Interaction-Induced Gradients Across a Confined Fermion Lattice

An imposed chemical potential gradient $A_uparrow=dmu_uparrow/dx$ on a single fermionic species (spin up) directly produces a gradient in the density $drho_uparrow/dx$ across a lattice. We study here the induced density inhomogeneity $drho_downarrow/dx$ in the second fermionic species (spin down) which results from fermionic interactions $U$, even in the absence of a chemical potential gradient $A_downarrow=0$ on that species. The magnitude of $drho_downarrow/dx$ acquired by the second species grows with $U$, while the magnitude of $drho_uparrow/dx$ remains relatively constant, that is, set only by $A_uparrow$. For a given $A_uparrow$, we find an interaction strength $U_*$ above which the two density gradients are equal in magnitude. We also evaluate the spin-spin correlations and show that, as expected, antiferromagnetism is most dominant at locations where the local density is half-filled. The spin polarization induced by sufficiently large gradients, in combination with $U$, drives ferromagnetic behavior. In the case of repulsive interactions, $drho_downarrow/dx = -drho_uparrow/dx$. A simple particle-hole transformation determines the related effect in the case of attractive interactions.