This paper studies how a specific kind of randomness, called mass disorder, changes the electronic behavior of particles on negatively curved, or hyperbolic, lattices. The authors focus on three examples of plaquette-centered hyperbolic lattices, labeled by their Schläfli symbols {10,3}, {8,3}, and {8,4}. In the clean (no-disorder) limit these three lattices show very different low-energy behavior: {10,3} has a vanishing density of states at zero energy (a “Dirac liquid”), {8,3} has a finite density of states (a “Fermi liquid”), and {8,4} has a diverging density of states (a “flat band”). The paper asks how these phases respond when mass disorder is added.

Density of states is a simple measure of how many electronic states are available at a given energy. The lattices studied here are bipartite (they can be split into two complementary sublattices). The authors implement mass disorder by assigning a random on-site potential to each unit cell, with the two sublattice sites in that cell carrying equal magnitude but opposite sign values. These random values are drawn from a box distribution and vary independently from cell to cell. Because of the bipartite structure, this disorder breaks some symmetries of the clean system but preserves time-reversal symmetry; the symmetry class of the problem is the Wigner–Dyson orthogonal class AI once disorder is present.

The calculations use a nearest-neighbor tight-binding model and the kernel polynomial method (KPM), a numerical technique that approximates densities of states using Chebyshev polynomials. The authors compute two related quantities. The average density of states (ADOS) is the usual disorder-averaged count of states. The typical density of states (TDOS) is a geometric-mean style measure that is sensitive to localization: TDOS tends to zero when states become localized even if ADOS remains finite. In the numerical work the authors average ADOS over 10 disorder realizations, TDOS over 60 realizations, and use stochastic traces with 12 random vectors when computing moments. They also checked convergence with respect to the number of Chebyshev moments.