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 ß 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.

Nonergodic extended states in the ß ensemble.

Matrix models showing a chaotic-integrable transition in the spectral statistics are important for understanding many-body localization (MBL) in physical systems. One such example is the ß ensemble, known for its structural simplicity. However, eigenvector properties of the ß ensemble remain largely unexplored, despite energy level correlations being thoroughly studied. In this work we numerically study the eigenvector properties of the ß ensemble and find that the Anderson transition occurs at γ=1 and ergodicity breaks down at γ=0 if we express the repulsion parameter as ß=N^{-γ}. Thus other than the Rosenzweig-Porter ensemble (RPE), the ß ensemble is another example where nonergodic extended (NEE) states are observed over a finite interval of parameter values (0<γ<1). We find that the chaotic-integrable transition coincides with the breaking of ergodicity in the ß ensemble but with the localization transition in the RPE or the 1D disordered spin-1/2 Heisenberg model. As a result, the dynamical timescales in the NEE regime of the ß ensemble behave differently than the latter models.

Transport in deformed centrosymmetric networks.

Centrosymmetry often mediates perfect state transfer (PST) in various complex systems ranging from quantum wires to photosynthetic networks. We introduce the deformed centrosymmetric ensemble (DCE) of random matrices H(λ)≡H_{+}+λH_{-}, where H_{+} is centrosymmetric while H_{-} is skew-centrosymmetric. The relative strength of the H_{±} prompts the system size scaling of the control parameter as λ=N^{-γ/2}. We propose two quantities, P and C, quantifying centro and skewcentrosymmetry, respectively, exhibiting second-order phase transitions at γ_{P}≡1 and γ_{C}≡-1. In addition, DCE posses an ergodic transition at γ_{E}≡0. Thus equipped with a precise control of the extent of centrosymmetry in DCE, we study the manifestation of γ on the transport properties of complex networks. We propose that such random networks can be constructed using the eigenvectors of H(λ) and establish that the maximum transfer fidelity F_{T} is equivalent to the degree of centrosymmetry P.