Anomalies in a slightly doped insulator with strong particle-hole asymmetry and a narrow gap: The case of SmB6

${\mathrm{SmB}}₆$, known to be a Kondo insulator, has received intense scrutiny in recent years due to its paradoxical experimental signatures: While some quantities show an insulating behavior, others point to a metallic state. This has led to the conjecture that ${\mathrm{SmB}}₆$ hosts nontrivial excitations within its bulk gap, and has spawned several theories to that effect. In principle, there exists an alternative possibility: The system is a metal but unusually with both metal- and insulatorlike properties. Inspired by this possibility, I consider a minimal model of a Kondo insulator---a flat band hybridized with a parabolic band---that is slightly electron doped, i.e., the chemical potential is in the conduction band but close to the band edge. I show that, at the phenomenological level, the dc conductivity, ac conductivity, specific heat, and quantum oscillations within this model exhibit unusual behaviors that are, surprisingly, qualitatively consistent with those observed experimentally in ${\mathrm{SmB}}₆$. The rapid change of band curvature around the chemical potential arising from the strong particle-hole asymmetry and the narrow gap in the model, a feature not usually encountered in the textbook cases of metals or insulators, is at the heart of the unusual behaviors.