The dilute Fermi gas via Bogoliubov theory

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. 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.

arXiv comment52 pages

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• Ground state energy of the dilute spin-polarized Fermi gas: Upper bound via cluster expansion
• Ground state energy of the polarized diluted gas of interacting spin $1/2$ fermions
• Bosonization of Fermionic Many-Body Dynamics
• Lower bounds on the energy of the Bose gas
• Almost optimal upper bound for the ground state energy of a dilute Fermi gas via cluster expansion
• Pressure of a dilute spin-polarized Fermi gas: Lower bound
• Bogoliubov Transformations Beyond Shale-Stinespring: Generic $v^{\ast }v$ for bosons
• Friedrichs Diagrams -- Bosonic and Fermionic
• Implementing Bogoliubov Transformations Beyond the Shale-Stinespring Condition
• The Huang-Yang formula for the low-density Fermi gas: upper bound
• Bosonized Momentum Distribution of a Fermi Gas via Friedrichs Diagrams
• Effective Dynamics of Interacting Fermions from Semiclassical Theory to the Random Phase Approximation
• A second order upper bound on the ground state energy of a Bose gas beyond the Gross-Pitaevskii regime
• Effective quantum dynamics for magnetic fermions