Localization Transition on Random Graphs with Chiral and Bogoliubov-de Gennes Symmetry Classes

TL;DR

This work investigates Anderson localization on graphs with chiral (BDI) and BdG symmetry using the cavity method to analyze DOS, LSD, and fractal dimensions. For chiral RRGs, self-consistent equations for Green functions yield the DOS and reveal zero modes when $|N-M|>0$, with $D_2(E,W)$ indicating a smoother transition than in ordinary RRG. In BdG ensembles on RRG, the BdG-Kesten-McKay distribution with a finite gap $ riangle_0$ characterizes the DOS, and diagonal disorder that conserves symmetry preserves the gap and lowers $D_2$, whereas symmetry-breaking disorder erodes the gap and pushes the system toward RRG-like behavior. Overall, the results quantify how symmetry and pairing affect localization transitions on random graphs and provide a framework for comparing critical properties across symmetry classes.

Abstract

We studied single-particle Anderson localization in ensembles of graphs that correspond to chiral and Bogoliubov-de Gennes (BdG) symmetry classes. For a random biregular bipartite graph with chiral symmetry, the density of states was found using the cavity approach. Calculating the fractal dimension shows the effects of disordered zero modes. For Bogoliubov-de Gennes ensembles with an underlying random regular graph (RRG), the density of states was calculated both numerically and analytically. The ensembles BdG-RRG with symmetry-conserving diagonal disorder in the delocalized phase have a smaller fractal dimension compared to the usual RRG.

Figures (2)

• Figure 1: Numerical calculation of (a) density of states and (b) level spacing distribution for bipartite biregular graph with partition $N = 256$, $M = 768$, $d_N = 12$, $d_M = 4$. The black dashed line shows DOS calculated using Eq. (\ref{['eq:DOS_ch']}). The purple and black dots indicate LSD at localized and delocalized regimes. Blue and red lines show Poisson and Wigner distribution. (c) Comparison of average fractal dimensions $D_2$ for different ensembles depending on diagonal disorder $W$. Panels (d)-(e) shows dependence of fractal dimension $D_2$ on energy $E$ and diagonal disorder $W$.
• Figure 2: The green histograms shows numerical calculation of density of states for Hamiltonian described by Eq. (\ref{['eq:BdG+A']}) without on-site disorder $\Delta_0 = 1$ (a) $d=3$, (b) $d=8$, the black dashed lines show DOS calculated by Eq. (\ref{['eq:BdG-KM']}). (c) Level spacing distribution, the purple and black dots indicate LSD at localized and delocalized regimes, blue and red lines show Poisson and Wigner distribution. (d) Comparison of average fractal dimensions $D_2$ for different ensembles depending on diagonal disorder $W$. Dependence of fractal dimension on energy $E$ and symmetry (c) conserve and (d) non-conserve diagonal disorder $W$.