arXiv:2102.05966 · Physical Review C

Effective field theory for dilute Fermi systems at fourth order

Corbinian Wellenhofer, C. Drischler, A. Schwenk

Published Jul 20, 2021• 14 citations

Abstract

We discuss high-order calculations in perturbative effective field theory for fermions at low energy scales. The Fermi-momentum or ${k}_{\mathrm{F}}{a}_{s}$ expansion for the ground-state energy of the dilute Fermi gas is calculated to fourth order, both in cutoff regularization and in dimensional regularization. For the case of spin one-half fermions we find from a Bayesian analysis that the expansion is well converged at this order for $|{k}_{\mathrm{F}}{a}_{s}|\ensuremath{\lesssim}0.5$. Furthermore, we show that Pad\'e-Borel resummations can improve the convergence for $|{k}_{\mathrm{F}}{a}_{s}|\ensuremath{\lesssim}1$. Our results provide important constraints for nonperturbative calculations of ultracold atoms and dilute neutron matter.

DOI: 10.1103/physrevc.104.014003

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arXiv:2102.05966 · Physical Review C

Effective field theory for dilute Fermi systems at fourth order

Corbinian Wellenhofer, C. Drischler, A. Schwenk

Published Jul 20, 2021• 14 citations

arXiv HTML PDF

Abstract

We discuss high-order calculations in perturbative effective field theory for fermions at low energy scales. The Fermi-momentum or

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

expansion for the ground-state energy of the dilute Fermi gas is calculated to fourth order, both in cutoff regularization and in dimensional regularization. For the case of spin one-half fermions we find from a Bayesian analysis that the expansion is well converged at this order for

|{k}_{\mathrm{F}}{a}_{s}|\ensuremath{\lesssim}0.5

. Furthermore, we show that Pad'e-Borel resummations can improve the convergence for

|{k}_{\mathrm{F}}{a}_{s}|\ensuremath{\lesssim}1

. Our results provide important constraints for nonperturbative calculations of ultracold atoms and dilute neutron matter.

DOI: 10.1103/physrevc.104.014003

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Dilute Fermi gas at fourth order in effective field theory

Corbinian Wellenhofer, C. Drischler, A. Schwenk

arXiv:1812.08444Physics Letters BJan 27, 202022 citations

Ground-state energy of the polarized dilute gas of interacting spin- 12 fermions

Piotr H. Chankowski, Jacek Wojtkiewicz

The effective field theory approach simplifies the perturbative computation of the ground-state energy of the diluted gas of repulsively interacting fermions allowing in the case of the unpolarized system to easily rederive the classic results up to the ${({k}_{\mathrm{F}}{a}_{0})}^{2}$ 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.The effective field theory approach simplifies the perturbative computation of the ground-state energy of the diluted gas of repulsively interacting fermions allowing in the case of the unpolarized system to easily rederive the classic results up to the ${({k}_{\mathrm{F}}{a}_{0})}^{2}$ 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.

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Oct 9, 20200 citations

We discuss high-order calculations in perturbative effective field theory for fermions at low energy scales. The Fermi-momentum or

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

|{k}_{\mathrm{F}}{a}_{s}|\ensuremath{\lesssim}0.5

. Furthermore, we show that Pad'e-Borel resummations can improve the convergence for

|{k}_{\mathrm{F}}{a}_{s}|\ensuremath{\lesssim}1

. Our results provide important constraints for nonperturbative calculations of ultracold atoms and dilute neutron matter.

{({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.

s

-wave scattering length, but only on momentum scales of the order of the Fermi momentum, we build effective low-momentum interactions that reproduce the scattering phase shifts of the contact interaction below some momentum cutoff. Inspired by recent successes of this method in nuclear structure theory, we use these interactions to describe the equation of state of the Fermi gas in the framework of Hartree-Fock-Bogliubov theory with perturbative corrections. In the BCS regime, there is a range of cutoffs where we obtain fully converged results. Near unitarity, convergence is not yet reached, but we obtain promising results for the ground-state energies close to the experimental ones. Limitations and possible extensions of the approach are discussed.