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

The stability of spinless Fermions with power-law hopping H_ij∝|i−j|^−α is investigated. It is shown that at precisely α_c=2, the dispersive inflection point coalesces with the band minimum and the charge carriers exhibit a transition into negative effective-mass regime, m_α^∗<0 characterized by retarded transport in the presence of an electric field. Moreover, bands with α≤2 must be accompanied by counter carriers with m_α^∗>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 (α=1). The analysis is generalized to regular lattices in dimensions d=1, 2, and 3.