Chiral symmetry and the orbital susceptibility of disordered honeycomb flakes

The orbital susceptibility of a small disordered grain is dominated by rare samples in which an occupied and an empty level nearly touch. In a chaotic grain such near-degeneracies are suppressed by level repulsion and must also coincide with the chemical potential, so the mean susceptibility stays finite. In a neutral honeycomb flake, chiral (sublattice) symmetry removes the second condition: It places the chemical potential in the middle of the smallest gap, in every disorder realization. Here we show that the mean orbital susceptibility then grows as $\ln (1\mathrm{/}T)$ at low temperature $T$ if disorder can close this gap while a magnetic field still mixes the two levels, and that the hopping geometry decides whether it can: Positive nearest-neighbor hopping on a compact hexagonal flake keeps the gap open, third-neighbor hopping allows it to close. In a chiral random-matrix model with $n$ level pairs the coefficient of the logarithm is $C_n=\sqrt{2\mathrm{/}n}\mathrm{\Gamma }[(n+1)\mathrm{/}2]\mathrm{/}\mathrm{\Gamma }(n\mathrm{/}2)$, close to unity, and a detuned chemical potential cuts the logarithm off.