Negative effective-mass transition and anomalous transport in power-law hopping bands

Abstract

The stability of spinless Fermions with power-law hopping ${H}_{ij}\ensuremath{\propto}{|i\ensuremath{-}j|}^{\ensuremath{-}\ensuremath{\alpha}}$ is investigated. It is shown that at precisely ${\ensuremath{\alpha}}_{c}=2$, the dispersive inflection point coalesces with the band minimum and the charge carriers exhibit a transition into negative effective-mass regime, ${m}_{\ensuremath{\alpha}}^{\ensuremath{\ast}}<0$ characterized by retarded transport in the presence of an electric field. Moreover, bands with $\ensuremath{\alpha}\ensuremath{\le}2$ must be accompanied by counter carriers with ${m}_{\ensuremath{\alpha}}^{\ensuremath{\ast}}>0$, having a positive band curvature, thus stabilizing the system in order to maintain equilibrium conditions and a proper electrical response. We further examine the semiclassical transport and response properties, finding an infrared-divergent conductivity for $1/r$ hopping $(\ensuremath{\alpha}=1)$. The analysis is generalized to regular lattices in dimensions $d=1$, 2, and 3.