Anderson transition and mobility edges on hyperbolic lattices with randomly connected boundaries

Hyperbolic lattices, formed by tessellating the hyperbolic plane with regular polygons, exhibit a diverse range of exotic physical phenomena beyond conventional Euclidean lattices. Here, we investigate the impact of disorder on hyperbolic lattices and reveal that the Anderson localization occurs at strong disorder strength, accompanied by the presence of mobility edges. Taking the hyperbolic $\{p,q\}=\{3,8\}$ and $\{p,q\}=\{4,8\}$ lattices as examples, we employ finite-size scaling of both spectral statistics and the inverse participation ratio to pinpoint the transition point and critical exponents. Our findings indicate that the transition points tend to increase with larger values of $\{p,q\}$ or curvature. In the limiting case of $\{\infty, q\}$, we further determine its Anderson transition using the cavity method, drawing parallels with the random regular graph. Our work lays the cornerstone for a comprehensive understanding of Anderson transition and mobility edges on hyperbolic lattices.