The low-frequency conductivity of a disordered Fermi gas in one spatial dimension is governed by the Mott-Berezinskii law
σ(ω)∝ω2lnω2. In a recent Letter [Phys. Rev. Lett. 84, 1760 (2000)] A. O. Gogolin claimed that this law is invalid, challenging our basic understanding of disordered systems and a massive amount of previous theoretical work. We point out two calculational errors in Gogolin's paper. Once we correct them, the Mott-Berezinskii formula is fully recovered. We also present numerical results supporting the Mott-Berezinskii formula but ruling out that of Gogolin.
The macroscopic transport properties in a disordered potential, namely diffusion and weak/strong localization, closely depend on the microscopic and statistical properties of the disorder itself.This dependence is rich in counter-intuitive consequences.It can be particularly exploited in matter wave experiments, where the disordered potential can be tailored and controlled, and anisotropies are naturally present.In this work, we apply a perturbative microscopic transport theory and the self-consistent theory of Anderson localization to study the transport properties of ultracold atoms in anisotropic two-dimensional (2D) and three-dimensional (3D) speckle potentials.In particular, we discuss the anisotropy of single-scattering, diffusion and localization.We also calculate disorder-induced shift of the energy states and propose a method to include it, which amounts to renormalizing energies in the standard on-shell approximation.We show that the renormalization of energies strongly affects the prediction for the 3D localization threshold (mobility edge).We illustrate the theoretical findings with examples which are relevant for current matter wave experiments, where the disorder is created with laser speckle.This paper provides a guideline for future experiments aiming at the precise location of the 3D mobility edge and study of anisotropic diffusion and localization effects in 2D and 3D.
In this work we study Anderson localization of quantum states in different kind of disordered media. First we apply the recently-developed theory of \textit{landscape localization} pioneered by Filoche and Mayboroda, giving rise to an effective potential which yields further insight into localization. We analyze this problem numerically in one, two and three dimensions with uniform and point-like potentials, aiming at inferring the properties of the localized states from the knowledge of the localization landscape. We also consider quasiperiodic potentials which induce a metal-insulator transition (extended to localized states), testing the prediction of the wave functions using the effective potential as a reference. Finally, we study how the localized states are distributed in the energy spectrum energy in one-dimension and how we can modify their localization varying the properties of the random potential. We find that for a uniform disordered potential, the density of states is symmetric around the mean energy. And for a point-like potential, the spectrum energy has a continuous part at low energies and a discrete part at high energies.
An important experimental signature of localization is the low-frequency ac conductivity, which typically vanishes as ${\ensuremath{\omega}}^{\ensuremath{\phi}}$. The exponent $\ensuremath{\phi}=2$ for Anderson insulators, whereas for many body-localized insulators $\ensuremath{\phi}$ is a continuously varying exponent, $1\ensuremath{\le}\ensuremath{\phi}\ensuremath{\le}2$. In this work, we study the low-frequency ac conductivity of localized superconductors, in which disorder is strong enough to localize all quasiparticles, while remaining weak enough to leave superconductivity intact. We find that, while the ac conductivity still follows the general form $\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\omega}}^{\ensuremath{\phi}}$, the exponent $\ensuremath{\phi}$ can be markedly different from the characteristic value for localized insulators. This difference occurs due to singularities in the low-energy density of states, permitted by the effective particle-hole symmetry around the Fermi level. In particular, in certain symmetry classes at zero temperature, we obtain $\ensuremath{\phi}>2$. We further identify an interesting temperature-dependent crossover in the scaling form of the ac conductivity, which could be useful for the experimental characterization of localized superconductors.
Anderson localization has evolved far beyond the conventional dichotomy between extended and localized states. Modern localization theory predicts a complete transport hierarchy comprising extended, critical, and localized phases together with all coexistence phases among them, forming a seven-phase Anderson-localization landscape. Despite its fundamental importance, this hierarchy has never been experimentally realized within a single system. Here we realize the complete seven-phase Anderson-localization landscape in a one-dimensional Floquet photonic lattice. By engineering quasiperiodic hopping profiles containing inhomogeneously distributed hopping zeros, we generate critical states and enable their coexistence with extended and localized sectors. The resulting transport regimes are directly resolved through their distinct spatiotemporal dynamics, including ballistic expansion, confined critical oscillations, and persistent localization. We observe all seven phases, including the elusive triply coexisting extended-critical-localized phase, and experimentally track the phase transitions connecting them. Our results establish the first complete experimental map of the Anderson-localization landscape and provide a unified platform for investigating mobility edges, multifractality, and programmable coherent transport.
This work is devoted to the development of a systematic method for studying electron localization. The developed method is Typical Medium Dynamical Cluster Approximation (TMDCA) using the Anderson-Hubbard model. The TMDCA incorporates non-local correlations beyond the local typical environment in a self-consistent way utilizing the momentum resolved typical-density-of-states and the non-local hybridization function to characterize the localization transition. For the (non-interacting) Anderson model, I show that the TMDCA provides a proper description of the Anderson localization transition in one, two, and three dimensions. In three-dimensions, as a function of cluster size, the TMDCA systematically recovers the re-entrance behavior of the mobility edge and obtains the correct critical disorder strength for the various disorder configurations and the associated \textit{universal order-parameter-critical-exponent}
β and in lower-dimensions, the well-knowing scaling relations are reproduced in agreement with numerical exact results. The TMDCA is also extended to treat diagonal and off-diagonal disorder by generalizing the local Blackman-Esterling-Berk and the importance of finite cluster is demonstrated. It was further generalized for multiband systems. Applying the TMDCA to weakly interaction electronic systems, I show that incorporating Coulomb interactions into disordered electron system result in two competing tendencies: the suppression of the current due to correlations and the screening of the disorder leading to the homogenizing of the system. It is shown that the critical disorder strength (
WcU), required to localize all states, increases with increasing interactions (
U); implying that the metallic phase is stabilized by interactions. Using the results, a soft pseudogap at the intermediate
W close to
WcU is predicted independent of filling and dimension, and I demonstrate in three-dimensions that the mobility edge is preserved as long as the chemical potential,
μ, is at or beyond the mobility edge energy (
ωϵ). A two-particle formalism of electron localization is also developed within the TMDCA and used to calculate the direct-current conductivity, enabling direct comparison with experiments. Note significantly, the TMDCA benchmarks well with numerical exact results with a dramatic reduction in computational cost, enabling the incorporation of material's specific details as such provide an avenue for the possibility of studying electron localization in real materials.