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).

Figures (9)

• Figure 1.1: Energies of dilute Fermi gasses. The curves and points labelled HC are for a Hard Core interaction of radius $a$. The curves and points labelled SC are for a Soft Core interaction of radius $2a$ and strength $V_0$ chosen so that it's scattering length is $a$. (Meaning $v(x) = V_0 \chi_{|x|\leq 2a}$, $\chi$ being the characteristic function.) The points (labelled QMC) are Quantum Monte Carlo simulations from Bertaina.Tarallo.ea.2023. The curves include the (conjectured) corrections up to the labelled order in $k_F = (6\pi^2\rho)^{1/3}$.
• Figure 3.1: A diagram $(\pi, G) \in \mathcal{D}_8^3$ decomposed into linked components. The dashed lines denote $g$-edges and the arrows $(i,j)$ denote that $\pi(i) = j$. Vertices labelled with $*$ denote external vertices.
• Figure 3.2: A linked diagram $(\pi, G)\in\mathcal{L}_{11}$ decomposed into clusters $G_1, G_2, G_3$. Dashed lines denote $g$-edges, and arrows $(i,j)$ denote that $\pi(i) = j$.
• Figure 3.3: An anchored tree $\tau$ (arrows) and trees $T_1,\ldots,T_6$ (dashed lines).
• Figure A.1: $g$-graphs of small diagrams of different types. For each diagram only the graph $G$ is drawn. The relevant diagrams come with permutations $\pi$ such that the diagrams are linked.

Theorems & Definitions (106)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Remark 1.4
• Proposition 1.5
• Remark 1.6: Numerical investigation
• Remark 1.7
• Remark 1.8
• Definition 1.9
• Theorem 1.10: Two dimensions