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arXiv:2006.00491 · Annales Henri Poincaré

The Dilute Fermi Gas via Bogoliubov Theory

Marco Falconi, Emanuela L. Giacomelli, Christian Hainzl, Marcello Porta

Published Apr 16, 2021• 26 citations
arXivHTMLPDF

Abstract

We study the ground state properties of interacting Fermi gases in the dilute regime, in three dimensions. We compute the ground state energy of the system, for positive interaction potentials. We recover a well-known expression for the ground state energy at second order in the particle density, which depends on the interaction potential only via its scattering length. The first proof of this result has been given by Lieb, Seiringer and Solovej (Phys Rev A 71:053605, 2005). In this paper, we give a new derivation of this formula, using a different method; it is inspired by Bogoliubov theory, and it makes use of the almost-bosonic nature of the low-energy excitations of the systems. With respect to previous work, our result applies to a more regular class of interaction potentials, but it comes with improved error estimates on the ground state energy asymptotics in the density.

DOI
10.1007/s00023-021-01031-6

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Recent developments in the physics of low-density trapped gases make it worthwhile to verify old, well-known results that, while plausible, were based on perturbation theory and assumptions about pseudopotentials. We use and extend recently developed techniques to give a rigorous derivation of the asymptotic formula for the ground-state energy of a dilute gas of NNN fermions interacting with a short-range, positive potential of scattering length aaa. For spin-1∕21∕21∕2 fermions, this is $E\ensuremath{\sim}{E}^{0}+({\ensuremath{\hbar}}^{2}∕2m)2\ensuremath{\pi}N\ensuremath{\varrho}a$, where E0{E}^{0}E0 is the energy of the noninteracting system and $\ensuremath{\varrho}$ is the density. A similar formula holds in two dimensions (2D), with $\ensuremath{\varrho}a$ replaced by $\ensuremath{\varrho}∕\ensuremath{\mid}\mathrm{ln}(\ensuremath{\varrho}{a}^{2})\ensuremath{\mid}$. Obviously this 2D energy is not the expectation value of a density-independent pseudopotential.
The textbook calculation of the ground-state energy of a dilute gas of Bose particles is examined in detail, and certain mathematical inconsistencies are pointed out. On the basis of this analysis, a refined approach suitable for soft interaction potentials which lend themselves to a low-order Born approximation is developed. This procedure emphasizes the low-density character of the resulting formula for the ground-state energy, and avoids all divergent expressions at intermediate stages of the computation. It is stressed that the standard Bogoliubov approximation, if not augmented by some additional device, leads to an error which manifests itself already in the lowest order of the density. - PACS 03.75.Hh, 05.30.Jp, 03.65.Nk.
We discuss recent progress in the mathematical analysis of dilute Bose gases. We review results in one to three dimensions, but the focus will be on three dimensions. In all dimensions we have a two term asymptotic expansion of the ground state energy density by an expression that depends only on the scattering length of the potential. In dimension three this is the celebrated Lee–Huang–Yang formula. In dimensions two and three the dilute limit is a weakly interacting regime whereas in dimension one it is rather strongly interacting. We sketch briefly the mathematical difficulties and review some remaining open problems in the field.
Our work establishes a three-term asymptotic expansion of the ground state energy of a dilute gas of spin 1/21/21/2 fermions with repulsive short-range interactions, validating a formula predicted by Huang and Yang in 1957. The formula is universal in the sense that it holds for a large class of interaction potentials and depends on those only via their scattering length. We have recently proved an upper bound on the ground state energy of the desired form, and the present work completes the program by proving the matching lower bound.