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arXiv:2504.19310 · Physical review. B./Physical review. B

Fluctuation-induced first-order superfluid transition in unitary SU( N ) Fermi gases

Georgii Kalagov

Published Jun 10, 2025• 0 citations
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Abstract

We investigate the superfluid phase transition in an $\mathrm{SU}(N)$-symmetric Fermi gas with $N$ distinct spin states using the functional renormalization group. To capture pairing phenomena beyond mean-field theory, we introduce an auxiliary bosonic field and employ the leading order of the derivative expansion of the partially bosonized effective average action. By discretizing the effective potential on a grid and numerically integrating the flow equations, we resolve the thermodynamic behavior near the transition. Our results reveal a fluctuation-induced first-order phase transition for $N \geq 4$, which is absent at the mean-field level. In the unitary regime, we provide quantitative predictions for the critical temperature, as well as for the discontinuities in the superfluid gap and entropy density as functions of $N$. With increasing $N$, the critical temperature decreases, while the discontinuities become more pronounced, indicating a stronger first-order transition.

DOI
10.1103/tncd-kbh8

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We study the exact renormalization group flow for ultracold Fermi gases in the unitary regime. We introduce a pairing field to describe the formation of the Cooper pairs and take a simple ansatz for the effective action. A set of approximate flow equations for effective couplings including boson and fermionic fluctuations is derived. At some value of the running scale, the system undergoes a phase transition to a gapped phase. The values of the energy density, chemical potential, pairing gap and the corresponding proportionality constants relating the interacting and non-interacting Fermi gases are calculated. Standard mean-field results are recovered if we omit the boson loops.
We reveal the critical properties of the phase transition towards superfluid order that has been proposed to occur in large spin fermionic systems. For this purpose, we consider the bosonic field theory for fluctuations of the complex skew-symmetric rank-2 tensor order parameter close to the transition. We then nonperturbatively determine the scale dependence of the couplings of the theory by means of the functional renormalization group. We established a fluctuation-induced first-order phase transition. In the weak-coupling regime, the jump in the order parameter is small and a new phase occurs almost continuously, while in the strong one the discontinuity of the transition is well detectable.
We investigate the superfluid transition of the unitary Fermi gas by means of the functional renormalization group, aiming at quantitative precision. We extract ${T}_{\mathrm{c}}/\ensuremath{\mu}=0.38(2)$ and $\ensuremath{\Delta}/\ensuremath{\mu}=1.04(15)$ for the critical temperature and the superfluid gap at zero temperature, respectively, within a systematic improvement of the truncation for the effective average action. The key ingredient in comparison to previous approaches consists in the use of regulators which cut off both frequencies and momenta. We incorporate renormalization effects on both the bosonic and the fermionic propagators, include higher-order bosonic scattering processes, and investigate the regulator and specification parameter dependence for an error estimate. The ratio $\ensuremath{\Delta}/{T}_{\mathrm{c}}=2.7(3)$ becomes less sensitive to the relative cutoff scale of bosons and fermions when improving the truncation. The techniques developed in this work are easily carried over to the cases of finite scattering length, lower dimensionality, and spin imbalance.
We present a comprehensive analysis of quantum fluctuation effects in the superfluid ground state of an attractively interacting Fermi system, employing the attractive Hubbard model as a prototype. The superfluid order parameter and fluctuations thereof are implemented by a bosonic Hubbard-Stratonovich field, which splits into two components corresponding to longitudinal and transverse (Goldstone) fluctuations. Physical properties of the system are computed from a set of approximate flow equations obtained by truncating the exact functional renormalization group flow of the coupled fermion-boson action. The equations capture the influence of fluctuations on nonuniversal quantities such as the fermionic gap, as well as the universal infrared asymptotics present in every fermionic superfluid. We solve the flow equations numerically in two dimensions and compute the asymptotic behavior analytically in two and three dimensions. The fermionic gap $\ensuremath{\Delta}$ is reduced significantly compared to the mean-field gap, and the bosonic order parameter $\ensuremath{\alpha}$, which is equivalent to $\ensuremath{\Delta}$ in mean-field theory, is suppressed to values below $\ensuremath{\Delta}$ by fluctuations. The fermion-boson vertex is only slightly renormalized. In the infrared regime, transverse order-parameter fluctuations associated with the Goldstone mode lead to a strong renormalization of longitudinal fluctuations: the longitudinal mass and the bosonic self-interaction vanish linearly as a function of the scale in two dimensions, and logarithmically in three dimensions, in agreement with the exact behavior of an interacting Bose gas.
Abstract At temperatures well below the Fermi temperature T F , the coupling of magnetic fluctuations to particle-hole excitations in a two-component Fermi gas leads to non-analytic corrections in the renormalized free energy and makes the transition to itinerant ferromagnetism first order. On the other hand, despite that larger symmetry often introduces larger degeneracies in the low-lying states, here we show that for a Fermi gas with SU(N > 2)-symmetry in three space dimensions the ferromagnetic phase transition is first order in agreement with the predictions of Landau’s mean-field theory in its minimal formulation, which contains a cubic term in the free-energy (Cazalilla et al 2009 New J. Phys. 11 103033). By performing unrestricted Hartree–Fock calculations for an SU(N > 2)-symmetric Fermi gas with short range interactions, we find the order parameter undergoes a finite jump across the transition. In addition, for SU(N > 2) we do not observe any tri-critical point up to temperatures T ≃ 0.5 T F , for which the thermal smearing of the Fermi surface and thermal fluctuations are substantial. Going beyond mean-field theory, we find that the coupling of magnetic fluctuations to particle-hole excitations makes the transition more abrupt and further enhances the tendency of the gas to become fully polarized for smaller values of N and the gas parameter k F a s . In our study, we also clarify the role of time reversal symmetry in the microscopic Hamiltonian and obtain the temperature dependence of Tan’s contact. For the latter, the presence of the tri-critical point for N = 2 leads to a more pronounced temperature dependence around the transition than for SU(N > 2)-symmetric gases. Although the results are obtained for a microscopic model relevant to ultracold atomic gases, they may also apply to models of solid state systems with SU(N > 2) symmetry provided Coulomb interactions are sufficiently well screened.
We investigate the phase structure of spin-imbalanced unitary Fermi gases beyond mean-field theory by means of the functional renormalization group. In this approach, quantum and thermal fluctuations are resolved in a systematic manner. The discretization of the effective potential on a grid allows us to accurately account for both first- and second-order phase transitions that are present on the mean-field level. We compute the full phase diagram in the plane of temperature and spin imbalance and discuss the existence of other conjectured phases such as the Sarma phase and a precondensation region. In addition, we explain on a qualitative level how we expect that in situ density images are affected by our findings and which experimental signatures may potentially be used to probe the phase structure.