 Particle-hole symmetry of a half-filled bipartite band

A nearest-neighbour tight-binding model on a bipartite lattice, with identical on-site energy ${ϵ}_0$, has paired single-particle energies ${ϵ}_0\pm x$. At half filling, $f({ϵ}_0+x)+f({ϵ}_0-x)=1$ for the Fermi-Dirac distribution with $\mu ={ϵ}_0$. Thus its fixed-density chemical potential equals ${ϵ}_0$ at all positive temperatures within the isolated-band model. An anisotropic rectangular nearest-neighbour band retains this property; additional same-sublattice hopping or unequal on-site energies can remove it.