Kitaev's honeycomb model on a buckyball

We study the effect of disclinations in the Kitaev's honeycomb model [1] by examining the effective tight binding hamiltonian of Majorana fermions on Buckminsterfullerene [2]. Disclinations are realized by the 12 pentagons which shape the buckyball. We found that the ground state of the system with isotropic nearest neighbor coupling $t_1$, corresponds to a uniform flux sector of the $Z_2$ gauge field, where hexagons are flux free and pentagons have the same fluxes. Inclusion of second neighbor couplings $t_2$, preserve the projective symmetries of the truncated icosahedron as long as fluxes through all plaquettes (triangles, pentagons, and hexagons) related by symmetries are the same. For $t_1/t_2$ smaller than $1/2$, the local density of states reorganizes suggesting that the zero energy Majorana modes localize at the disclinations. The robustness of this quantum state against noise is examined. \\[4pt] [1] A. Kitaev, Ann. Phys. 321, 2 (2006).\\[0pt] [2] Kroto, Harold W., et al., Nature 318.6042 (1985).