Finite-size energy of non-interacting Fermi gases

TL;DR

This paper rigorously derives the finite-size energy corrections for non-interacting Fermi gases on a half line, revealing dependence on the thermodynamic limit details and scattering phase shifts, refining previous heuristic claims.

Contribution

It provides a precise asymptotic analysis of the finite-size energy for non-interacting Fermi gases, including subdominant terms up to order 1/L, and clarifies their dependence on system specifics.

Findings

Finite-size energy asymptotics are derived up to order 1/L.

Finite-size energy depends on thermodynamic limit details.

Finite-size energy includes a linear term in the scattering phase shift.

Abstract

We prove the asymptotics of the difference of the ground-state energies of two non-interacting $N$-particle Fermi gases on the half line of length $L$ in the thermodynamic limit up to order $1/L$. We are particularly interested in subdominant terms proportional to $1/L$, called finite-size energy. In the nineties Affleck and co-authors [Aff97, ZA97, AL94] claimed that the finite-size energy equals the decay exponent occuring in Anderson's orthogonality catastrophe. It turns out that the finite-size energy depends on the details of the thermodynamic limit and typically also includes a linear term in the scattering phase shift.