arXiv:2307.01113 · Forum of Mathematics Sigma

Pressure of a dilute spin-polarized Fermi gas: Lower bound

Asbjørn Bækgaard Lauritsen, Robert Seiringer

Published Jan 1, 2024• 4 citations

Abstract

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).

DOI: 10.1017/fms.2024.56

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arXiv:2307.01113 · Forum of Mathematics Sigma

Pressure of a dilute spin-polarized Fermi gas: Lower bound

Asbjørn Bækgaard Lauritsen, Robert Seiringer

Published Jan 1, 2024• 4 citations

arXiv HTML PDF

Abstract

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).

DOI: 10.1017/fms.2024.56

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Asbjørn Bækgaard Lauritsen, Robert Seiringer

arXiv:2301.04894Journal of Functional AnalysisJan 11, 202411 citations

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Asbjørn Bækgaard Lauritsen, Robert Seiringer

We prove an upper bound on the pressure of a dilute fully spin-polarized Fermi gas capturing the leading correction to the pressure of a free gas resulting from repulsive interactions. This correction is of order $a^3ρ^{8/3}$ , 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].We prove an upper bound on the pressure of a dilute fully spin-polarized Fermi gas capturing the leading correction to the pressure of a free gas resulting from repulsive interactions. This correction is of order $a^3ρ^{8/3}$, 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].

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The second order Huang-Yang approximation to the Fermi thermodynamic pressure

Xuwen Chen, Jiahao Wu, Zhifei Zhang

We consider a dilute Fermi gas in the thermodynamic limit with interaction potential scattering length $\mathfrak{a}_0$ at temperature $T>0$ . We prove the 2nd order Huang-Yang approximation for the Fermi pressure of the system, in which there is a 2nd order term carrying the positive temperature efffect.Our formula is valid up to the temperature $T<ρ^{\frac{2}{3}+\frac{1}{6}}$ , which is, by scaling, also necessary for the Huang-Yang formula to hold. Here, $T_F\simρ^{\frac{2}{3}}$ is the Fermi temperature. We also establish during the course of the proof, a conjecture regarding the second order approximation of density $ρ$ by R. Seiringer \cite{FermithermoTpositive}. Our proof uses frequency localization techniques from the analysis of nonlinear PDEs and does not involve spatial localization or Bosonization. In particular, our method covers the classical Huang-Yang formula at zero temperature.We consider a dilute Fermi gas in the thermodynamic limit with interaction potential scattering length $\mathfrak{a}_0$ at temperature $T>0$. We prove the 2nd order Huang-Yang approximation for the Fermi pressure of the system, in which there is a 2nd order term carrying the positive temperature efffect.Our formula is valid up to the temperature $T<ρ^{\frac{2}{3}+\frac{1}{6}}$, which is, by scaling, also necessary for the Huang-Yang formula to hold. Here, $T_F\simρ^{\frac{2}{3}}$ is the Fermi temperature. We also establish during the course of the proof, a conjecture regarding the second order approximation of density $ρ$ by R. Seiringer \cite{FermithermoTpositive}. Our proof uses frequency localization techniques from the analysis of nonlinear PDEs and does not involve spatial localization or Bosonization. In particular, our method covers the classical Huang-Yang formula at zero temperature.

arXiv:2505.23136May 29, 20250 citations

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

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.

p

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

We consider a dilute Fermi gas in the thermodynamic limit with interaction potential scattering length

\mathfrak{a}_0

at temperature

T>0

. We prove the 2nd order Huang-Yang approximation for the Fermi pressure of the system, in which there is a 2nd order term carrying the positive temperature efffect.Our formula is valid up to the temperature

T<ρ^{\frac{2}{3}+\frac{1}{6}}

, which is, by scaling, also necessary for the Huang-Yang formula to hold. Here,

T_F\simρ^{\frac{2}{3}}

is the Fermi temperature. We also establish during the course of the proof, a conjecture regarding the second order approximation of density

ρ

by R. Seiringer \cite{FermithermoTpositive}. Our proof uses frequency localization techniques from the analysis of nonlinear PDEs and does not involve spatial localization or Bosonization. In particular, our method covers the classical Huang-Yang formula at zero temperature.