Ground state energy of the dilute spin-polarized Fermi gas: Upper bound via cluster expansion

Abstract

We prove an upper bound on the ground state energy of the dilute spin-polarized Fermi gas capturing the leading correction to the kinetic energy resulting from repulsive interactions. One of the main ingredients in the proof is a rigorous implementation of the fermionic cluster expansion of Gaudin, Gillespie and Ripka (Nucl. Phys. A, 176.2 (1971), pp. 237-260).

Almost optimal upper bound for the ground state energy of a dilute Fermi gas via cluster expansion

“…A central ingredient in the proof is to prove a rigorous version of a fermionic cluster expansion adapted from [GGR71]. This is analogous to what is done in [LS23] for spin-polarized fermions.…”

“…This term may be understood as coming from the energy of a pair of opposite-spin fermions times the number of such pairs. The energy correction arising from interactions between fermions of the same spin is of order a 3 p ρ 8/3 , where a p denotes the p-wave scattering length (see [LS23]) and so much smaller.…”

“…The paper is structured as follows. In Section 2 we give some preliminary computations and recall some properties of the scattering functions and Fermi polyhedron from [LS23;LY01]. Next, in Section 3 we give the calculation of a fermionic cluster expansion adapted from [GGR71].…”

“…In the smaller boxes we will need to use Dirichlet boundary conditions, however in Section 5.4 we will construct trial states with Dirichlet boundary conditions out of trial states with periodic boundary conditions. (See also [LS23].) Thus, we will use periodic boundary conditions in the box Λ…”

“…The polyhedron P σ approximates the unit ball in the sense that any point on the surface ∂P σ has radial coordinate 1 + O(s −1 σ ). The polyhedron P σ is moreover symmetric under the maps (k 1 , k 2 , k 3 ) → (±k 1 , ±k 2 , ±k 3 ) and "almost symmetric" under the maps (k 1 , k 2 , k 3 ) → (k a , k b , k c ) for (a, b, c) = (1, 2, 3), see [LS23,Lemma 2.11].…”