Energy gap and averaged inversion symmetry of tight-binding electrons on generalized honeycomb lattice

We study the condition to open a finite gap in tight-binding electrons on an extended honeycomb lattice with the next-nearest-neighbor transfer integrals t(2a), t(2b), t(2c), t(2d), t(2e), and t(2f), where t(2a), t(2b), and t(2c) are transfer integrals between the sublattice A and t(2d), t(2e), and t(2f) are transfer integrals between the sublattice B. If the system has the inversion symmetry in this model, i.e., the sublattices A and B have the same on-site potential (epsilon(A) = epsilon(B)), the gap is zero. We find that although the finite gap is generally opened by inversion-symmetry breaking, the gap remains zero if the averaged inversion symmetry, which is defined as the sum of the transfer integrals and the on-site potentials of the sublattices are the same (t(2a) + t(2b) + t(2c) + epsilon(A) = t(2d) + t(2e) + t(2f) + epsilon(B)), is conserved.