Interaction-Induced Gradients Across a Confined Fermion Lattice

An imposed chemical potential gradient $A_{\uparrow }=d{\mu }_{\uparrow }\mathrm{/}dx$ on a single fermionic species (“spin up”) directly produces a gradient in the density $d{\rho }_{\uparrow }\mathrm{/}dx$ across a lattice. We study here the induced density inhomogeneity $d{\rho }_{\downarrow }\mathrm{/}dx$ 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 $d{\rho }_{\downarrow }\mathrm{/}dx$ acquired by the second species grows with $U$, while the magnitude of $d{\rho }_{\uparrow }\mathrm{/}dx$ remains relatively constant, that is, set only by $A_{\uparrow }$. For a given $A_{\uparrow }$, we find an interaction strength $U_{\ast }$ 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, $d{\rho }_{\downarrow }\mathrm{/}dx=-d{\rho }_{\uparrow }\mathrm{/}dx$. A simple particle-hole transformation determines the related effect in the case of attractive interactions.