New theory nails the thermodynamics of strongly interacting superfluid Fermi gases

Ultracold atomic gases are among the coldest matter in the universe, yet they reproduce some of the hottest problems in physics. When fermionic atoms are cooled through a Feshbach resonance, their interactions can be tuned continuously from weakly bound Cooper pairs, as in conventional superconductors, to tightly bound molecules that condense like bosons. At the point where the scattering length diverges, the so-called unitary limit, the gas becomes scale invariant and its thermodynamics is governed by a single parameter, the reduced temperature. A new theoretical study now reports that an improved strong-coupling calculation, extending the widely used Gaussian pair fluctuation approach, quantitatively reproduces the measured thermodynamics of this strongly interacting superfluid across the Bardeen-Cooper-Schrieffer to Bose-Einstein condensate crossover, provided the temperature is low enough.

The work, published in AAPPS Bulletin by Brendan Mulkerin, Xing-Can Yao, Yoji Ohashi, Xia-Ji Liu and Hui Hu, extends a theory that was previously applied only to the normal, non-superfluid state. The central object is the thermodynamic potential, from which the chemical potential, internal energy, entropy and Tan contact parameter all follow. In the standard Gaussian pair fluctuation (GPF) treatment, quantum fluctuations of Cooper pairs are included only at the lowest, quadratic order around the mean-field saddle point. The new calculation goes one step further, adding the quartic fluctuation contribution evaluated through a partial re-summation of Feynman diagrams guided by a dimensional epsilon-expansion near four dimensions, keeping terms up to second order in the expansion parameter.

The motivation for this particular scheme comes from a long-standing dilemma in many-body theory. Because strongly interacting fermions admit no small expansion parameter, perturbative approximations are formally uncontrolled, and researchers often rely on quantum Monte Carlo simulations instead. Among field-theoretical approaches, the family of many-body T-matrix theories differs mainly in how much self-consistency is imposed. The fully self-consistent Luttinger-Ward theory performs well in the unitary regime, predicting a Bertsch parameter of 0.360, close to experiment, but it fails in the BEC limit, where it yields an incorrect molecular scattering length of twice the atom-atom scattering length instead of the exact value of roughly 0.6 times. Almost all T-matrix theories share this failure; the GPF theory is the notable exception, giving about 0.57 times the atomic scattering length.

The beyond-GPF theory is designed to satisfy both benchmark criteria simultaneously. In the BEC limit the authors show that the second-order fluctuation contribution becomes progressively less important, so the theory inherits the accurate molecular scattering length of the GPF approximation. On the BCS side, where plain GPF systematically overestimates the chemical potential relative to experiment, the added higher-order diagrams pull the prediction back into agreement with the data within experimental uncertainty. The comparison uses ground-state measurements from the University of Tokyo at a reduced temperature of 0.06, low enough that temperature dependence is expected to be negligible, so the results effectively probe zero-temperature crossover physics.

The quantitative test extends beyond the chemical potential. The authors compute the superfluid gap parameter and compare it with spectroscopic measurements from MIT and from Swinburne University of Technology, noting a slight but systematic improvement of the beyond-GPF prediction over plain GPF, while cautioning that current gap measurements are less accurate than thermodynamic ones. They also calculate the internal energy, expressed relative to the molecular binding energy, and Tan contact parameter, which encodes the short-range, large-momentum behaviour of the gas through an adiabatic relation linking it to the response of the thermodynamic potential to changes in the scattering length. On the BEC side, where no recent experimental data exist, the beyond-GPF and GPF theories converge to the same internal energy, and the authors argue that future measurements there would be a valuable test of their conjecture that the improved molecular scattering length survives.

The most striking results concern the unitary Fermi gas at finite temperature. Here the authors benchmark against the latest experiment from the University of Science and Technology of China, Shanghai Branch, which exploited a large homogeneous gas confined in a box potential together with high-resolution Bragg spectroscopy of the isentropic speed of sound, removing the local density approximation required by earlier measurements. Below about one tenth of the Fermi temperature, the beyond-GPF predictions for the chemical potential, internal energy, entropy and contact parameter agree closely with both the experimental data and finite-temperature quantum Monte Carlo simulations. At zero temperature the theory yields a Bertsch parameter of approximately 0.365, sitting within the error bars of the latest quantum Monte Carlo value of 0.367 with an uncertainty of 0.007 and the experimental value of 0.367 with an uncertainty of 0.009.

The agreement is not universal, and the authors are explicit about where the theory breaks down. Above roughly 0.1 times the Fermi temperature, the beyond-GPF chemical potential develops an unphysical maximum near 0.15 times the Fermi temperature and then decreases, the entropy is strongly underestimated, and the contact parameter fails to drop as sharply as the experiments show near the superfluid transition. The authors attribute these failures to large critical thermal fluctuations close to the transition temperature, which the truncated diagrammatic expansion cannot capture. Indeed, the theory cannot even locate the transition, because its chemical potential is too large to drive the gap parameter to zero; the authors suggest that a future calculation of the superfluid density might resolve this.

The practical upshot is a theory with a well-defined domain of validity. In the unitary limit, the reliable low-temperature window extends up to about 0.1 times the Fermi temperature, equivalent to roughly 0.6 times the transition temperature of about 0.17 times the Fermi temperature, while a previously published extension for the normal state works well above about half the Fermi temperature, where thermal fluctuations are suppressed. The awkward intermediate window, from 0.1 to 0.5 times the Fermi temperature, remains challenging. As a rough guide, a Ginzburg-Levanyuk number analysis within the GPF framework suggests that on the BCS side at an interaction parameter of minus 0.5, where the transition temperature is about 0.1 times the Fermi temperature, fluctuations dominate within a band of roughly 0.08 to 0.12 times the Fermi temperature around the transition.

The authors attribute the wide applicability of their approach to the empirically useful partial re-summation of diagrams selected by their scaling near the upper critical dimension of four, which yields accurate predictions at second order in the epsilon-expansion. They point to two natural extensions: performing the calculation directly near four dimensions with a small expansion parameter, which would suppress the large thermal fluctuations that currently spoil the critical region, and systematically incorporating large-loop Feynman diagrams at higher orders, potentially with the help of the bold diagrammatic Monte Carlo technique that has already achieved few-percent accuracy for the normal-state equation of state. They also propose extending the framework to dynamical properties, such as the density and spin response functions, by introducing external source fields. For now, the study establishes that a controlled-looking diagrammatic improvement of a decades-old approximation can match the precision of state-of-the-art experiments and simulations in the low-temperature superfluid regime, a regime that is difficult both for experiments, because of thermometry limitations, and for Monte Carlo methods, whose computational cost grows rapidly as temperatures fall.

Subject of Research: Thermodynamic properties of strongly interacting superfluid Fermi gases at the BEC-BCS crossover calculated with a beyond-Gaussian pair fluctuation theory

Article Title: Beyond Gaussian pair fluctuation theory for strongly interacting Fermi gases II: the broken-symmetry phase

Article References: Mulkerin, B. C., Yao, X.-C., Ohashi, Y., Liu, X.-J., & Hu, H. (2026). Beyond Gaussian pair fluctuation theory for strongly interacting Fermi gases II: the broken-symmetry phase. AAPPS Bulletin, 36(1), Article 15. https://doi.org/10.1007/s43673-026-00195-9

Image Credits: AI Generated

DOI: 10.1007/s43673-026-00195-9

Keywords: ultracold Fermi gases, BEC-BCS crossover, Gaussian pair fluctuation theory, unitary Fermi gas, Bertsch parameter, Tan contact, superfluid transition, quantum Monte Carlo, dimensional epsilon-expansion, strongly interacting fermions, thermodynamics, Feshbach resonance