The central open question is the finite-temperature normal-state problem of the crossover: does a pairing temperature T∗T^*, at which the single-particle gap closes without global phase coherence, exist distinctly above TcT_c at unitarity; how does T∗T^* vary across the crossover; and which many-body theory quantitatively reproduces the now-measured pseudogap spectra while remaining consistent with the settled equation of state. A 2024 momentum-resolved microwave-spectroscopy experiment observed a large pseudogap above TcT_c, while continuum-limit quantum Monte Carlo finds a narrow pairing regime, T∗≈0.2 TFT^* \approx 0.2\,T_F versus Tc=0.16(1) TFT_c = 0.16(1)\,T_F — an active quantitative conflict that defines the open problem.

Background

A two-component ultracold Fermi gas is a mixture of fermionic atoms (the workhorse species are 6^6Li and 40^{40}K) in two internal hyperfine states, cooled far below the Fermi temperature TFT_F (the Fermi energy EF=kBTFE_F = k_B T_F set by the density, with kFk_F the Fermi wavevector). A magnetic Feshbach resonance — a tunable coupling between open scattering channels and a closed bound molecular channel — lets the experimenter tune the ss-wave scattering length aa between the two spin states at will, so the dimensionless interaction parameter 1/kFa1/k_F a sweeps continuously from −∞-\infty (weak attraction, Cooper pairs of the Bardeen-Cooper-Schrieffer, BCS, type) through 1/kFa=01/k_F a = 0 (unitarity, where ∣a∣|a| diverges and the gas is scale-invariant and universal) to +∞+\infty (tightly bound diatomic molecules undergoing Bose-Einstein condensation, BEC). This BCS-BEC crossover, realized in 2003-2005 (Regal's JILA thesis is a founding document; see also the review Giorgini, Pitaevskii and Stringari 2008), created a clean laboratory model of strongly interacting fermion pairing relevant to condensed matter and nuclear physics. In this system two distinct energy scales compete at finite temperature: the superfluid transition temperature TcT_c, at which pairs achieve long-range phase coherence, and the pairing temperature T∗T^*, at which attractive interactions first open a gap in the single-particle spectrum. Between TcT_c and T∗T^* the gas would host so-called preformed pairs — bound or correlated pairs without phase coherence — producing a pseudogap, a partial suppression of low-energy spectral weight in the normal (non-superfluid) phase, conceptually parallel to the pseudogap debated in the cuprate high-temperature superconductors. Regal's thesis raised this finite-temperature question explicitly among the unresolved issues of the young field: the simplest finite-temperature crossover theory available then (Nozières-Schmitt-Rink, NSR, a Gaussian fluctuation extension of mean-field BCS theory) already failed quantitatively — it gave the chemical potential at unitarity as μ=(1+β)EF\mu = (1+\beta)E_F with β=−0.41\beta = -0.41 versus the quantum Monte Carlo (QMC) value β≈−0.58\beta \approx -0.58, and predicted a molecule-molecules scattering length amm=2aa_{mm} = 2a versus the exact four-body result 0.6a0.6a — and no direct, model-independent thermometry existed through the crossover to map TcT_c and any pseudogap regime. Much of this has since been settled. Precision measurements of the equation of state (EoS) in trapped, locally uniform gases (Navon et al. 2010; Ku et al. 2012) determined the universal thermodynamics with no free parameters: the Bertsch parameter ξ=0.376(4)\xi = 0.376(4) (the ground-state energy E=ξ⋅(3/5)NEFE = \xi \cdot (3/5) N E_F of the unitary gas), a lambda-like anomaly in the heat capacity at Tc/TF=0.167(13)T_c/T_F = 0.167(13), and entropy-based thermometry valid through the crossover; diagrammatic Monte Carlo — a stochastic summation of the Feynman diagrams of many-body perturbation theory (Van Houcke et al. 2012) — reproduces the EoS at the few-percent level, and continuum-extrapolated auxiliary-field quantum Monte Carlo, a lattice simulation that decouples the contact interaction through a fluctuating auxiliary field (Jensen, Gilbreth and Alhassid 2024), independently finds Tc=0.16(1) TFT_c = 0.16(1)\,T_F. What remains unsettled is the dynamical, spectral content of the normal state: early radio-frequency and photoemission spectroscopy (Stewart, Gaebler and Jin 2008) was clouded by final-state interactions — the residual interaction of the atom with the internal state it is transferred into, which distorts the measured lineshape — and trap averaging; the 2024 measurement of Li et al., using momentum-resolved microwave spectroscopy (a hyperfine-transition probe that resolves the transferred atoms' momentum, the cold-atom analogue of angle-resolved photoemission) of a homogeneous unitary 6^6Li gas, observed a pair-fluctuation-driven pseudogap with a large gap surviving well above TcT_c and an inverse pair lifetime that is thermally activated; yet finite-temperature QMC analyses find the pairing signatures confined to a narrow window, T∗≈0.2 TFT^* \approx 0.2\,T_F, and state openly that the extent of the pseudogap regime is still debated. Theoretical spectral functions are under active construction (Johansen, Frank and Lang 2024; Mulkerin et al. 2026), and no current many-body scheme simultaneously matches the settled thermodynamics and the measured spectra above TcT_c. The unresolved target is therefore a quantitative account of the normal phase of the crossover above TcT_c: whether it hosts pairing without phase coherence, over what range of temperature and interaction strength, and by what microscopic mechanism.

Problem Statement

Determine, quantitatively and in agreement with the existing precision data on the crossover, the finite-temperature pairing structure of the normal phase of the two-component Fermi gas across the BCS-BEC crossover. Concretely: (i) establish whether a pairing temperature T∗T^* — the temperature at which the single-particle (pseudo)gap of the normal phase closes — is distinct from the superfluid transition temperature TcT_c at unitarity (1/kFa=01/k_F a = 0), and determine how T∗T^* varies with 1/kFa1/k_F a across the crossover on both the BCS and BEC sides, stating explicitly the operational definition of T∗T^* adopted (the observable and the criterion taken for gap closure) and reporting the T∗(1/kFa)T^*(1/k_F a) curve under that definition; (ii) determine the microscopic origin of the normal-state pairing correlations — whether they reflect long-lived preformed pairs, strong scattering without well-defined pair degrees of freedom, or another mechanism — with the temperature dependence of the pairing gap, the pair lifetime and the single-particle scattering rate among the evidence that can discriminate between these possibilities; and (iii) construct (or identify) a many-body theory or controlled simulation methodology that quantitatively accounts for the normal-state structure established in (i) and (ii) — reproducing the single-particle spectral function in the pairing regime above TcT_c if such a regime is established, or, if it is not, accounting quantitatively for the reported above-TcT_c gap signatures and their extent — while remaining consistent with the already-settled equilibrium thermodynamics of the crossover. These three deliverables jointly serve one objective — a validated characterization of the crossover's normal state above TcT_c, including whether it hosts pairing without phase coherence — and are not independent problems: the normal-state structure established in (i) and the mechanism identified in (ii) fix what the theory or methodology of (iii) has to account for.

References

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