Absence of Mobility Edge in Short-Range Uncorrelated Disordered Model: Coexistence of Localized and Extended States

Unlike the well-known Mott’s argument that extended and localized states should not coexist at the same energy in a generic random potential, we formulate the main principles and provide an example of a nearest-neighbor tight-binding disordered model which carries both localized and extended states without forming the mobility edge. Unexpectedly, this example appears to be given by a well-studied $\beta$ ensemble with independently distributed random diagonal potential and inhomogeneous kinetic hopping terms. In order to analytically tackle the problem, we locally map the above model to the 1D Anderson model with matrix-size- and position-dependent hopping and confirm the coexistence of localized and extended states, which is shown to be robust to the perturbations of both potential and kinetic terms due to the separation of the above states in space. In addition, the mapping shows that the extended states are nonergodic and allows one to analytically estimate their fractal dimensions.

Figure 1

(a) Schematic of $\beta$ ensemble, given by 1D lattice in Eq. (1). The hopping increases along the lattice (thickness, color of links) and yields localized (left) and extended (right) states, coexisting at the same energy, but living in spatially different system parts. (b) Phase diagram of the $\beta$ ensemble, with three distinct phases along with typical fractal dimension in the NEE phase, $D_2^{\mathrm{typ}}\approx 1-\gamma$, and the number $N_{\mathrm{loc}}\sim N^{\gamma }$ of strongly localized states, $|{\mathrm{\Psi }}_{\mathrm{loc}}⟩$. (c) Joint density $\mathrm{P}(I,E^{\prime })$ of IPR, $I$ and the energy, $E^{\prime }=E/{\epsilon }_{\beta }$, rescaled to the bandwidth ${\epsilon }_{\beta }$ for $\gamma =0.7$. The colorbar indicates the values of joint density in ${\log }_N$ scale, and $(-E_{\mathrm{G}}^{\prime },E_{\mathrm{G}}^{\prime })$ is the rescaled energy band for coexistent states. Numerical results are for $N=8192$ and 128 realizations.

Figure 2

(a) IPR, (b) fractal dimension $D_2$, (c) mean level-spacing ratio $r$ for $N=8192$ in the $\gamma \text{−}{E}^{\prime }$ plane. Solid black lines indicate $(-E_{\mathrm{G}}^{\prime },E_{\mathrm{G}}^{\prime })$, the energy band of localized states.

Figure 3

Ensemble-averaged power spectrum vs dimensionless frequency for $\gamma =0.4$ in a randomly chosen energy window with width $\mathrm{\Delta }\mathcal{E}=(N/8)$. Dashed (dotted) lines show ${\omega }^{-1}$ (${\omega }^{-2}$) fits, and the star denotes the critical frequency.

Figure 4

Left: threshold-filtered covariance matrix $\tilde{M}$, Eq. (8), for $N=1024$, $\gamma =0.5$, and $\delta M=0.5$ The red dashed lines denote spatial blocks of the form ${\mathrm{\Delta }}_l$ [Eq. (6)] with $\delta \zeta =0.1$ and $l=0,1,\dots ,5$, while the cyan lines show $N^{\gamma }$ spatial sub-blocks in the last block. Right: energies of spatially ordered eigenstates. The blue dashed vertical lines denote $(-E_{\mathrm{G}}^{\prime },E_{\mathrm{G}}^{\prime })$, i.e., energy bound of localized states.