Semi-fractality and localization on a chiral Cayley tree

This paper investigates the statistical properties of quantum wave functions on an infinite Cayley tree with random hopping amplitudes. Specifically, the authors seek to understand how the absence of on-site disorder and the presence of chiral symmetry (due to the bipartite nature of the tree) influence the localization and fractal properties of eigenstates at the band center.

The authors employ the cavity method, solved via population dynamics, to analyze the distribution of the local density of states (LDoS) in the thermodynamic limit. By calculating the LDoS distribution, they derive the multifractal spectrum of the wave functions and identify a regime they term "semi-fractality." They further validate these findings using exact diagonalization on finite random regular graphs and examine the rank-ordered hierarchy of eigenstate weights to interpret the physical nature of these states.

The study reveals that the LDoS distribution develops power-law tails, which lead to a non-analytic behavior in the fractal dimensions of the wave functions. In the semi-fractal phase, the support of the wave function is extensive (fractal dimension D1 = 1), yet higher-order moments show multifractal scaling (Dq < 1 for q > 1). The authors show that the symmetry properties of the LDoS distribution are not universal but vary continuously with the disorder parameter. As this parameter is tuned, the system undergoes a transition from this semi-fractal regime to a localized phase. At the critical point, the system reaches a "semi-localized" state where the fractal dimension jumps abruptly from 1 to 0 at q = 1.

This work provides a theoretical framework for understanding non-ergodic, extended phases that fall outside the standard classification of Anderson localization. By linking chiral symmetry to semi-fractality, the authors offer a microscopic explanation for similar phenomena recently observed in quantum computing experiments and random matrix ensembles, suggesting that such states are a robust consequence of specific symmetry classes in high-dimensional systems.