Ground States of Nonlinear Fermionic Systems: From Power-Law Interactions to Logarithmic Sobolev Inequality

We consider ground states of a two-component logarithmic fermionic system in ${\mathbb{R}}^d$, where $d\ge 1$ is arbitrary. We prove that up to translations and scalings, ground states of the logarithmic system are the $L^{\mathrm{\infty }}$-limits of ground states for a two-component $2p-1$ power-law fermionic system as $p\searrow 1$. As a byproduct, we also establish a sharp logarithmic Sobolev inequality for orthonormal functions in ${\mathbb{R}}^d$, whose optimizers are, up to scalings, the minimizers of a constraint variational problem associated with the logarithmic system.