Dirac fermions on wires confined to the graphene Möbius strip

We investigate the effects of the curved geometry on a massless relativistic electron constrained to a graphene strip with a M\"obius strip shape. The anisotropic and parity violating geometry of the M\"obius band produces a geometric potential that inherits these features. By considering wires along the strip width and the strip length, we find exact solutions for the Dirac equation and the effects of the geometric potential on the electron were explored. In both cases, the geometric potential yields to a geometric phase on the wave function. Along the strip width, the density of states depends on the direction chosen for the wire, a consequence of the lack of axial symmetry. Moreover, the breaking of the parity symmetry enables the electronic states to be concentrated on the inner or on the outer portion of the strip. For wires along the strip length, the nontrivial topology influences the eigenfunctions by modifying their periodicity. It turns out that the ground state has a period of $4\ensuremath{\pi}$ whereas the first excited state is a $2\ensuremath{\pi}$ periodic function. Moreover, we found that the energy levels are half-integer multiples of the energy of the ground state.