperature in the model in a nontrivial way. A similar mapping with a gravitational analogue model has been constructed for a Bose-Einstein condensate (BEC) pushed to flow at a velocity higher than its speed of sound, with Hawking radiation as sound waves propagating over the cold atoms.

Jumps in current-voltage characteristics in disordered films.

We argue that giant jumps of current at finite voltages observed in disordered films of InO, TiN, and YSi manifest a bistability caused by the overheating of electrons. One of the stable states is overheated and thus low resistive, while the other, high-resistive state is heated much less by the same voltage. The bistability occurs provided that cooling of electrons is inefficient and the temperature dependence of the equilibrium resistance R(T) is steep enough.

Eigenfunction fractality and pseudogap state near the superconductor-insulator transition.

We develop a theory of a pseudogap state appearing near the superconductor-insulator (SI) transition in strongly disordered metals with an attractive interaction. We show that such an interaction combined with the fractal nature of the single-particle wave functions near the mobility edge leads to an anomalously large single-particle gap in the superconducting state near SI transition that persists and even increases in the insulating state long after the superconductivity is destroyed. We give analytic expressions for the value of the pseudogap in terms of the inverse participation ratio of the corresponding localization problem.

Dynamic localization and the Coulomb blockade in quantum dots under ac pumping.

We study conductance through a quantum dot under Coulomb blockade conditions in the presence of an external periodic perturbation. The stationary state is determined by the balance between the heating of the dot electrons by the perturbation and cooling by electron exchange with the cold contacts. We show that the Coulomb blockade peak can have a peculiar shape if heating is affected by dynamic localization, which can be an experimental signature of this effect.

Density of states for almost-diagonal random matrices.

We study the density of states (DOS) for disordered systems whose spectral statistics can be described by a Gaussian ensemble of almost-diagonal Hermitian random matrices. The matrices have independent random entries H(i > or =j) with small off-diagonal elements: <|H(i not equal to j)|2> << <|H(ii)|2> approximately 1. Using the recently suggested method of a virial expansion in the number of interacting energy levels [J.

Dynamic localization in quantum dots: analytical theory.

We analyze the response of a complex quantum-mechanical system (e.g., a quantum dot) to a time-dependent perturbation phi(t).

Aharonov-bohm magnetization of mesoscopic rings caused by inelastic relaxation.

The magnetization of a system of many mesoscopic rings under nonequilibrium conditions is considered. The corresponding disorder-averaged current in a ring I(phi) is shown to be a sum of the "thermodynamic" and "kinetic" contributions both resulting from the electron-electron interaction. The thermodynamic part can be expressed through the diagonal matrix elements J(micro micro) of the current operator in the basis of exact many-body eigenstates and is a generalization of the equilibrium persistent current.

Relationship between the noise-induced persistent current and the dephasing rate.

ac noise in disordered conductors causes both dephasing of the electron wave functions and a dc current around a mesoscopic ring. We demonstrate that the dephasing rate tau(-1)(varphi) in long wires and the dc current , induced by the same noise and averaged over an ensemble of small rings, are connected in a remarkably simple way: tau(varphi) = C(beta)e. Here e is an electron charge, and the constant C(beta) approximately 1 depends on the Dyson symmetry class.