On the Correlation Energy of Interacting Fermionic Systems in the Mean-Field Regime

Abstract

We consider a system of $N\gg 1$ interacting fermionic particles in three dimensions, confined in a periodic box of volume 1, in the mean-field scaling. We assume that the interaction potential is bounded and small enough. We prove upper and lower bounds for the correlation energy, which are optimal in their N-dependence. Moreover, we compute the correlation energy at leading order in the interaction potential, recovering the prediction of second order perturbation theory. The proof is based on the combination of methods recently introduced for the study of fermionic many-body quantum dynamics together with a rigorous version of second-order perturbation theory, developed in the context of non-relativistic QED.

Appendices

A Proof of Lemma 4.7

We start by writing:

$$\begin{aligned}&\sum _{k} \Vert b_{k+p} c_{k} \varphi \Vert = \sum _{k} \frac{1}{\sqrt{e(k+p) + e(k)}} \sqrt{e(k+p) + e(k)} \Vert b_{k+p} c_{k} \varphi \Vert \nonumber \\&\quad \le \left( \sum _{k:\, k+p\notin \mathcal {B}_{\mu },\, k\in \mathcal {B}_{\mu }} \frac{1}{e(k+p) + e(k)}\right) ^{\frac{1}{2}} \left( \sum _{k} (e(k+p) + e(k)) \Vert b_{k+p} c_{k} \Vert ^{2}\right) ^{\frac{1}{2}}\nonumber \\&\quad \le N^{\frac{1}{2}} I_{\mu }(p)^{\frac{1}{2}} \Vert \mathbb {H}_{0}^{\frac{1}{2}} \varphi \Vert , \end{aligned}$$

(A.1)

where we defined:

$$\begin{aligned} I_{\mu }(p) = \frac{1}{N} \sum _{k:\, k+p\notin \mathcal {B}_{\mu },\, k\in \mathcal {B}_{\mu }} \frac{1}{e(k+p) + e(k)}. \end{aligned}$$

(A.2)

We claim that, for $|p| \le N^{\zeta }$ and $\zeta >0$ to be determined below:

$$\begin{aligned} I_{\mu }(p) \le C, \end{aligned}$$

(A.3)

which would conclude the proof of Lemma 4.7. To prove this, we shall split the sum in three terms:

$$\begin{aligned} I_{\mu }(p) = I^{\text {A}}_{\mu }(p) + I^{\text {B}}_{\mu }(p) + I^{\text {C}}_{\mu }(p) \end{aligned}$$

(A.4)

with, for A large enough (independent of |p| and N), and with $\gamma = 131/208$ (the motivation for this choice will be clear later on):

$$\begin{aligned} I^{\text {A}}_{\mu }(p)= & {} \frac{1}{N} \sum _{\begin{array}{c} k:\, k+p\notin \mathcal {B}_{\mu },\, k\in \mathcal {B}_{\mu } \\ e(k) + e(k+p) \le A|p|^{2}\varepsilon ^{2} \end{array}} \frac{1}{e(k+p) + e(k)}\nonumber \\ I^{\text {B}}_{\mu }(p)= & {} \frac{1}{N} \sum _{\begin{array}{c} k:\, k+p\notin \mathcal {B}_{\mu },\, k\in \mathcal {B}_{\mu } \\ A|p|^{2}\varepsilon ^{2}< e(k) + e(k+p) \le \varepsilon ^{2 - \gamma } \end{array}} \frac{1}{e(k+p) + e(k)}\nonumber \\ I^{\text {C}}_{\mu }(p)= & {} \frac{1}{N} \sum _{\begin{array}{c} k:\, k+p\notin \mathcal {B}_{\mu },\, k\in \mathcal {B}_{\mu } \\ \varepsilon ^{2-\gamma } < e(k) + e(k+p) \end{array}} \frac{1}{e(k+p) + e(k)}. \end{aligned}$$

(A.5)

We shall study the three terms separately. As we shall see, the main contribution to $I_{\mu }(p)$ will come from $I^{\text {C}}_{\mu }(p)$. The other two terms will be proven to be o(1) as $\varepsilon \rightarrow 0$.

${\underline{\hbox {Term }A}.}$ Using that $e(k+p) + e(k) \ge \varepsilon ^{2}$, we estimate:

$$\begin{aligned} I^{\text {A}}_{\mu }(p) \le \frac{1}{N \varepsilon ^{2}} |\mathcal {R}_{p}| \end{aligned}$$

(A.6)

where $\mathcal {R}_{p}$ is the set:

$$\begin{aligned} \mathcal {R}_{p} = \big \{ k\in \mathcal {B}_{\mu }\mid k+p\notin \mathcal {B}_{\mu },\; e(k+p) + e(k) \le A|p|^{2}\varepsilon ^{2} \big \}. \end{aligned}$$

(A.7)

Recall that $e(k) = |\varepsilon ^{2} |k|^{2} - \mu |$. The constraint $e(k) \le A|p|^{2}\varepsilon ^{2}$ is equivalent to $||k|^{2} - N^{\frac{2}{3}} \mu | \le A|p|^{2}$, that is $||k| - N^{\frac{1}{3}} \sqrt{\mu }|\le C|p|^{2}N^{-\frac{1}{3}}$: the lattice points in $\mathcal {R}_{p}$ are at a distance at most $C |p|^{2} N^{-\frac{1}{3}}$ from the Fermi surface. Also, the constraints $k\in \mathcal {B}_{\mu }$, $k+p\notin \mathcal {B}_{\mu }$ imply that:

$$\begin{aligned} e(k+p) + e(k) = \varepsilon ^{2} |k+p|^{2} - \mu + \mu - \varepsilon ^{2} |k|^{2} = \varepsilon ^{2}|p|^{2} + 2\varepsilon ^{2}k\cdot p. \end{aligned}$$

(A.8)

Let $\theta _{k}$ the relative angle of k and p, $k\cdot p = |k| |p| \cos \theta _{k}$. Since $|k| = O(N^{\frac{1}{3}})$, the constraint $e(k+p) + e(k) \le A|p|^{2}\varepsilon ^{2}$ together with Eq. (A.8) implies that

$$\begin{aligned} |\cos \theta _{k}| \le CA|p| \varepsilon . \end{aligned}$$

(A.9)

Hence, the set $\mathcal {R}_{p}$ has width bounded by CA|p| in the direction of p.

To count the lattice points in $\mathcal {R}_{p}$, we slice $\mathcal {R}_{p}$ with planes orthogonal to the vector p. Let $\{ e_{1}, e_{2}, e_{3} \}$ be the standard orthogonal basis for $(2\pi ) \mathbb {Z}^{3}$, with $|e_{i}| = 2\pi$. If p is parallel to one of the basis vectors, say $e_{1}$, the lattice points intersected by these planes form subsets of $(2\pi )\mathbb {Z}^{2}$. We write:

$$\begin{aligned} \mathcal {R}_{p}= & {} \bigcup _{n} \big \{ k\in \mathcal {R}_{p} \mid k\cdot e_{1} = (2\pi )^{2} n \big \} \nonumber \\\equiv & {} \bigcup _{n} \mathcal {R}_{p, n}. \end{aligned}$$

(A.10)

Since the width of $\mathcal {R}_{p}$ in the direction of p is bounded by CA|p|, to count the number of lattice points in $\mathcal {R}_{p}$ it will be enough to count the number of lattice points in a single slice $\mathcal {R}_{p, n}$. This number is bounded by the number of lattice points lying in the difference of two disks:

$$\begin{aligned} | \mathcal {R}_{p,n} | \le | \{ k\in (2\pi ) \mathbb {Z}^{2}\mid N^{\frac{1}{3}}\nu - C|p|^{2}N^{-\frac{1}{3}} \le |k| \le N^{\frac{1}{3}}\nu \} |, \end{aligned}$$

(A.11)

for some $\nu = \sqrt{\mu } - O(N^{-\frac{1}{3}})$. To estimate the right-hand side of Eq. (A.11) we proceed as follows. Let $\mathcal {C}\subset \mathbb {R}^{2}$ be a convex body with smooth boundary (e.g. a disk). Then, for any $\delta > 0$ and for $\gamma = 131/208$, [22]:

$$\begin{aligned} | \{ x\in \mathbb {Z}^{2}\mid x/R \in \mathcal {C} \} | = R^{2} \mu (\mathcal {C}) + E_{2}(R),\quad |E_{2}(R)| \le C_{\delta } R^{\gamma + \delta } \end{aligned}$$

(A.12)

where $\mu (\mathcal {C})$ is the area of $\mathcal {C}$. Therefore, thanks to Eq. (A.12), it is easy to estimate $|\mathcal {R}_{p,n}|$ as:

$$\begin{aligned} | \mathcal {R}_{p,n} | \le | \{ k\in (2\pi ) \mathbb {Z}^{2}\mid N^{\frac{1}{3}}\nu - C|p|^{2}N^{-\frac{1}{3}} \le |k| \le N^{\frac{1}{3}}\nu \} | \le K_{\delta } N^{\frac{1}{3} (\gamma + \delta )},\qquad \end{aligned}$$

(A.13)

where we used that $|p|\le N^{\zeta }$ with $\zeta$ small enough, such that $|p|^{2}\le N^{\frac{1}{3}(\gamma +\delta )}$. Hence, Eq. (A.13) implies that

$$\begin{aligned} I^{\text {A}}_{\mu }(p) \le \frac{1}{N^{\frac{1}{3}}}|\mathcal {R}_{p}| \le CA|p| K_{\delta } N^{\frac{1}{3}(\gamma + \delta ) - \frac{1}{3}}, \end{aligned}$$

(A.14)

which is o(1) for $|p| \le N^{\zeta }$ with $\zeta < \frac{1}{3} - \frac{1}{3}(\gamma + \delta )$.

Suppose now that p is not aligned with any basis vector $e_{1}, e_{2}, e_{3}$. We introduce a new orthogonal basis $\{ b_{1}, b_{2}, b_{3} \}$, with $b_{1} = (2\pi ) p/|p|$ and $|b_{i}| = 2\pi$. For instance, let:

$$\begin{aligned} p_{\perp } = (0, -p_{3}, p_{2}),\qquad p_{\perp }' = ( -p_{2}^{2} - p_{3}^{2}, p_{1} p_{2}, p_{1} p_{3}). \end{aligned}$$

(A.15)

Then, we set:

$$\begin{aligned} b_{1} = (2\pi ) \frac{p}{|p|},\qquad b_{2} = (2\pi ) \frac{p_{\perp }}{|p_{\perp }|},\qquad b_{3} = (2\pi ) \frac{p_{\perp }'}{|p_{\perp }'|}. \end{aligned}$$

(A.16)

For later use, notice that $|p_{\perp }'| = |p| |p_{\perp }| \le |p|^{2}$. We then slice the ribbon $\mathcal {R}_{p}$ with planes orthogonal to $b_{1}$:

$$\begin{aligned} \mathcal {R}_{p}= & {} \bigcup _{\alpha } \big \{ k\in \mathcal {R}_{p} \mid k\cdot b_{1} = (2\pi )^{2}\alpha \big \} \nonumber \\\equiv & {} \bigcup _{\alpha } \mathcal {R}_{p,\alpha }. \end{aligned}$$

(A.17)

Since $\alpha = \frac{1}{(2\pi )|p|}\sum _{i} k_{i} p_{i}$ with $k, p\in (2\pi )\mathbb {Z}^{3}$, the smallest increment of $\alpha$ is bounded below by $(\text {const.}) |p|^{-1}$. Also, since the width of $\mathcal {R}_{p}$ in the direction of p is bounded by CA|p|, the number of sets $\mathcal {R}_{p,\alpha }$ in the union in Eq. (A.17) is bounded by $CA|p|^{2}$. To bound the number of lattice points in $\mathcal {R}_{p}$ it is again enough to bound the number of lattice points in a given $\mathcal {R}_{p,\alpha }$.

In contrast to the previous case, the number of lattice points in $\mathcal {R}_{p,\alpha }$ is not a subset of $\mathbb {Z}^{2}$. However, it can be estimated with the number of lattice points in a suitable subset of $\mathbb {Z}^{2}$, proceeding as follows. Any lattice point belonging to $\mathcal {R}_{p,\alpha }$ can be written as $k = \alpha b_{1} + \beta b_{2} + \gamma b_{3}$, with

$$\begin{aligned} \alpha= & {} \frac{1}{(2\pi )^{2}} k\cdot b_{1} = \frac{1}{(2\pi )} \frac{k\cdot p}{|p|}\nonumber \\ \beta= & {} \frac{1}{(2\pi )^{2}} k\cdot b_{2} = \frac{1}{(2\pi )} \frac{k\cdot p_{\perp }}{|p_{\perp }|} \nonumber \\ \gamma= & {} \frac{1}{(2\pi )^{2}} k\cdot b_{3} = \frac{1}{(2\pi )} \frac{k\cdot p'_{\perp }}{|p'_{\perp }|}. \end{aligned}$$

(A.18)

Different points in $\mathcal {R}_{p,\alpha }$ correspond to different values of $\beta$ and $\gamma$, for $\alpha$ fixed. The coefficients $\beta$, $\gamma$ are not integers, in general. However, they become integer valued if properly rescaled:

$$\begin{aligned} n_{2} = \frac{1}{(2\pi )}|p_{\perp }| \beta \in \mathbb {Z},\qquad n_{3} = \frac{1}{(2\pi )^{2}}|p'_{\perp }| \gamma \in \mathbb {Z}. \end{aligned}$$

(A.19)

Therefore, the number of lattice points in $\mathcal {R}_{p,\alpha }$ is estimated by the number of lattice points in:

$$\begin{aligned}&\Big \{ (n_{2}, n_{3}) \in \mathbb {Z}^{2}\, \Big |\, N^{\frac{2}{3}} {\tilde{\nu }} - C|p|^{2}\le \frac{(2\pi )^{4}}{|p_{\perp }|^{2}} n_{2}^{2} + \frac{(2\pi )^{6}}{|p_{\perp }'|^{2}} n_{3}^{2} \le N^{\frac{2}{3}} {\tilde{\nu }}\Big \}\nonumber \\&\quad = \Big \{ (n_{2}, n_{3}) \in \mathbb {Z}^{2}\, \Big |\, K_{p} N^{\frac{2}{3}} {\tilde{\nu }} - C_{p}|p|^{2}\le \frac{|p|^{2}}{(2\pi )^{2}} n_{2}^{2} + n_{3}^{2} \le K_{p} N^{\frac{2}{3}} {\tilde{\nu }}\Big \}\nonumber \\ \end{aligned}$$

(A.20)

with $N^{\frac{2}{3}} {\tilde{\nu }} = N^{\frac{2}{3}} \nu - \alpha ^{2} (2\pi )^{2}$, and where $K_{p} = |p_{\perp }'|^{2}/(2\pi )^{6}$, $C_{p} = C |p_{\perp }'|^{2}/(2\pi )^{6}$. The number of lattice points in (A.20) is less or equal than the number of elements of:

$$\begin{aligned} \Big \{ (n_{2}, n_{3}) \in \mathbb {Z}^{2}\, \Big |\, K_{p} N^{\frac{2}{3}} {\tilde{\nu }} - C_{p}|p|^{2}\le \frac{|p|^{2}}{(2\pi )^{2}P^{2}} n_{2}^{2} + n_{3}^{2} \le K_{p} N^{\frac{2}{3}} {\tilde{\nu }}\Big \} \end{aligned}$$

(A.21)

where $P = \lceil |p| \rceil$. This is the number of lattice points lying in the difference of two smooth convex bodies, with elliptic boundary. To estimate the number of lattice points in (A.21), we use again Eq. (A.12) (thanks to the factor P, the curvature of the ellipses is bounded uniformly in |p|). We get, for all $\delta > 0$ and for p-independent constants K and $C_{\delta }$:

$$\begin{aligned} |\mathcal {R}_{p,\alpha }|\le & {} K |p|^{6} + C_{\delta } ( |p|^{2} N^{\frac{1}{3}} {\tilde{\nu }} )^{\gamma + \delta } \nonumber \\\le & {} {{\widetilde{C}}}_{\delta } ( |p|^{2} N^{\frac{1}{3}} {\tilde{\nu }} )^{\gamma + \delta }, \end{aligned}$$

(A.22)

for $|p| \le N^{\zeta }$, with $\zeta < \frac{\gamma + \delta }{18 - 6(\gamma + \delta )}$. Recalling that the number of sets in the union in Eq. (A.17) is bounded by $C A|p|^{2}$, we get:

$$\begin{aligned} I^{A}_{\mu }(p) \le \frac{1}{N^{\frac{1}{3}}}|\mathcal {R}_{p}|\le & {} K_{\delta } N^{\frac{1}{3}(\gamma + \delta ) - \frac{1}{3}} |p|^{2 + 2(\gamma + \delta )}, \end{aligned}$$

(A.23)

which is o(1) for $|p| \le N^{\zeta }$ with $\zeta < \frac{1}{1+ \gamma + \delta } ( \frac{1}{6} - \frac{1}{6}(\gamma + \delta ) )$.

${\underline{\hbox {Term }B}.}$ Let us consider the term $I^{B}_{\mu }(p)$. We can parametrize every point k by an angle $\theta _{k} \in [0,\pi ]$ such that $k\cdot p = |k| |p| \cos \theta _{k}$, and a remaining angle $\phi _{k} \in [0, 2\pi ]$. We rewrite the argument of the sum as:

$$\begin{aligned} \frac{1}{e(k+p) + e(k)}= & {} \frac{1}{\varepsilon ^{2} |p|^{2} + 2\varepsilon ^{2} k\cdot p}\nonumber \\= & {} \frac{1}{\varepsilon ^{2} |p|^{2} + 2\varepsilon ^{2} |k| |p| \cos \theta _{k}}. \end{aligned}$$

(A.24)

For A large enough, the constraints $A|p|^{2}\varepsilon ^{2} \le e(k+p) + e(k)\le \varepsilon ^{2-\gamma }$ and the fact that $|k| = O(N^{\frac{1}{3}})$ imply that $a\varepsilon \le \cos \theta _{k} \le c\varepsilon ^{1-\gamma }$ for suitable $a, c>0$. Using that $||k| - N^{\frac{1}{3}} \sqrt{\mu }| \le C$ for all points in the sum, we estimate:

$$\begin{aligned} \frac{1}{e(k+p) + e(k)} \le \frac{C}{\varepsilon ^{2}|p|^{2} + 2\varepsilon \sqrt{\mu } |p| \cos \theta _{k}}. \end{aligned}$$

(A.25)

Let us now introduce a family of angles $\{\eta _{i}\}$ partitioning $[0, \pi /2]$, such that

$$\begin{aligned} |\eta _{i} - \eta _{i+1}| = \kappa \varepsilon , \end{aligned}$$

(A.26)

for some $\kappa >0$ of order 1. We write:

$$\begin{aligned} I^{\text {B}}_{\mu }(p)\le & {} \frac{1}{N} \sum _{a\varepsilon \le \cos \eta _{i} \le c\varepsilon ^{1-\gamma }} \sum _{\begin{array}{c} k:\, k+p\notin \mathcal {B}_{\mu },\, k\in \mathcal {B}_{\mu } \\ A|p|^{2}\varepsilon ^{2} \le e(k+p) + e(k) \le \varepsilon ^{2 - \gamma } \\ |\theta _{k} - \eta _{i}| \le \frac{\kappa }{2}\varepsilon \end{array}} \frac{C}{\varepsilon ^{2}|p|^{2} + 2\varepsilon \sqrt{\mu } |p| \cos \theta _{k}} \\\le & {} \frac{1}{N} \sum _{a\varepsilon \le \cos \eta _{i} \le c\varepsilon ^{1-\gamma }} \frac{D}{\varepsilon ^{2}|p|^{2} + 2\varepsilon \sqrt{\mu } |p| \cos \eta _{i}} \sum _{\begin{array}{c} k:\, k+p\notin \mathcal {B}_{\mu },\, k\in \mathcal {B}_{\mu } \\ |\theta _{k} - \eta _{i}| \le \frac{\kappa }{2}\varepsilon \end{array}} 1. \end{aligned}$$

The innermost sum is the number of lattice points in:

$$\begin{aligned} \mathcal {S}_{p,i} = \Big \{ k\in \mathcal {B}_{\mu } \mid k+p\notin \mathcal {B}_{\mu },\; | \theta _{k} - \eta _{i} |\le \frac{\kappa }{2}\varepsilon \Big \}. \end{aligned}$$

(A.27)

To estimate the number of points in this set, we proceed as in the discussion of the term $I^{A}_{\mu }(p)$. We introduce the orthogonal basis $\{ b_{1}, b_{2}, b_{3} \}$ as in Eq. (A.18), and we slice the set $\mathcal {S}_{p,i}$ with planes with fixed $b_{1}$ component; we then estimate the number of lattice points in each slice, and use that $\mathcal {S}_{p,i}$ is sliced by a number of planes bounded by C|p| (since the width of $\mathcal {S}_{p,i}$ is the direction of p is bounded by a constant, and since the spacing between two planes is bounded below by C/|p|). We write:

$$\begin{aligned} \mathcal {S}_{p,i}= & {} \bigcup _{\alpha } \mathcal {S}_{p,i,\alpha } \nonumber \\ \mathcal {S}_{p,i,\alpha }= & {} \big \{ k\in \mathcal {S}_{p,i} \mid k\cdot b_{1} = (2\pi )^{2}\alpha \big \}. \end{aligned}$$

(A.28)

A simple trigonometric argument shows that the conditions $k\in \mathcal {B}_{\mu }$, $k+p\notin \mathcal {B}_{\mu }$, together with $| \cos \theta _{k} - \cos \eta _{i} |\le C\kappa \varepsilon$ and $\cos \eta _{i} \ge a\varepsilon$, imply that the width of the set $\mathcal {S}_{p,i,\alpha }$ in any direction orthogonal to p is bounded above by $|p| \cos \eta _{i}$.

Proceeding as in the analysis of $I^{A}_{\mu }(p)$, following the steps to arrive at Eq. (A.22), the number of lattice points in $\mathcal {S}_{p,i,\alpha }$ is bounded by:

$$\begin{aligned} K |p|^{6} N^{\frac{1}{3}}\cos \eta _{i} + C_{\delta } ( |p|^{2} N^{\frac{1}{3}} {\tilde{\nu }} )^{\gamma + \delta }. \end{aligned}$$

(A.29)

Since $\cos \eta _{i} \le c\varepsilon ^{1-\gamma }$, the second term dominates for $|p| \le N^{\zeta }$ with $\zeta = \frac{\delta }{18 - 6(\gamma + \delta )}$. We then have, taking into account that the number of sets in the union in Eq. (A.28) is bounded by C|p|:

$$\begin{aligned} I^{\text {B}}_{\mu }(p) \le \frac{K}{N^{\frac{1}{3}}} \sum _{i: a\varepsilon \le \cos \eta _{i} \le c \varepsilon ^{1-\gamma -\delta }} \frac{|p|^{2(\gamma + \delta ) + 1} \varepsilon ^{1-\gamma -\delta } }{\varepsilon |p|^{2} + 2 \sqrt{\mu } |p| \cos \eta _{i}}. \end{aligned}$$

(A.30)

This sum can be estimated as:

$$\begin{aligned} I^{\text {B}}_{\mu }(p)\le & {} {\widetilde{K}} |p|^{2(\gamma + \delta )} \int _{0}^{\frac{\pi }{2}} dx\, \frac{\varepsilon ^{1-\gamma - \delta }}{\varepsilon + \cos x}\nonumber \\\le & {} C |p|^{2(\gamma + \delta )} \varepsilon ^{1-\gamma - \delta } |\log \varepsilon |, \end{aligned}$$

(A.31)

which is o(1) for $|p| \le N^{\zeta }$ with $\zeta < \frac{1-\gamma -\delta }{6(\gamma + \delta )}$.

Term C. To conclude, let us consider $I^{\text {C}}_{\mu }(p)$. Up to o(1) error terms, we can replace the third constraint in the sum by:

$$\begin{aligned} C\varepsilon ^{2-\gamma } < \varepsilon ^{2} |k| |p| \cos \theta _{k}\;; \end{aligned}$$

(A.32)

again up o(1) error terms, we can replace Eq. (A.32) by:

$$\begin{aligned} \frac{C}{|p|} \varepsilon ^{1-\gamma } < \cos \theta _{k}. \end{aligned}$$

(A.33)

Next, consider the summand. We replace |k| with $N^{\frac{1}{3}}\sqrt{\mu }$, up to a small error:

$$\begin{aligned} \frac{1}{e(k) + e(k+p)} = \frac{1}{\varepsilon ^{2} |p|^{2} + 2\varepsilon \sqrt{\mu } |p| \cos \theta _{k}} (1 + r_{k,p}) \end{aligned}$$

(A.34)

with $|r_{k,p}| \le C\varepsilon |p|$. Then, we introduce angles $\xi _{i}$ partitioning $[0,\pi /2]$ such that:

$$\begin{aligned} |\xi _{i} - \xi _{i+1}| = \kappa \varepsilon ^{1 - \gamma + \alpha } \end{aligned}$$

(A.35)

for $\alpha >0$ small enough, to be chosen later, and $\kappa = O(1)$. For $|\theta _{k} - \xi _{i}| \le \frac{\kappa }{2} \varepsilon ^{1-\gamma + \alpha }$, we further rewrite the summand as:

$$\begin{aligned} \frac{1}{\varepsilon ^{2} |p|^{2} + 2\varepsilon \sqrt{\mu } |p| \cos \theta _{k}} = \frac{1}{\varepsilon ^{2} |p|^{2} + 2\varepsilon \sqrt{\mu } |p| \cos \xi _{i}} (1 + r'_{k,p}), \end{aligned}$$

(A.36)

where $|r'_{k,p}| \le C\varepsilon ^{\alpha }$. Neglecting for the moment the error terms, we are left with computing:

$$\begin{aligned} \frac{1}{N} \sum _{\frac{C}{|p|}\varepsilon ^{1-\gamma } \le \cos \xi _{i}}\frac{1}{\varepsilon ^{2} |p|^{2} + 2\varepsilon \sqrt{\mu } \cos \xi _{i}} \sum _{\begin{array}{c} k:\, k+p\notin \mathcal {B}_{\mu },\, k\in \mathcal {B}_{\mu } \\ |\theta _{k} - \xi _{i}| \le \frac{\kappa }{2}\varepsilon ^{1-\gamma + \alpha } \end{array}} 1. \end{aligned}$$

(A.37)

This sum can be bounded by a constant, uniformly in $\varepsilon$, repeating the slicing and counting argument used for $I^{B}_{\mu }(p)$. We shall however adopt a different strategy, that will allow us to explicitly compute (A.37) as $\varepsilon \rightarrow 0$. The reason is that the same argument can be used to prove Eq. (3.7), about the $\varepsilon \rightarrow 0$ limit of the correlation energy at second order in the potential, see Appendix B.1.

The innermost sum can be studied as in Section 6 of [6]. With respect to [6], the only difference is that instead of counting lattice points in a patch around the Fermi surface, Eq. (6.1) of [6], we are interested in the number of lattice points in the set:

$$\begin{aligned} \mathcal {C}_{i} = \Big \{ k\in \mathcal {B}_{\mu } \mid k+p\notin \mathcal {B}_{\mu },\; | \theta _{k} - \xi _{i}| \le \frac{\kappa }{2}\varepsilon ^{1-\gamma + \alpha } \Big \}. \end{aligned}$$

(A.38)

Let us denote by $C_{i}$ the corresponding subset of $N^{1/3} \sqrt{\mu }\, \mathbb {S}^{2}$:

$$\begin{aligned} C_{i} = \{ x\in N^{\frac{1}{3}} \sqrt{\mu }\, \mathbb {S}^{2} \mid |\theta - \xi _{i}| \le \frac{\kappa }{2}\varepsilon ^{1-\gamma + \alpha } \}, \end{aligned}$$

(A.39)

where $\theta$ is such that $x\cdot p = |x| |p| \cos \theta$. Let us denote by $C_{i,p}$ the projection of $C_{i}$ along p onto $\mathbb {R}^{2}\times \{0\}$, see Fig. 6 of [6]. Without loss of generality, we are assuming that $p_{3} \ne 0$; otherwise, we pick another component of p, and project over another coordinate plane. Then, see Eq. (6.4) of [6]:

$$\begin{aligned} |\mathcal {C}_{i}| = \mu (C_{i,p}) p_{3} + \mathfrak {e}_{i,p}, \end{aligned}$$

(A.40)

where $\mu$ is the Lebesgue measure on the plane and the error term $\mathfrak {e}_{i,p}$ is bounded proportionally to $|p| |\partial C_{i,p}|$, which is estimated by $C |p|^{2} N^{\frac{1}{3}} \sin \xi _{i}$, see discussion after Eq. (6.4) of [6]. Finally, the main term in Eq. (A.40) is, see Eq. (6.3) of [6]:

$$\begin{aligned} \mu (C_{i,p}) = N^{\frac{2}{3}} \mu \frac{|p|}{p_{3}} \cos \xi _{i} (1 + O(\varepsilon ^{\alpha })) \sigma (c_{i}), \end{aligned}$$

(A.41)

where $c_{i}$ is the subset of the unit sphere $\mathbb {S}^{2}$, such that $N^{\frac{1}{3}}\sqrt{\mu } c_{i} = C_{i}$ and $\sigma (c_{i})$ is the surface area of $c_{i}$. Referring to the notations of Eq. (6.3) of [6], the factor $M^{-1/2}$ of [6] is here replaced by $\varepsilon ^{1-\gamma + \alpha }$ (corresponding to the largest variation of the angles $\theta _{k}$ with respect to $\xi _{i}$), and the factor $N^{-\delta }$ of [6] is here replaced by $\varepsilon ^{1-\gamma }$ (corresponding to the lower bound for $\cos \xi _{i}$).

Consider the first term in the right-hand side of Eq. (A.40). The corresponding contribution to $I^{C}_{\mu }(p)$ is:

$$\begin{aligned} \mu |p| \sum _{\frac{C}{|p|}\varepsilon ^{1-\gamma } \le \cos \xi _{i}}\frac{\cos \xi _{i}}{\varepsilon |p|^{2} + 2\sqrt{\mu } |p|\cos \xi _{i}} \sigma (c_{i}), \end{aligned}$$

(A.42)

which converges to a bounded integral as $\varepsilon \rightarrow 0$:

$$\begin{aligned} (A.42)= & {} 2\pi \mu \int _{0}^{\pi /2} d\xi \, \frac{\sin \xi \cos \xi }{\varepsilon |p| + 2\sqrt{\mu } \cos \xi }(1 + o(1)) \nonumber \\ |(A.42)|\le & {} C. \end{aligned}$$

(A.43)

To conclude, consider now the contribution to $I^{C}_{\mu }(p)$ due to the second term in the right-hand side of Eq. (A.40). It is bounded as, using that $\sigma (c_{i}) = O(\varepsilon ^{1 - \gamma + \alpha } \sin \xi _{i})$:

$$\begin{aligned}&\frac{C|p|^{2} N^{\frac{1}{3}}}{N^{\frac{2}{3}}}\sum _{\frac{\widetilde{C}}{|p|}\varepsilon ^{1-\gamma } \le \cos \xi _{i}}\frac{\sin \xi _{i}}{\varepsilon |p|^{2} + 2\sqrt{\mu }|p| \cos \xi _{i}}\nonumber \\&\quad \le K |p|^{2} N^{-\frac{1}{3} + \frac{1}{3}( 1 - \gamma + \alpha )} \sum _{\frac{{{\widetilde{C}}}}{|p|}\varepsilon ^{1-\gamma } \le \cos \xi _{i}}\frac{1}{\varepsilon |p|^{2} + 2\sqrt{\mu } |p|\cos \xi _{i}} \sigma (c_{i}) \end{aligned}$$

(A.44)

$$\begin{aligned}&\quad \le {\widetilde{K}} |p|^{2} N^{\frac{1}{3}(-\gamma + \alpha )} |\log \varepsilon |, \end{aligned}$$

(A.45)

where in the last step we bounded the sum with the corresponding integral. Thus, the second term in Eq. (A.40) gives rise to a o(1) contribution to $I^{C}_{\mu }(p)$, for $|p| \le N^{\zeta }$ with $\zeta < \frac{1}{6}(\gamma - \alpha )$.

Finally, it is easy to see that the error terms due to $r_{k,p}$ and $r'_{k,p}$ in Eqs. (A.34), (A.36) give rise to o(1) contributions to $I_{\mu }(p)$. All in all, for $|p| \le N^{\zeta }$:

$$\begin{aligned} I^{C}_{\mu }(p) \le C. \end{aligned}$$

(A.46)

Conclusion. Putting together Eqs. (A.23), (A.31), (A.46) we see that, for $|p| \le N^{\zeta }$ with $\zeta = \frac{\delta }{18 - 6(\gamma + \delta )}$ and $\delta >0$ small enough, the main contribution to $I_{\mu }(p)$ is given by the integral in Eq. (A.43),

$$\begin{aligned} I_{\mu }(p) = 2\pi \mu \int _{0}^{\pi /2} d\xi \, \frac{\sin \xi \cos \xi }{\varepsilon |p| + 2\sqrt{\mu } \cos \xi }(1 + o(1)). \end{aligned}$$

(A.47)

In particular, $I_{\mu }(p) \le C$. This concludes the proof of Lemma 4.7. $\quad \square$

B Explicit computations

1.1 B.1 Computation of $\mathcal {C}^{(2)}_{N}$ in the $N\rightarrow \infty$ limit

In this appendix we shall compute $\mathcal {C}_{N}^{(2)}$ in the $N\rightarrow \infty$ limit, and we shall prove Eq. (3.7). Arguing as in the proof of Lemma 4.7, one gets, for $|p|\le N^{\zeta }$ with $\zeta = \frac{\delta }{18 - 6(\gamma + \delta )}$ and $\delta > 0$ small enough:

$$\begin{aligned}&\frac{1}{2N^{2}} \sum _{\begin{array}{c} k,\, k'\in \mathcal {B}_{\mu }\nonumber \\ k+p,\, k'-p \in \mathcal {B}_{\mu }^{c} \end{array}} \frac{1}{e(k+p) + e(k) + e(k'-p) + e(k')} \nonumber \\&\quad \simeq \frac{1}{4}\varepsilon \mu ^{2} |p| (2\pi )^{2} \int _{0}^{\frac{\pi }{2}} d\theta \int _{0}^{\frac{\pi }{2}} d\theta ' \sin \theta \sin \theta ' \frac{\cos \theta \cos \theta '}{\varepsilon |p| + \sqrt{\mu } (\cos \theta + \cos \theta ')},\qquad \quad \end{aligned}$$

(B.1)

up to o(1) as $N\rightarrow \infty$. Let us compute the integral. We rewrite it as:

$$\begin{aligned} \frac{1}{4}\varepsilon \mu ^{3/2} |p| (2\pi )^{2} \int _{0}^{1} dx \int _{0}^{1}dy\, \frac{x y}{a + x + y}, \end{aligned}$$

(B.2)

where $a = \varepsilon |p| / \sqrt{\mu }$. Explicit evaluation of the integral gives:

$$\begin{aligned} (B.2) = \varepsilon (2\pi )^{3} |p| \frac{\pi }{2} (1 - \log 2) + o(\varepsilon ). \end{aligned}$$

(B.3)

Therefore,

$$\begin{aligned} \mathcal {C}_{N}^{(2)}= & {} -\frac{1}{2 N^{2}} \sum _{p}\sum _{\begin{array}{c} k, k':\, k, k'\in \mathcal {B}_{\mu } \\ k + p, k'-p \in \mathcal {B}_{\mu }^{c} \end{array}} \frac{{\hat{v}}(p)^{2}}{e(k+p) + e(k) + e(k'-p) + e(k')} + o(\varepsilon ) \nonumber \\= & {} -\varepsilon (2\pi )^{3} \frac{\pi }{2} (1 - \log 2) \sum _{p} |p| {{\hat{v}}}_{<}(p)^{2} + o(\varepsilon )\nonumber \\= & {} -\varepsilon (2\pi )^{3} \frac{\pi }{2} (1 - \log 2) \sum _{p} |p| {{\hat{v}}}(p)^{2} + o(\varepsilon ), \end{aligned}$$

(B.4)

where we used the assumption on the decay of ${{\hat{v}}}(p)$. This concludes the check of Eq. (3.7).

1.2 B.2 Normalization of the trial state

Here we shall prove that $|M_{\mathbb {F}} - 1|\le C\Vert {\hat{v}}\Vert _{1}^{2}$, Eq. (4.90). We have:

$$\begin{aligned} M_{\mathbb {F}}^{2}= & {} 1 + \langle \Omega , \mathbb {F} \mathbb {H}_{0}^{-2} \mathbb {F}^{*} \Omega \rangle \nonumber \\= & {} 1 + \frac{1}{4 N^{2}} \sum _{p, k, k'} \sum _{q, r, r'} {\hat{v}}(p) {\hat{v}}(q) \langle \Omega , c_{k} c_{k'} b_{k'-p} b_{k+p} b^{*}_{r+q} b^{*}_{r'-q} c^{*}_{r'} c^{*}_{r}\nonumber \\&\cdot \frac{1}{(\mathbb {H}_{0} + e(r+q) + e(r'-q) + e(r) + e(r'))^{2}}\Omega \rangle \nonumber \\= & {} 1+ \frac{1}{2 N^{2}} \sum _{p}\sum _{\begin{array}{c} k, k':\, k, k'\in \mathcal {B}_{\mu } \\ k + p, k'-p \in \mathcal {B}_{\mu }^{c} \end{array}} \frac{{\hat{v}}(p)^{2}}{( e(k+p) + e(k) + e(k'-p) + e(k') )^{2}}\nonumber \\&-\frac{1}{2N^{2}} \sum _{p}\sum _{\begin{array}{c} k, k':\, k, k'\in \mathcal {B}_{\mu } \\ k + p, k'-p \in \mathcal {B}_{\mu }^{c} \end{array}} \frac{{\hat{v}}(p) {\hat{v}}(p - k'+k)}{( e(k+p) + e(k) + e(k'-p) + e(k') )^{2}} \end{aligned}$$

(B.5)

$$\begin{aligned}\equiv & {} 1 + J_{1} + J_{2}. \end{aligned}$$

(B.6)

Therefore, for N large enough:

$$\begin{aligned} J_{1}\le & {} \frac{1}{2 N^{2}} \sum _{p} {\hat{v}}(p)^{2} \sum _{\begin{array}{c} k, k':\, k, k'\in \mathcal {B}_{\mu } \\ k + p, k'-p \in \mathcal {B}_{\mu }^{c} \end{array}} \frac{1}{e(k+p) + e(k)} \frac{1}{e(k'-p) + e(k')}\nonumber \\\equiv & {} \frac{1}{2} \sum _{p} {\hat{v}}(p)^{2} I_{\mu }(p) I_{\mu }(-p) \le C\Vert {{\hat{v}}}\Vert _{1}^{2} + \mathfrak {e}_{K}, \end{aligned}$$

(B.7)

with C independent of N. In the last step we used the bound $I_{\mu }(p)\le C$ for $|p|\le N^{\zeta }$, proven in Appendix A, and the quantity $\mathfrak {e}_{K}$ takes into account the large |p| dependence of ${{\hat{v}}}(p)$, recall Eq. (4.15). Similarly,

$$\begin{aligned} |J_{2}|\le & {} \frac{\Vert {\hat{v}}\Vert _{\infty }}{2N^{2}} \sum _{p} {\hat{v}}(p) \sum _{\begin{array}{c} k, k':\, k, k'\in \mathcal {B}_{\mu } \\ k + p, k'-p \in \mathcal {B}_{\mu }^{c} \end{array}} \frac{1}{e(k+p) + e(k)} \frac{1}{e(k'-p) + e(k')}\nonumber \\\le & {} \frac{\Vert {\hat{v}}\Vert _{\infty }}{2} \sum _{p} {\hat{v}}(p) I_{\mu }(p) I_{\mu }(-p) \le C\Vert {{\hat{v}}}\Vert _{1}^{2} + \mathfrak {e}_{K}. \end{aligned}$$

(B.8)

This concludes the check of the boundedness of the normalization $M_{\mathbb {F}}$.

C Proof of Eq. (4.5)

Here we report the details of the computation leading to Eq. (4.5). We have:

$$\begin{aligned}&R^{*}_{\omega _{N}} a^{*}_{k+p} a^{*}_{k'-p} a_{k'} a_{k} R_{\omega _{N}} = (b^{*}_{k+p} + c_{k+p}) (b^{*}_{k' - p} + c_{k'-p})(b_{k'} + c^{*}_{k'})(b_{k} + c^{*}_{k})\nonumber \\&\quad = (b^{*}_{k+p}b^{*}_{k'-p} + b^{*}_{k+p} c_{k'-p} + c_{k+p} b^{*}_{k'-p} + c_{k+p} c_{k'-p} )\nonumber \\&\qquad \cdot (b_{k'} b_{k} + b_{k'} c^{*}_{k} + c^{*}_{k'} b_{k} + c^{*}_{k'} c^{*}_{k}). \end{aligned}$$

(C.1)

We take the products and put the operators in normal order, using that the only nontrivial anticommutators are:

$$\begin{aligned} \{ b^{*}_{k}, b_{k'} \} = \delta _{k,k'} \chi (k \notin \mathcal {B}_{\mu }),\qquad \{ c^{*}_{k}, c_{k'} \} = \delta _{k,k'}\chi (k\in \mathcal {B}_{\mu }). \end{aligned}$$

(C.2)

We get, collecting together terms that are equal after performing $k\leftrightarrow k'$ and $p\rightarrow -p$ (recall that ${\hat{v}}(p) = {\hat{v}}(-p)$ by assumption):

$$\begin{aligned} (C.1) = \text {(quartic terms)} + \text {(quadratic terms)} + \text {(constant terms);} \end{aligned}$$

(C.3)

the quartic terms are:

$$\begin{aligned}&\text {(quartic terms)} = b^{*}_{k+p} b^{*}_{k'-p} b_{k'} b_{k} - 2 b^{*}_{k+p} b^{*}_{k'-p} c^{*}_{k} b_{k'} + b^{*}_{k+p} b^{*}_{k'-p} c^{*}_{k'} c^{*}_{k}\\&\quad + 2 b^{*}_{k+p} c_{k' - p} b_{k'} b_{k} + 2 b^{*}_{k+p} c^{*}_{k} c_{k'-p} b_{k'} - 2 b^{*}_{k+p} c^{*}_{k'} c_{k'-p} b_{k}\\&\quad + 2 b^{*}_{k+p} c^{*}_{k'} c^{*}_{k} c_{k'-p} + c_{k+p} c_{k'-p} b_{k'} b_{k} - 2 c^{*}_{k} c_{k+p} c_{k'-p} b_{k'}\\&\quad + c^{*}_{k'} c^{*}_{k} c_{k+p} c_{k'-p}, \end{aligned}$$

the quadratic terms are:

$$\begin{aligned}&\text {(quadratic terms)} = 2 b^{*}_{k'-p} b_{k'}\delta _{k+p, k}\chi (k\in \mathcal {B}_{\mu }) - 2 b^{*}_{k+p} b_{k'}\delta _{k, k'-p}\chi (k\in \mathcal {B}_{\mu })\nonumber \\&\quad - 2 c^{*}_{k} c_{k+p} \delta _{k'-p,k'} \chi (k'\in \mathcal {B}_{\mu }) + 2c^{*}_{k'} c_{k+p}\delta _{k'-p, k} \chi (k \in \mathcal {B}_{\mu }), \end{aligned}$$

(C.4)

while the constant terms are:

$$\begin{aligned} \text {(constant terms)}= & {} \delta _{k'-p, k'}\delta _{k+p, k}\chi (k'\in \mathcal {B}_{\mu })\chi (k\in \mathcal {B}_{\mu })\nonumber \\&- \delta _{k+p, k'} \delta _{k'-p, k}\chi (k'\in \mathcal {B}_{\mu })\chi (k\in \mathcal {B}_{\mu }). \end{aligned}$$

(C.5)

In performing this computation we used that $b_{k}c_{k} = 0$, to cancel some quadratic terms produced after the normal ordering. Consider Eq. (C.4). We get, separating the particle-number conserving terms from the rest:

$$\begin{aligned}&\frac{1}{2N} \sum _{k, k', p} {\hat{v}}(p)\times \text {(quartic terms)} \nonumber \\&\quad = \frac{1}{2N} \sum _{k, k', p} {\hat{v}}(p)\big ( b^{*}_{k+p} b^{*}_{k'-p} b_{k'} b_{k} + c^{*}_{k'} c^{*}_{k} c_{k+p} c_{k'-p} + 2 b^{*}_{k+p} c^{*}_{k} c_{k'-p} b_{k'}\nonumber \\&\qquad - 2 b^{*}_{k+p} c^{*}_{k'} c_{k'-p} b_{k}\big )\nonumber \\&\qquad + \frac{1}{2N} \sum _{k,k',p} {\hat{v}}(p) \big ( - 2 b^{*}_{k+p} b^{*}_{k'-p} c^{*}_{k} b_{k'} + b^{*}_{k+p} b^{*}_{k'-p} c^{*}_{k'} c^{*}_{k} + 2 b^{*}_{k+p} c^{*}_{k'} c^{*}_{k} c_{k'-p}\big )\nonumber \\&\qquad + \frac{1}{2N} \sum _{k,k', p} {\hat{v}}(p) \big ( 2 b^{*}_{k+p} c_{k' - p} b_{k'} b_{k} + c_{k+p} c_{k'-p} b_{k'} b_{k} - 2 c^{*}_{k} c_{k+p} c_{k'-p} b_{k'}\big ).\qquad \quad \end{aligned}$$

(C.6)

Notice that the last two terms are one the adjoint of the other. Equation (C.6) reproduces $\mathbb {Q}$ in Eq. (4.2). Consider now the quadratic terms. We get:

$$\begin{aligned}&\frac{1}{2N} \sum _{k, k', p} {\hat{v}}(p) \times \text {(quadratic terms)}\nonumber \\&\quad = \frac{1}{N} \sum _{k, k' \in \mathcal {B}_{\mu }} {\hat{v}}(0) (b^{*}_{k} b_{k} - c^{*}_{k} c_{k}) - \frac{1}{N} \sum _{p} {\hat{v}}(p) \sum _{k\in \mathcal {B}_{\mu }} (b^{*}_{k+p} b_{k+p} - c^{*}_{k+p} c_{k+p})\nonumber \\&\equiv \mathbb {D} + \mathbb {X}, \end{aligned}$$

(C.7)

with $\mathbb {X}$ defined in Eq. (4.2), and $\mathbb {D}$ defined in Eq. (4.6) (recall that $\sum _{k \in \mathcal {B}_{\mu }} = N$). Finally, consider the constant terms. We get:

$$\begin{aligned}&\frac{1}{2N} \sum _{k,k',p} {\hat{v}}(p)\times \text {(constant terms)}\nonumber \\&\quad = \frac{1}{2N} \sum _{k,k'\in \mathcal {B}_{\mu }} {\hat{v}}(0) - \frac{1}{2N} \sum _{k,k' \in \mathcal {B}_{\mu }} {\hat{v}}(k-k')\nonumber \\&\quad = \frac{N}{2} {\hat{v}}(0) - \frac{1}{2N} \sum _{k,k' \in \mathcal {B}_{\mu }} {\hat{v}}(k-k'). \end{aligned}$$

(C.8)

These last two terms reproduce the Hartree–Fock interaction. This concludes the check of Eq. (4.5).