Ground-state pairing structure and equation of state of the mass-imbalanced resonant $^6$Li--$^{40}$K Fermi gas across the BCS--BEC crossover

Fixed-node diffusion Monte Carlo calculations (Gandolfi, Curry, Gezerlis, 2024) for the unpolarized two-component Fermi gas corresponding to the $^6$Li--$^{40}$K mixture (simulated at mass ratio $m_h/m_l = 6.5$) find a Bertsch parameter $\xi \approx 0.368(1)$ slightly below the equal-mass value, conjecturally due to nonzero-total-momentum paired components in the ground state, while the effective-range slope and higher-order near-unitarity coefficients closely match the equal-mass gas for unexplained reasons. The open problem is a bias-controlled, ab initio determination of the zero-temperature ground state of this mass-imbalanced resonant gas: whether finite-momentum (LOFF-like) or exotic pairing is genuinely present or energetically competitive, whether the observed deviations from the equal-mass gas are physical rather than fixed-node or finite-range artifacts, and a quantified account of how mass imbalance modifies the pairing structure and equation of state across the crossover.

Background

Two-component ultracold Fermi gases tuned through a Feshbach resonance (a magnetically tunable scattering resonance that lets the interaction strength be swept at ultralow temperatures) realize the BCS--BEC crossover: the interspecies s-wave (zero-angular-momentum) scattering length $a$ is swept from weak attraction (BCS superfluidity) through unitarity ($|a| \to \infty$) to bosonic dimers forming a Bose--Einstein condensate. At unitarity the gas is scale-invariant and its ground-state energy per particle is a universal constant times the energy $E_{FG}$ of the noninteracting gas at the same density; this constant is the Bertsch parameter $\xi$. For equal masses, quantum Monte Carlo (QMC) calculations bracket $\xi \approx 0.372$--$0.383$ depending on method (lattice auxiliary-field QMC by Carlson, Gandolfi, Schmidt, and Zhang gives 0.372(3); fixed-node diffusion Monte Carlo by Gandolfi, Schmidt, and Carlson gives 0.383(1)). Fixed-node diffusion Monte Carlo (FN-DMC) projects an optimized variational trial state in imaginary time while constraining the fermion nodes to those of the trial function; it yields a variational upper bound whose residual bias is hard to quantify. For unequal fermion masses an additional obstacle appears: exact projection methods then suffer a fermion sign problem, in which Monte Carlo contributions of opposite sign cancel exponentially, so unbiased ground-state calculations are unavailable and bias-controlled methods are the standard. Extra physics also appears: with majority light atoms the Chandrasekhar--Clogston limit (the polarization threshold for losing superfluidity) collapses toward zero, Efimov-type few-body nonuniversality (an infinite ladder of three-body bound states at resonant short-range interaction, which breaks the universality of the two-body scattering description) can occur at larger mass ratios, and spatially modulated pairing states become conceivable. The Larkin--Ovchinnikov--Fulde--Ferrell (LOFF) phase is a superfluid in which Cooper pairs carry nonzero total momentum, producing a periodically modulated pair field; induced p-wave pairing denotes order in a higher angular-momentum channel generated by fluctuations of the underlying s-wave pairs. The canonical experimental candidate is the $^6$Li--$^{40}$K mixture with mass ratio $\kappa \approx 6.67$, well below the three-body stability limit near $\kappa \approx 12.3$ but large enough that mean-field intuition fails. The prior literature is: Gezerlis, Gandolfi, Schmidt, and Carlson (Phys. Rev. Lett. 103, 060403 (2009)) performed the first FN-DMC study at this mass ratio, finding the unpolarized equation of state nearly mass-independent and conjecturing that small residual differences trace to nonzero-total-momentum pairs; Diener and Randeria (Phys. Rev. A 81, 033608 (2010)), at mean-field level plus Gaussian fluctuations, obtained a weak but nontrivial mass-ratio dependence at unitarity in excellent agreement with the QMC result for the $^{40}$K--$^6$Li mixture at mass ratio 6.67, whereas pure mean-field calculations predict no difference between equal and unequal masses; Blume and Daily (Phys. Rev. Lett. 105, 170403 (2010)) showed in exact few-body calculations that unequal-mass gases at unitarity need not be universal, with surprisingly large effective-range effects; Forbes, Gandolfi, and Gezerlis (Phys. Rev. A 86, 053603 (2012)) established the effective-range corrections of the equal-mass gas, whose leading slope $c$ enters $E/E_{FG} = \xi + c\, k_F r_e$ with $k_F$ the Fermi momentum and $r_e$ the effective range; Braun, Drut, and Roscher (Phys. Rev. Lett. 114, 050404 (2015)) obtained a sign-problem-free lattice equation of state by simulating imaginary mass imbalance and analytically continuing; Roscher, Braun, and Drut (Phys. Rev. A 91, 053611 (2015)) mapped, via renormalization group methods, where inhomogeneous phases may occur in mass- and spin-imbalanced unitary gases; and Rammelmüller, Porter, Drut, and Braun (Phys. Rev. D 96, 094506 (2017)) demonstrated complex Langevin sampling as a route around the sign problem for mass-imbalanced fermions. In this landscape, Gandolfi, Curry, and Gezerlis (Phys. Rev. A 110, 043320 (2024)) computed the full zero-temperature crossover of the unpolarized $^6$Li--$^{40}$K mixture with FN-DMC: near unitarity they fit $E/E_{FG} = \xi - \zeta/(k_F a) - 5\nu/(3 (k_F a)^2)$ obtaining $\xi = 0.3726(6)$, $\zeta = 0.900(2)$, $\nu = 0.46(1)$; extrapolating to zero effective range at unitarity they obtained $\xi = 0.368(1)$ and slope $c = 0.13(1)$; in the BEC limit the dimer-dimer scattering length $a_{dd}/a = 0.886(4)$. They also computed pair-distribution functions, momentum distributions, quasiparticle spectra, and the contact (the universal amplitude setting the high-momentum tail of the pair momentum distribution), and pointed to the open issues that define the present problem.

Problem Statement

Determine, from ab initio methods with quantified or eliminated fixed-node/sign bias, the zero-temperature ground state of the unpolarized two-component resonant Fermi gas with mass ratio $m_K/m_L \approx 6.67$ (the $^6$Li--$^{40}$K mixture) as a function of scattering length across the BCS--BEC crossover. The objective is a bias-controlled account of how mass imbalance alone modifies the pairing structure and equation of state. The following sub-claims are independently refutable: (i) whether the ground state at and near unitarity contains nonzero-total-momentum (LOFF-like) paired components, whether the slightly smaller Bertsch parameter $\xi \approx 0.368(1)$ (from the 2024 FN-DMC simulation at $m_h/m_l = 6.5$) relative to equal-mass values $\approx 0.372$ is a physical effect (arising either from such nonzero-total-momentum components or from ordinary mass-imbalance corrections to the homogeneous s-wave superfluid) rather than a fixed-node or finite-range artifact, and how any such components modify the pairing structure and thermodynamics; (ii) whether exotic phases, such as spatially modulated LOFF states or induced p-wave pairing, appear, are stable, or are energetically competitive with the homogeneous s-wave paired superfluid anywhere in the zero-temperature phase diagram at this mass ratio; (iii) why the finite-range slope $c$ and the higher-order near-unitarity coefficients $\zeta, \nu$ so closely match the equal-mass values while $\xi$ differs, and whether these similarities persist at other mass ratios, interaction models, and smaller effective ranges. The minimum sufficient deliverable is a bias-controlled determination of the near-unitarity equation of state and pairing structure, i.e. sub-claim (i), together with a quantification of the methodological bias of the calculation (fixed-node error or sign-problem control) for energy coefficients, contact, and structural observables; the equation of state across the full crossover, sub-claims (ii) and (iii), and the connection of the zero-range predictions to the finite effective range and width of the experimental Li-K resonance may be sampled rather than exhaustively resolved.

References

1. S. Gandolfi, R. Curry, A. Gezerlis, 'BCS-BEC crossover of the strongly interacting 6Li-40K mixture', Phys. Rev. A 110, 043320 (2024), DOI 10.1103/PhysRevA.110.043320, arXiv:2407.12102, https://arxiv.org/abs/2407.12102
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