 Orbital phase and the sign of one-dimensional hopping

In an orthogonal one-orbital nearest-neighbor tight-binding model, the hopping integral is $t=⟨{\varphi }_j|H|{\varphi }_{j+1}⟩$ and the dispersion is the displayed cosine. Using the same orbital-phase convention at every site, equal s-orbital coefficients usually make a bonding state at the zone center, so $t<0$ and $k=0$ is a band minimum. For $p_x$ orbitals along the chain, with every positive lobe pointing right, equal site coefficients bring opposite-sign lobes together between atoms. This is antibonding; alternating coefficients are bonding, giving the usual $t_{\sigma }>0$ and a band maximum at zero. Changing conventions requires transforming the Bloch basis consistently. The physical statement concerns the curvature and orbital character of a specified Bloch state, not an isolated hopping sign divorced from its basis.