Showing papers by "Boris I Shklovskii published in 2020"

Anderson transition in three-dimensional systems with non-Hermitian disorder

Abstract: We study the Anderson transition for three-dimensional (3D) $N\ifmmode\times\else\texttimes\fi{}N\ifmmode\times\else\texttimes\fi{}N$ tightly bound cubic lattices where both real and imaginary parts of on-site energies are independent random variables distributed uniformly between $\ensuremath{-}W/2$ and $W/2$. Such a non-Hermitian analog of the Anderson model is used to describe random-laser medium with local loss and amplification. We employ eigenvalue statistics to search for the Anderson transition. For 25% smallest-modulus complex eigenvalues we find the average ratio $r$ of distances to the first and the second nearest neighbor as a function of $W$. For a given $N$ the function $r(W)$ crosses from 0.72 to 2/3 with a growing $W$ demonstrating a transition from delocalized to localized states. When plotted at different $N$ all $r(W)$ cross at ${W}_{c}=6.0\ifmmode\pm\else\textpm\fi{}0.1$ (in units of nearest-neighbor overlap integral) clearly demonstrating the 3D Anderson transition. We find that in the non-Hermitian 2D Anderson model, the transition is replaced by a crossover.

Spectral rigidity of non-Hermitian symmetric random matrices near the Anderson transition

Abstract: We study the spectral rigidity of the non-Hermitian analog of the Anderson model suggested by Tzortzakakis, Makris, and Economou (TME). This is a $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ tightly bound cubic lattice, where both real and imaginary parts of onsite energies are independent random variables uniformly distributed between $\ensuremath{-}W/2$ and $W/2$. The TME model may be used to describe a random laser. In a recent paper we proved that this model has the Anderson transition at $W={W}_{c}\ensuremath{\simeq}6$ in three dimension. Here we numerically diagonalize TME $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ cubic lattice matrices and calculate the number variance of eigenvalues in a disk of their complex plane. We show that on the metallic side $Wl6$ of the Anderson transition, complex eigenvalues repel each other as strongly as in the complex Ginibre ensemble only in a disk containing ${N}_{c}(L,W)$ eigenvalues. We find that ${N}_{c}(L,W)$ is proportional to $L$ and grows with decreasing $W$ similarly to the number of energy levels ${N}_{c}$ in the Thouless energy band of the Anderson model.

Isotropically conducting (hidden) quantum Hall stripe phases in a two-dimensional electron gas

Abstract: Quantum Hall stripe (QHS) phases, predicted by the Hartree-Fock theory, are manifested in GaAs-based two-dimensional electron gases as giant resistance anisotropies. Here, we predict a ``hidden'' QHS phase which exhibits isotropic resistivity whose value, determined by the density of states of QHS, is independent of the Landau index $N$ and is inversely proportional to the Drude conductivity at zero magnetic field. At high enough $N$, this phase yields to an Ando-Unemura/Coleridge-Zawadski-Sachrajda phase in which the resistivity is proportional to $1/N$ and to the ratio of quantum and transport lifetimes. Experimental observation of this border can provide an alternative way to obtain quantum relaxation time.

Hidden Quantum Hall Stripes in Al x Ga 1 − x As / Al 0.24 Ga 0.76 As Quantum Wells

Abstract: We report on transport signatures of hidden quantum Hall stripe (hQHS) phases in high (N>2) half-filled Landau levels of Al_{x}Ga_{1-x}As/Al_{0.24}Ga_{0.76}As quantum wells with varying Al mole fraction x<10^{-3}. Residing between the conventional stripe phases (lower N) and the isotropic liquid phases (higher N), where resistivity decreases as 1/N, these hQHS phases exhibit isotropic and N-independent resistivity. Using the experimental phase diagram, we establish that the stripe phases are more robust than theoretically predicted, calling for improved theoretical treatment. We also show that, unlike conventional stripe phases, the hQHS phases do not occur in ultrahigh mobility GaAs quantum wells but are likely to be found in other systems.

Spectral rigidity of non-Hermitian symmetric random matrices

Abstract: We study the spectral rigidity of the non-Hermitian analog of the Anderson model suggested by Tzortzakakis, Makris, and Economou (TME). This is a $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ tightly bound cubic lattice, where both real and imaginary parts of onsite energies are independent random variables uniformly distributed between $\ensuremath{-}W/2$ and $W/2$. The TME model may be used to describe a random laser. In a recent paper we proved that this model has the Anderson transition at $W={W}_{c}\ensuremath{\simeq}6$ in three dimension. Here we numerically diagonalize TME $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ cubic lattice matrices and calculate the number variance of eigenvalues in a disk of their complex plane. We show that on the metallic side $Wl6$ of the Anderson transition, complex eigenvalues repel each other as strongly as in the complex Ginibre ensemble only in a disk containing ${N}_{c}(L,W)$ eigenvalues. We find that ${N}_{c}(L,W)$ is proportional to $L$ and grows with decreasing $W$ similarly to the number of energy levels ${N}_{c}$ in the Thouless energy band of the Anderson model.