The Ground State Energy of a Mean-Field Fermi Gas in Two Dimensions

TL;DR

This paper proves a precise formula for the ground-state energy of a two-dimensional mean-field Fermi gas in the regime $\hbar=N^{-1/2}$, $\lambda=N^{-1}$, with potentials whose Fourier transform satisfies $\hat{V}(k)|k|$ summability (and a stronger variant allowing Coulomb-like interactions). Adapting a patch-based bosonization to 2D, the authors construct roughly bosonic pair operators on a discretized Fermi surface, apply two pseudo-Bogoliubov transformations, and linearize the kinetic energy to extract the leading correlation energy. They prove a lower and an upper bound matching up to $o(N^{-1/2})$ corrections, yielding $E_{GS}=E_{FS}+E^{\mathrm{RPA}}+o(N^{-1/2})$ with an explicit RPA term $E^{\mathrm{RPA}}$, and they provide detailed bounds on error terms via three-scale decompositions and number-theoretic lattice estimates. The work extends rigorous mean-field correlation-energy results from 3D to 2D, addressing the unique 2D challenges posed by a relative coupling of order one and refined control of low-energy excitations.

Abstract

We rigorously establish a formula for the correlation energy of a two-dimensional Fermi gas in the mean-field regime for potentials whose Fourier transform $\hat{V}$ satisfies $\hat{V}(\cdot) | \cdot | \in \ell^1$. Further, we establish the analogous upper bound for $\hat{V}(\cdot)^2 | \cdot |^{1 + \varepsilon} \in \ell^1$, which includes the Coulomb potential $\hat{V}(k) \sim |k|^{-2}$. The proof is based on an approximate bosonization using slowly growing patches around the Fermi surface. In contrast to recent proofs in the three-dimensional case, we need a refined analysis of low-energy excitations, as they are less numerous, but carry larger contributions.

Figures (6)

• Figure 1: Example for a patch decomposition around the Northern Hemisphere of the Fermi surface. Here, half of all $M=14$ patches $B_{\alpha}$ are shown. The patches have thickness $2R$ and are separated by corridors of size $2R$, where $R$ grows slowly with increasing $N$.
• Figure 2: Left: Depiction of the set $Y_k$ and geometric considerations for determining $s_{\min}$. Right: We decompose the set $Y_k$ into planes $Y_{k,m}$ parallel to $k$. For a point $p \in Y_{k,m}$, the pair excitation energy is then $\lambda_{k,p} = \hbar^2 |k| s(p)$, which is conveniently lower-bounded for $|m| \neq m^*$ using $s(p) \geq s_{\min}$.
• Figure 3: Left: For $|k| \approx 2 k_{\textnormal{F}}$, the lune $L_k$ is almost identical to the shifted Fermi ball $B_{\textnormal{F}}+k$; only a small cap around $s=0$ is cut away. Right: The intersection length of $B_{\textnormal{F}}^c \cap (B_{\textnormal{F}} + k)$ with a vertical line at fixed $s< s'$ is given by $y_1(s)+y_2(s)$, which starts off at 0 for $s=0$ and then grows rapidly. It therefore has to be estimated carefully, using the properties of $y_1(s)$, $y_2(s)$ and $y_3(s)$.
• Figure 4: Left: The angle $\alpha$ is given by $\sin(\alpha) = \frac{|k|}{2 k_{\textnormal{F}}}$. For $|k| < k_{\textnormal{F}}$, we have $\alpha \le \frac{\pi}{6}$. Right: The opening angle of the lune $L_k$ is $2 \alpha$ and for $|k| < k_{\textnormal{F}}$, the tip $L^{\textrm{Tip}}_k$ is linearly approximated by a triangle.
• Figure 5: Left: For $1 \le |k| < 2 k_{\textnormal{F}}$, the intersection of the two annuli $\mathcal{A} \cap (\mathcal{A}+k)$ amounts to two areas, each bordered by four arcs between four points $P_1$, $P_2$, $P_3$ and $P_4$. Right: The coordinates $x_j$ and $y_j$ are defined by putting the intersection points $P_j$ into the coordinate system spanned by $\hat{k}$ and $\hat{k}^\perp$

Theorems & Definitions (55)

• Theorem 1.1: Upper and lower bound on the ground state energy
• proof
• Definition 3.1
• Lemma 3.2: Onsager bound
• proof
• Definition 3.3
• Lemma 3.4: Bound on $\mathcal{G}_\delta$ and gapped conversion
• proof
• Lemma 3.5: Naive bounds on $b$ and $d$
• proof