The semi-classical limit of large fermionic systems

Abstract

We study a system of N fermions in the regime where the intensity of the interaction scales as 1 / N and with an effective semi-classical parameter $\hbar =N^{-1/d}$ where d is the space dimension. For a large class of interaction potentials and of external electromagnetic fields, we prove the convergence to the Thomas–Fermi minimizers in the limit $N\rightarrow \infty$. The limit is expressed using many-particle coherent states and Wigner functions. The method of proof is based on a fermionic de Finetti–Hewitt–Savage theorem in phase space and on a careful analysis of the possible lack of compactness at infinity.

Appendix: Proof of Lemma 3.2

We first get a uniform estimate on $\rho _{\gamma _N}$. Indeed, using that

$$\begin{aligned} \gamma _N={\mathbb {1}}\left( \left( -i N^{-1/d}\nabla + A\right) ^2_{C_R}-c_{\mathrm{TF}}\rho \leqslant 0\right) \leqslant e^{-\beta ((-i N^{-1/d}\nabla + A)^2_{C_R}-c_{\mathrm{TF}}\rho )}, \end{aligned}$$

for any $\beta >0$, we deduce that

$$\begin{aligned} \rho _{\gamma _N}(x)\leqslant |e^{-\beta ((-i N^{-1/d}\nabla + A)^2_{C_R}-c_{\mathrm{TF}}\rho )}(x,x)|. \end{aligned}$$

From the Feynman–Kac formula and the diamagnetic inequality, we have

$$\begin{aligned} |e^{-\beta ((-i N^{-1/d}\nabla + A)^2_{C_R}-c_{\mathrm{TF}}\rho )}(x,y)|\leqslant e^{c_{\mathrm{TF}}\beta \left| \! \left| \rho \right| \! \right| _{L^\infty }}e^{N^{-2/d}\beta \Delta _{C_R}}(x,y), \end{aligned}$$

where $\Delta _{C_R}$ is the (non-magnetic) Dirichlet Laplacian on $C_R$, and hence

$$\begin{aligned} \rho _{\gamma _N}(x)&\leqslant \left( \frac{2}{R}\right) ^{d}e^{c_{\mathrm{TF}}\beta \left| \! \left| \rho \right| \! \right| _{L^\infty }}\sum _{k_1,\ldots ,k_d\in (\pi /R){\mathbb {N}}\setminus \{0\}} e^{-N^{-2/d}\beta \sum _{j=1}^d|k_j|^2}\times \\&\quad \times \prod _{j=1}^d\sin ^2\left( k_j (x_j-R/2)\right) \\&\leqslant \left( \frac{2}{R}\right) ^{d}e^{c_{\mathrm{TF}}\beta \left| \! \left| \rho \right| \! \right| _{L^\infty }}\left( \sum _{k\in \frac{\pi }{RN^{1/d}}{\mathbb {N}}\setminus \{0\}} e^{-\beta |k|^2}\right) ^d\\&\leqslant e^{c_{\mathrm{TF}}\beta \left| \! \left| \rho \right| \! \right| _{L^\infty }}N\pi ^{-d}\beta ^{-d/2}\left( \int _{\mathbb {R}}e^{-|k|^2}\,dk\right) ^d. \end{aligned}$$

Optimizing over $\beta$ gives

$$\begin{aligned} \rho _{\gamma _N}(x)\leqslant \left( \frac{2ec\pi }{\pi d}\right) ^{d/2}N \left| \! \left| \rho \right| \! \right| _{L^\infty ({\mathbb {R}}^d)}^{d/2}. \end{aligned}$$

(A.1)

We have therefore shown that $\rho _{\gamma _N}/N$ is bounded in $L^\infty (C_R)$. Up to extraction of a subsequence, we may assume that $\rho _{\gamma _N}/N\rightharpoonup \widetilde{\rho }$ weakly. Now we use that

$$\begin{aligned} 0&\geqslant -\frac{1}{N}\mathrm{Tr}\,\big ((-i N^{-1/d}\nabla + A)^2_{C_R}-c_{\mathrm{TF}}\rho \big )_-\\&=\frac{1}{N}\mathrm{Tr}\,\left( (-i N^{-1/d}\nabla + A)^2-c_{\mathrm{TF}}\rho \right) \gamma _N\\&=\frac{1}{N}\mathrm{Tr}\,\left( -i N^{-1/d}\nabla + A\right) ^2_{C_R}\gamma _N-c_{\mathrm{TF}}\frac{1}{N}\int _{C_R}\rho \rho _{\gamma _N} \end{aligned}$$

from which we deduce that

$$\begin{aligned} \frac{1}{N}\mathrm{Tr}\,\left( -i N^{-1/d}\nabla + A\right) ^2_{C_R}\gamma _N\leqslant c_{\mathrm{TF}}\frac{1}{N}\int _{C_R}\rho \rho _{\gamma _N}\leqslant C \left| \! \left| \rho \right| \! \right| _{L^1} \left| \! \left| \rho \right| \! \right| _{L^\infty }^{d/2}, \end{aligned}$$

due to the uniform upper bound on $\rho _{\gamma _N}/N$. We then introduce the semi-classical measure $m_N(x,p)={\left\langle f^\hbar _{x,p},\gamma _Nf^\hbar _{x,p} \right\rangle }$ and call its weak limit m. Arguing as before, we obtain

$$\begin{aligned}&\liminf _{N\rightarrow \infty }\left( \frac{1}{N}\mathrm{Tr}\,(-i N^{-1/d}\nabla + A)^2_{C_R}\gamma _N-c_{\mathrm{TF}}\frac{1}{N}\int _{C_R}\rho \rho _{\gamma _N}\right) \\&\quad \geqslant \frac{1}{(2\pi )^d}\int _{{\mathbb {R}}^d}\int _{{\mathbb {R}}^d}\big (|p+A(x)|^2-c_{\mathrm{TF}}\rho (x)\big )m(x,p)\,dp\,dx. \end{aligned}$$

We now use the well-known Weyl asymptotics for the energy, i.e.

$$\begin{aligned}&\lim _{N\rightarrow \infty } -\frac{1}{N}\mathrm{Tr}\,\big ((-i N^{-1/d}\nabla + A)^2_{C_R}-c_{\mathrm{TF}}\rho (x)\big )_- \nonumber \\&\quad =-\frac{1}{(2\pi )^d}\int _{{\mathbb {R}}^d}\int _{{\mathbb {R}}^d}\big (|p+A(x)|^2-c_{\mathrm{TF}}\rho (x)\big )_-\,dx\,dp. \end{aligned}$$

(A.2)

The result (A.2) is standard for smooth vector potentials A. Let $A_{\varepsilon }$ be a smooth approximation of A in $L^2$. Using the inequality

$$\begin{aligned}&(1-\delta ) (-i N^{-1/d}\nabla + A_{\varepsilon })^2 - \delta ^{-2} |A - A_{\varepsilon }|^2\\&\quad \leqslant (-i N^{-1/d}\nabla + A)^2 \nonumber \\&\quad \leqslant (1+\delta ) (-i N^{-1/d}\nabla + A_{\varepsilon })^2 + \delta ^{-2} |A - A_{\varepsilon }|^2, \end{aligned}$$

and the uniform upper bound on $\rho _{\gamma _N}/N$, the result follows for general A.

So we consider the Weyl asymptotics,

$$\begin{aligned}&\lim _{N\rightarrow \infty } -\frac{1}{N}\mathrm{Tr}\,\big ((-i N^{-1/d}\nabla + A)^2_{C_R}-c_{\mathrm{TF}}\rho (x)\big )_-\\&\quad =-\frac{1}{(2\pi )^d}\int _{{\mathbb {R}}^d}\int _{{\mathbb {R}}^d}\big (|p+A(x)|^2-c_{\mathrm{TF}}\rho (x)\big )_-\,dx\,dp\\&\quad =\inf _{0\leqslant m'\leqslant 1} \frac{1}{(2\pi )^d}\int _{{\mathbb {R}}^d}\int _{{\mathbb {R}}^d}\big (|p+A(x)|^2-c_{\mathrm{TF}}\rho (x)\big )m'(x,p)\,dx\,dp. \end{aligned}$$

with unique minimizer $m'(x,p) = {\mathbb {1}}(|p+A(x)|^2-c_{\mathrm{TF}}\rho (x)\leqslant 0)$ in $L^\infty ({\mathbb {R}}^d\times {\mathbb {R}}^d)$, we conclude that $m={\mathbb {1}}(|p+A(x)|^2-c_{\mathrm{TF}}\rho (x)\leqslant 0)$ a.e. This gives in particular that

$$\begin{aligned} N^{-1}\rho _{\gamma _N}(x)\rightharpoonup \rho _m(x)=\frac{1}{(2\pi )^d}\int _{{\mathbb {R}}^d}{\mathbb {1}}(|p+A(x)|^2-c_{\mathrm{TF}}\rho (x)\big )\,dp=\rho (x) \end{aligned}$$

weakly in $L^1\cap L^\infty$, hence that

$$\begin{aligned} N^{-1}\mathrm{Tr}\,(\gamma _N)=N^{-1}\int _{C_R}\rho _{\gamma _N}\rightarrow \int _{C_R}\rho =1. \end{aligned}$$

(A.3)

The latter says that $(-i N^{-1/d}\nabla + A)^2_{C_R}-c_{\mathrm{TF}}\rho$ has $N+o(N)$ negative eigenvalues. From the above limits we also have as desired

$$\begin{aligned} \lim _{N\rightarrow \infty } \frac{1}{N}\mathrm{Tr}\,(-i N^{-1/d}\nabla + A)^2_{C_R}\gamma _N&=\frac{1}{(2\pi )^d}\int _{{\mathbb {R}}^d}\int _{{\mathbb {R}}^d}|p|^2{\mathbb {1}}(|p|^2-c_{\mathrm{TF}}\rho (x)\big )\,dx\,dp\\&=\frac{d}{d+2}4\pi ^2\left( \frac{d}{|S^{d-1}|}\right) ^{2/d}\int _{{\mathbb {R}}^d}\rho (x)^{1+2/d}\,dx. \end{aligned}$$

Finally, the results are all the same for an orthogonal projection $\widetilde{\gamma }_N$ on N first eigenfunctions of $(-i N^{-1/d}\nabla + A)^2_{C_R}-c_{\mathrm{TF}}\rho$ since $\mathrm{Tr}\,|\gamma _N-\widetilde{\gamma }_N|=o(N)$ by (A.3) and therefore $\Vert \rho _{\gamma _N}-\rho _{\widetilde{\gamma }_N}\Vert _{L^1}=o(N)$. For N large we have

$$\begin{aligned} {\mathbb {1}}\big ((-i N^{-1/d}\nabla + A)^2_{C_R}\leqslant \rho (x)-\varepsilon \big )\leqslant \widetilde{\gamma }_N\leqslant {\mathbb {1}}\big ((-i N^{-1/d}\nabla + A)^2_{C_R}\leqslant \rho (x)+\varepsilon \big ) \end{aligned}$$

since, by the above arguments with $\rho$ replaced by $\rho \pm \varepsilon$,

$$\begin{aligned} \mathrm{Tr}\,{\mathbb {1}}\big ((-i N^{-1/d}\nabla + A)^2_{C_R}\leqslant \rho (x)\pm \varepsilon \big )\sim (1\pm \varepsilon R^d)N. \end{aligned}$$

From the above estimates we conclude that $\rho _{\widetilde{\gamma }_N}/N$ is bounded in $L^\infty$, and therefore $\rho _{\widetilde{\gamma }_N}/N$ has the same weak limit as $\rho _{\gamma _N}$ in $L^1\cap L^\infty$. We also have

$$\begin{aligned} \mathrm{Tr}\,((-i N^{-1/d}\nabla + A)^2_{C_R}-c_{\mathrm{TF}}\rho )(\gamma _N-\widetilde{\gamma }_N)=o(N) \end{aligned}$$

which implies that

$$\begin{aligned} \mathrm{Tr}\,(-i N^{-1/d}\nabla + A)^2_{C_R}(\gamma _N-\widetilde{\gamma }_N)=o(N^{1+2/d}) \end{aligned}$$

and concludes the proof of Lemma 3.2. $\square$