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. Moreover, we show that our model is an example of a new universality class of models with power-law hopping, characterized by a duality between systems with long-range hops (a<1) and short-range hops (a>1), in which the wave function amplitude falls off algebraically with the same power γ from the localization center.

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. By extrapolation of the results of both approaches to N→∞ we obtain the fractal dimensions D_{1}(W) and D_{2}(W) as well as the population dynamics exponent D(W) with the accuracy sufficient to claim that they are nontrivial in the broad interval of disorder strength W_{E}10^{5} reveals a singularity in D_{1,2}(W) dependencies which provides clear evidence for the first order transition between the two delocalized phases on RRG at W_{E}≈10.0. We discuss the implications of these results for quantum and classical nonintegrable and many-body systems.

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. Here we exploit the analogy between the statistics of work dissipated in a driven single-electron box and that of random multifractal wavefunction amplitudes, and uncover new relations that generalize the Jarzynski equality. We checked the new relations theoretically using the rate equations for sequential tunnelling of electrons and experimentally by measuring the dissipated work in a driven single-electron box and found a remarkable correspondence. The results represent an important universal feature of the work statistics in systems out of equilibrium and help to understand the nature of the symmetry of multifractal exponents in the theory of Anderson localization.

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.e., the ergodicity can be reached only in the absence of disorder. The one-particle multifractality on the Bethe lattice signals on a possible inapplicability of the equipartition law to a generic many-body quantum system as long as it remains isolated.

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. We show that in strongly disordered systems with time-reversal symmetry broken by external or intrinsic exchange magnetic field the cooling power can be greatly enhanced. The enhancement factor as large as 10(2) at magnetic field B~10 T in 2D InSb films is predicted. A similar enhancement is found for the ultrasound attenuation.

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.

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.

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). Assuming the dot to be described by random-matrix theory for the Gaussian orthogonal ensemble, we find the quantum correction to the energy absorption rate as a function of the dephasing time t(phi). If phi(t) is a sum of d harmonics with incommensurate frequencies, the correction behaves similarly to that for the conductivity deltasigma(d)(t(phi)) in the d-dimensional Anderson model of the orthogonal symmetry class. For a generic periodic perturbation, the leading quantum correction is absent as in the systems of the unitary symmetry class, unless phi(-t+tau)=phi(t+tau) for some tau, which falls into the quasi-1D orthogonal universality class.

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. The novel kinetic part is present only out of equilibrium and is governed by the off-diagonal matrix elements J(micro nu). It has drastically different temperature and magnetic field behavior.