Spectral statistics, finite-size scaling and multifractal analysis of quasiperiodic chain with p-wave pairing

We study the spectral and wavefunction properties of a one-dimensional incommensurate system with p-wave pairing and unveil that the system demonstrates a series of particular properties in its ciritical region. By studying the spectral statistics, we show that the bandwidth distribution and level spacing distribution in the critical region follow inverse power laws, which however break down in the extended and localized regions. By performing a finite-size scaling analysis, we can obtain some critical exponents of the system and find these exponents fulfilling a hyperscaling law in the whole critical region. We also carry out a multifractal analysis on system's wavefuntions by using a box-counting method and unveil the wavefuntions displaying different behaviors in the critical, extended and localized regions.Comment: 9 pages, 8 figure

Localization of eigenvectors in random graphs

Using exact numerical diagonalization, we investigate localization in two classes of random matrices corresponding to random graphs. The first class comprises the adjacency matrices of Erdős-Rényi (ER) random graphs. The second one corresponds to random cubic graphs, with Gaussian random variables on the diagonal. We establish the position of the mobility edge, applying the finite-size analysis of the inverse participation ratio. The fraction of localized states is rather small on the ER graphs and decreases when the average degree increases. On the contrary, on cubic graphs the fraction of localized states is large and tends to 1 when the strength of the disorder increases, implying that for sufficiently strong disorder all states are localized. The distribution of the inverse participation ratio in localized phase has finite width when the system size tends to infinity and exhibits complicated multi-peak structure. We also confirm that the statistics of level spacings is Poissonian in the localized regime, while for extended states it corresponds to the Gaussian orthogonal ensemble