Nous montrons que la théorie de champ moyen minimale à utiliser pour le calcul des fonctions de distribution de paires $g_{\sigma \sigma ^{\prime }}(\mathbf{r},\mathbf{r}^{\prime })$ d’un gaz superfluide de fermions de spin 1/2 spatialement homogène non polarisé n’est pas la théorie BCS statique ordinaire, mais la théorie BCS dépendant du temps linéarisée, mise en œuvre par le truchement du théorème de fluctuation-dissipation. En effet, la première ignore totalement la branche d’excitation acoustique — les phonons — du superfluide, alors que la seconde en tient compte explicitement, ainsi que des fluctuations quantiques induites par le continuum de paires brisées. Contrairement à la première, la seconde théorie (i) répercute l’effet de ces excitations collectives sur l’équation d’état du système, y compris à température nulle, (ii) permet à la fonction $g_{\uparrow \downarrow }(\mathbf{r},\mathbf{r}^{\prime })$ de descendre à distance assez grande strictement en dessous de sa valeur asymptotique $(\rho /2)^2$ où $\rho $ est la densité du gaz, comme il se doit d’après l’hydrodynamique quantique de Landau et Khalatnikov à basse température, et (iii) prédit dans la fonction $g_{\uparrow \uparrow }(\mathbf{r},\mathbf{r}^{\prime })$ à courte distance des contributions sous-dominantes en $\vert \mathbf{r}-\mathbf{r}^{\prime }\vert ^2\ln \vert \mathbf{r}-\mathbf{r}^{\prime }\vert $ à 3D et en $\vert \mathbf{r}-\mathbf{r}^{\prime }\vert ^2\ln \bigl (-\ln \vert \mathbf{r}-\mathbf{r}^{\prime }\vert \bigr )$ à 2D, à côté des contributions dominantes en $\vert \mathbf{r}-\mathbf{r}^{\prime }\vert $ à 3D et en $\vert \mathbf{r}-\mathbf{r}^{\prime }\vert ^2\ln \vert \mathbf{r}-\mathbf{r}^{\prime }\vert $ à 2D déjà présentes dans la théorie BCS statique mais avec un coefficient plus faible. Cette discussion est pertinente pour les travaux théoriques récents d’Obeso-Jureidini et de Romero-Rochín, et pour les expériences en cours sur les gaz d’atomes froids à l’ENS et au MIT.
We show that the minimal mean-field theory to use for calculating the pair distribution functions $g_{\sigma \sigma ^{\prime }}(\mathbf{r},\mathbf{r}^{\prime })$ of a spatially homogeneous, unpolarized spin-1/2 superfluid Fermi gas is not the ordinary static BCS theory, but the linearized time-dependent BCS theory implemented via the fluctuation-dissipation theorem. Indeed, the former completely ignores the acoustic excitation branch — the phonons — of the superfluid, while the latter explicitly takes it into account, as well as the quantum fluctuations induced by the broken-pair continuum. Unlike the first, the second theory (i) reflects the effect of these collective excitations on the system’s equation of state, including at zero temperature, (ii) allows the function $g_{\uparrow \downarrow }(\mathbf{r},\mathbf{r}^{\prime })$ to go at sufficiently large distances strictly below its asymptotic value $(\rho /2)^2$ where $\rho $ is the gas density, as expected according to the quantum hydrodynamics of Landau and Khalatnikov at low temperatures, and (iii) predicts in the function $g_{\uparrow \uparrow }(\mathbf{r},\mathbf{r}^{\prime })$ at short distances subdominant contributions $\vert \mathbf{r}-\mathbf{r}^{\prime }\vert ^2\ln \vert \mathbf{r}-\mathbf{r}^{\prime }\vert $ in 3D and $\vert \mathbf{r}-\mathbf{r}^{\prime }\vert ^2\ln \bigl (-\ln \vert \mathbf{r}-\mathbf{r}^{\prime }\vert \bigr )$ in 2D, alongside the dominant contributions $\vert \mathbf{r}-\mathbf{r}^{\prime }\vert $ in 3D and $\vert \mathbf{r}-\mathbf{r}^{\prime }\vert ^2\ln \vert \mathbf{r}-\mathbf{r}^{\prime }\vert $ in 2D already present in static BCS theory but with a lower coefficient. This discussion is relevant to the recent theoretical work of Obeso-Jureidini and Romero-Rochín, and to the ongoing experiments on cold atomic gases at ENS and MIT.
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Keywords: Fermi gases, pair density, structure factor, fluctuation-dissipation theorem, superfluidity, unitary limit, contact interactions
Yvan Castin 1
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@article{CRPHYS_2026__27_G1_101_0,
author = {Yvan Castin},
title = {Fonctions de distribution de paires d{\textquoteright}un gaz superfluide de fermions de spin 1/2 en interaction de contact dans la th\'eorie {BCS} d\'ependant du temps lin\'earis\'ee},
journal = {Comptes Rendus. Physique},
pages = {101--160},
year = {2026},
publisher = {Acad\'emie des sciences, Paris},
volume = {27},
doi = {10.5802/crphys.276},
language = {fr},
}
TY - JOUR AU - Yvan Castin TI - Fonctions de distribution de paires d’un gaz superfluide de fermions de spin 1/2 en interaction de contact dans la théorie BCS dépendant du temps linéarisée JO - Comptes Rendus. Physique PY - 2026 SP - 101 EP - 160 VL - 27 PB - Académie des sciences, Paris DO - 10.5802/crphys.276 LA - fr ID - CRPHYS_2026__27_G1_101_0 ER -
%0 Journal Article %A Yvan Castin %T Fonctions de distribution de paires d’un gaz superfluide de fermions de spin 1/2 en interaction de contact dans la théorie BCS dépendant du temps linéarisée %J Comptes Rendus. Physique %D 2026 %P 101-160 %V 27 %I Académie des sciences, Paris %R 10.5802/crphys.276 %G fr %F CRPHYS_2026__27_G1_101_0
Yvan Castin. Fonctions de distribution de paires d’un gaz superfluide de fermions de spin 1/2 en interaction de contact dans la théorie BCS dépendant du temps linéarisée. Comptes Rendus. Physique, Volume 27 (2026), pp. 101-160. doi: 10.5802/crphys.276
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