arXiv:2402.17558

Ground state energy of the dilute spin-polarized Fermi gas: Lower bound

Asbjørn Bækgaard Lauritsen, Robert Seiringer

Published Feb 27, 2024• 1 citation

Abstract

We prove a lower bound on the ground state energy of the dilute spin-polarized Fermi gas capturing the leading correction to the kinetic energy resulting from repulsive interactions. This correction depends on the $p$-wave scattering length of the interaction and matches the corresponding upper bound in [J. Funct. Anal. 286.7 (2024), p. 110320].

DOI: 10.48550/arxiv.2402.17558

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arXiv:2402.17558

Ground state energy of the dilute spin-polarized Fermi gas: Lower bound

Asbjørn Bækgaard Lauritsen, Robert Seiringer

Published Feb 27, 2024• 1 citation

arXiv HTML PDF

Abstract

p

-wave scattering length of the interaction and matches the corresponding upper bound in [J. Funct. Anal. 286.7 (2024), p. 110320].

DOI: 10.48550/arxiv.2402.17558

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p

-wave scattering length of the interaction and matches the corresponding upper bound in [J. Funct. Anal. 286.7 (2024), p. 110320].

a^3ρ^{8/3}

, with

a

the

p

-wave scattering length of the interaction and

ρ

the particle density, depends on the temperature and matches the corresponding lower bound of [arXiv:2307.01113].

Abstract We prove an upper bound on the energy density of the dilute spin-

\frac{1}{2}

1 2 Fermi gas capturing the leading correction to the kinetic energy

8\pi a \rho _\uparrow \rho _\downarrow

8 π a ρ ↑ ρ ↓ with an error of size smaller than

a\rho ^{2}(a^3\rho )^{1/3-\varepsilon }

a ρ 2 ( a 3 ρ ) 1 / 3 - ε for any

\varepsilon > 0

ε > 0 , where a denotes the scattering length of the interaction. The result is valid for a large class of interactions including interactions with a hard core. A central ingredient in the proof is a rigorous version of a fermionic cluster expansion adapted from the formal expansion of Gaudin et al. (Nucl Phys A 176(2):237–260, 1971. https://doi.org/10.1016/0375-9474(71)90267-3 ).

{({k}_{\mathrm{F}}{a}_{0})}^{2}

order (where

{k}_{\mathrm{F}}

is the system's Fermi momentum and

{a}_{0}

the

s

-wave scattering length) and (with more labor) to extend it up to the order

{({k}_{\mathrm{F}}{a}_{0})}^{4}

. The analogous expansion of the ground-state energy of the polarized gas of spin

1/2

fermions is known only up to the

{k}_{\mathrm{F}}{a}_{0}

order (where

{k}_{\mathrm{F}}

stands for ${k}_{\mathrm{F}\ensuremath{\uparrow}}$ or ${k}_{\mathrm{F}\ensuremath{\downarrow}}$); the order

{({k}_{\mathrm{F}}{a}_{0})}^{2}

contribution has been computed (analytically) only using a specific (hard-core type) interaction potential. Here we show that the effective field theory method also allows to easily obtain the order

{({k}_{\mathrm{F}}{a}_{0})}^{2}

correction to the ground state of the polarized gas in a way applicable to all repulsive interactions.

Our work establishes a three-term asymptotic expansion of the ground state energy of a dilute gas of spin

1/2

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.

Abstract We consider a dilute fully spin-polarized Fermi gas at positive temperature in dimensions

d\in \{1,2,3\}

. We show that the pressure of the interacting gas is bounded from below by that of the free gas plus, to leading order, an explicit term of order

a^d\rho ^{2+2/d}

, where a is the p-wave scattering length of the repulsive interaction and $\rho $ is the particle density. The results are valid for a wide range of repulsive interactions, including that of a hard core, and uniform in temperatures at most of the order of the Fermi temperature. A central ingredient in the proof is a rigorous implementation of the fermionic cluster expansion of Gaudin, Gillespie and Ripka (Nucl. Phys. A, 176.2 (1971), pp. 237–260).