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arXiv:2610.20434

Universal enhancement of the pairing gap in a dilute Fermi superfluid

Silas R. Beane, Roland C. Farrell, Cameron P. Shotwell

cond-mat.quant-gascond-mat.supr-connucl-thquant-phSubmitted Oct 14, 2026• Announced Oct 15, 2026• 0 citations
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Abstract

At zero temperature, a dilute Fermi gas with an attractive contact interaction forms a superfluid with gapped fermionic excitations. We show that the gap at weak coupling can be organized as a trans-series expansion in the interaction strength and the asymptotic Gor'kov-Melik-Barkhudarov gap. Low-energy virtual quasiparticles generate corrections that are linear in the gap with a leading coefficient nearly 25$\times$ larger than the other known coefficients. This substantially increases the gap at intermediate densities, highlighting that systematic corrections are crucial for quantitative predictions.

Comments
47 pages, 6 figures

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At zero temperature, a dilute Fermi gas with an attractive contact interaction forms a superfluid with gapped fermionic excitations. We show that the gap at weak coupling can be organized as a trans-series expansion in the interaction strength and the asymptotic Gor'kov-Melik-Barkhudarov gap. Low-energy virtual quasiparticles generate corrections that are linear in the gap with a leading coefficient nearly 25×\times× larger than the other known coefficients. This substantially increases the gap at intermediate densities, highlighting that systematic corrections are crucial for quantitative predictions.
A framework for computing the superfluid gap in an effective field theory (EFT) of fermions interacting via momentum-independent contact forces is developed. The leading universal corrections in the EFT are one-loop in-medium effects at the Fermi surface, and reproduce the well-known Gor'kov-Melik-Barkhudarov result. The complete subleading universal corrections are presented here, and include one-loop effects away from the Fermi surface, two-loop in-medium effects, as well as modifications to the fermion propagator. Together, these effects are found to reduce the gap at low densities. Applications to neutron superfluidity in neutron stars are also discussed.
We report results of quantum Monte Carlo calculations of the ground state of dilute Fermi gases with attractive short-range two-body interactions. The strength of the interaction is varied to study different pairing regimes which are characterized by the product of the sss-wave scattering length and the Fermi wave vector, akFa{k}_{F}akF​. We report results for the ground-state energy, the pairing gap $\ensuremath{\Delta}$, and the quasiparticle spectrum. In the weak-coupling regime, $1∕a{k}_{F}<\ensuremath{-}1$, we obtain Bardeen-Cooper-Schrieffer (BCS) superfluid and the energy gap $\ensuremath{\Delta}$ is much smaller than the Fermi gas energy EFG{E}_{\mathrm{FG}}EFG​. When a>0a>0a>0, the interaction is strong enough to form bound molecules with energy Emol{E}_{\mathrm{mol}}Emol​. For $1∕a{k}_{F}\ensuremath{\gtrsim}0.5$, we find that weakly interacting composite bosons are formed in the superfluid gas with $\ensuremath{\Delta}$ and gas energy per particle approaching $\ensuremath{\mid}{E}_{\mathrm{mol}}\ensuremath{\mid}∕2$. In this region, we seem to have Bose-Einstein condensation (BEC) of molecules. The behavior of the energy and the gap in the BCS-to-BEC transition region, $\ensuremath{-}0.5<1∕a{k}_{F}<0.5$, is discussed.
In this paper we consider a two-component gas of fermions on the BCS side of the BCS-BEC crossover at zero temperature. We use a momentum dependent interaction that reproduces the s-wave scattering phase shifts of a contact interaction up to a momentum cutoff that is scaled with the Fermi momentum. Using a diagrammatic formulation of Bogoliubov many-body perturbation theory, suitably augmented by self-consistency conditions, we obtain the Hartree shift and the pairing gap to third order. In the weak-coupling regime, our results are not only well-converged but also agree with the well-established Gor'kov-Melik-Barkhudarov corrections for the gap and the Galitskii result for the Hartree shift. Near the unitary regime, our results for the Nambu-Gor'kov self-energy are less converged, but there is still reasonable agreement with experiments as well as with quantum Monte-Carlo results. Perspectives for improvements and applications of this approach to neutron matter are discussed.
This article follows up on arxiv:2511.15938, in which we developed an Hamiltonian renormalization scheme to construct a quantized theory of Fermi liquids. Here, we apply this formalism to a low-temperature atomic Fermi gas where the short-range interactions are fully parametrized by the s-wave scattering length aaa. We benchmark our renormalized theory by recovering known perturbative results on the static properties of the Fermi gas, such as the Lee-Huang-Yang expansion of the equation of state, the Galitskii expansion of the momentum distribution, and the Gor'kov-Melik Barkhudarov preexponential correction to the critical temperature. We then turn to the transport dynamics and demonstrate the presence of a preexponential correction to the speed of zero sound when including corrections of second order in aaa. Finally, we develop an efficient numerical method to solve the transport equation exactly, and we apply it to study the crossover from the collisionless to the hydrodynamic regime in the density and polarisation response functions.
The ultracold and weakly coupled Fermi gas in two spatial dimensions is studied in an effective field theory framework. It has long been observed that universal corrections to the energy density to two orders in the interaction strength do not agree with Monte Carlo simulations in the weak-coupling regime. Here, universal corrections to three orders in the interaction strength are obtained, and are shown to provide agreement between theory and simulation. Special consideration is given to the scale ambiguity associated with the nontrivial renormalization of the singular contact interactions. The isotropic superfluid gap is obtained to next-to-leading order, and nonuniversal contributions to the energy density due to effective range effects, ppp-wave interactions, and three-body forces are computed. Results are compared with precise Monte Carlo simulations of the energy density and the contact in the weakly coupled attractive and repulsive Fermi-liquid regimes. In addition, the known all-orders sum of ladder and ring diagrams is compared with Monte Carlo simulations at weak coupling and beyond.
It is demonstrated that the interaction between fluctuations of the pairing field in a two-dimensional (2D) dilute attractive Fermi gas stabilizes the low-temperature phase of the system. This result, which is valid for all values of the "coupling parameter" $\frac{{\ensuremath{\varepsilon}}_{0}}{{\ensuremath{\varepsilon}}_{F}}$ (where ${\ensuremath{\varepsilon}}_{0}$ is the two-fermion binding energy and ${\ensuremath{\varepsilon}}_{F}$ is the Fermi energy), then allows one to conclude that a 2D dilute attractive Fermi gas is superfluid at small but finite temperature.