Tail states and unusual localization transition in low-dimensional Anderson model with power-law hopping

Abstract

We study deterministic power-law quantum hopping model with an amplitude $J(r)\propto -r^{-\beta }$ and local Gaussian disorder in low dimensions $d=1,2$ under the condition $d<\beta <3d/2$. We demonstrate unusual combination of exponentially decreasing density of the "tail states" and localization-delocalization transition (as function of disorder strength $w$) pertinent to a small (vanishing in thermodynamic limit) fraction of eigenstates. At sub-critical disorder $w<w_c$ delocalized eigenstates with energies near the bare band edge co-exist with a strongly localized eigenstates in the same energy window. At higher disorder $w>w_c$ all eigenstates are localized. In a broad range of parameters density of states $\nu (E)$ decays into the tail region $E<0$ as simple exponential, $\nu (E)={\nu }_0{e}^{E/E_0}$, while characteristic energy $E_0$ varies smoothly across edge localization transition. We develop simple analytic theory which describes $E_0$ dependence on power-law exponent $\beta$, dimensionality $d$ and disorder strength $W$, and compare its predictions with exact diagonalization results. At low energies within the bare "conduction band", all eigenstates are localized due to strong quantum interference at $d=1,2$; however localization length grows fast with energy decrease, contrary to the case of usual Schrodinger equation with local disorder.

References in corpus (4)

• Correlation-induced localization
• Unconventional localisation transition in high dimensions
• Duality in power-law localization in disordered one-dimensional systems
• Non-ergodic extended phase of the Quantum Random Energy model