Interaction-Induced Gradients Across a Confined Fermion Lattice

An imposed chemical potential gradient $A_\uparrow=d\mu_\uparrow/dx$ on a single fermionic species ("spin up") directly produces a gradient in the density $d\rho_\uparrow/dx$ across a lattice. We study here the induced density inhomogeneity $d\rho_\downarrow/dx$ 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 $d\rho_\downarrow/dx$ acquired by the second species grows with $U$, while the magnitude of $d\rho_\uparrow/dx$ 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, $d\rho_\downarrow/dx = -d\rho_\uparrow/dx$. A simple particle-hole transformation determines the related effect in the case of attractive interactions.