Many-body theory of the three-hyperfine-species narrow Feshbach resonance beyond the perturbative interchannel-Pauli regime

An ultracold Fermi gas across a narrow Feshbach resonance whose open channel (the pair of hyperfine species $a$ and $b$) and closed channel (the pair $a$ and $c$) share one hyperfine species exhibits an interchannel Pauli exclusion effect with no two-body counterpart. Zhu and Leggett (Phys. Rev. A 87, 023627 (2013)) built a perturbative treatment of this effect controlled by the small parameter $\zeta=\Delta_2^2/(\Delta_1\eta)\sim a_c/a_0$ (with $\Delta_1$ and $\Delta_2$ the open- and closed-channel pairing gaps, $a_c$ the closed-channel bound-state size, $a_0$ the interparticle spacing, and $\eta$ the Zeeman energy difference between the two channels), showing that the exclusion enters the gap and number equations through two density-linear overlap parameters $\lambda_1$ and $\lambda_2$ and shifts the fermionic spectrum and collective modes at first order in $\zeta$. They explicitly identified as open: (i) the genuine three-species many-body problem in the nonperturbative regime $a_c\gtrsim a_0$, where the $\zeta$-expansion and the short-range projection of the closed-channel correlation both fail and no solution is available; (ii) rebuilding the theory on an improved beyond-mean-field zeroth-order crossover solution; and (iii) microscopic determination or experimental calibration of $\lambda_1$ and $\lambda_2$. Their collective-mode analysis is limited to leading nontrivial order in $\zeta$ and contains no damping computation. The problem remains open.

Background

A magnetic Feshbach resonance in an ultracold dilute gas couples an open scattering channel of two atoms to a closed channel supporting a bound molecular state, with the energy difference between the two channels (the Zeeman energy difference $\eta$ in a magnetic field) tunable through resonance; sweeping it drives the BEC-BCS crossover, the continuous evolution from a condensate of tightly bound molecules (BEC side) to a BCS superfluid of long-range Cooper pairs (BCS side) as the s-wave scattering length $a_s$ passes through infinity (see the reviews of Giorgini, Pitaevskii, and Stringari, and of Chin, Grimm, Julienne, and Tiesinga). The two-body theory of the resonance has a characteristic energy scale $\delta_c$, the detuning at which the bound-state weight shifts from predominantly open-channel to predominantly closed-channel. For $E_F\ll\delta_c$ (broad resonance, with $E_F$ the Fermi energy of the free atoms) the closed-channel population is negligible at the many-body level and single-channel universal theories apply; for $E_F\gtrsim\delta_c$ (narrow resonance) the closed channel must be retained, which is the regime addressed by Zhu and Leggett. Real alkali resonances, such as $^6$Li at 834 G, have a further structural feature: the open-channel pair $(a,b)$ and the closed-channel pair $(a,c)$ share one of three hyperfine species, $a$. The two channels then have overlapping single-particle phase space for species $a$, and the Pauli exclusion principle forbids simultaneous occupation of the same single-particle states by atoms of the two channels. This interchannel Pauli exclusion has no counterpart in the two-body problem and is the central subject of the source work.

Two many-body effects of a non-negligible closed channel were known before it: pairing is measured relative to the Fermi surface rather than zero, shifting the effective detuning from resonance by $2\mu$ ($\mu$ the chemical potential), and particle number must be conserved separately in each channel, giving a second number equation. Zhu and Leggett added the shared-species effect. They used a two-channel BCS-type mean-field theory with a two-component order parameter $(\Delta_1,\Delta_2)$ for the open and closed channels, and treated the interchannel Pauli exclusion perturbatively. The small parameter is $\zeta=\Delta_2^2/(\Delta_1\eta)\sim(E_F/\eta)^{1/2}$, of order the ratio between the spatial extent $a_c$ of the uncoupled closed-channel bound state and the interparticle spacing $a_0$; the perturbation is justified only when $a_c\ll a_0$, so that the closed-channel two-body correlation is essentially the free bound-state wave function $\phi$. The exclusion then enters through a detuning shift $\lambda_1$ and a gap-equation correction $\lambda_2$, both overlap integrals of $\phi$ with open-channel occupation factors, both linear in the density, and therefore left as parameters to be calibrated from experiments at different densities. The fermionic excitation spectrum is the standard BCS quasiparticle (Bogoliubov) spectrum corrected at order $\zeta$. The collective modes (computed to leading nontrivial order in $\zeta$) comprise two gapped amplitude modes, an in-sync phase mode that is the Goldstone sound mode of the spontaneously broken global phase symmetry (its velocity follows the single-channel structure with an $O(\zeta)$ correction), and a gapped out-of-sync phase mode (gap of order the pair-breaking energy) that is unique to two-channel systems and reflects oscillation of the relative phase of the two order parameters.

The framework fails exactly in the regime it was designed to exclude. When $a_c\gtrsim a_0$, the closed-channel bound state spans the many-body length scale, its internal structure is deformed by the surrounding atoms, the short-range projection of the closed-channel correlation onto $\phi$ is invalid, and the expansion in $\zeta$ breaks down; in the authors' words the resonance then becomes a genuine three-species many-body problem with no simple solution available to their knowledge. In addition, the zeroth-order theory on which the perturbation is built is the simple BCS mean-field single-channel crossover, which is known not to be quantitatively adequate across the whole BEC-BCS crossover, and the collective modes are evaluated only at leading order without any damping analysis.

Problem Statement

Construct a quantitatively controlled many-body theory of the BEC-BCS crossover for a narrow Feshbach resonance whose open channel $(a,b)$ and closed channel $(a,c)$ share one hyperfine species, incorporating the interchannel Pauli exclusion effect without the perturbative small-parameter assumption. The single objective subsuming the directions explicitly identified by the source work is a quantitatively reliable description of this three-species two-channel system; its components are:

1. In the regime where the spatial extent $a_c$ of the closed-channel bound state is comparable to or larger than the interparticle spacing $a_0$ (equivalently, when the ratio $a_c/a_0$ or $E_F/\eta$ is not small), the short-range projection of the closed-channel correlation onto the two-body bound-state wave function and the expansion in $\zeta=\Delta_2^2/(\Delta_1\eta)\sim a_c/a_0$ both fail, and the genuine three-species many-body problem has no available solution. Provide a controlled solution (nonperturbative, or based on an alternative controlled expansion) of the coupled gap and channel number equations and of the fermionic excitation spectrum of the three-species two-channel model in this regime.
2. Replace the BCS-type mean-field (single-channel crossover) zeroth-order solution by an improved beyond-mean-field or numerically exact crossover solution, and re-evaluate the interchannel Pauli corrections on top of it, determining whether the perturbative construction of the narrow-resonance result remains valid on the improved zeroth-order theory.
3. Determine the interchannel Pauli shifts $\lambda_1$ (the detuning shift) and $\lambda_2$ (the gap-equation correction), defined as overlap integrals of the closed-channel bound-state wave function with open-channel occupation factors, from microscopic calculation beyond the approximations of the source work, or from a documented quantitative experimental calibration, in place of treating them as phenomenological density-linear parameters.

These components are jointly linked: each removes a stated limitation of the same perturbative framework for the same physical system. The sought solution must be a controlled solution valid when $a_c\gtrsim a_0$ whose $\zeta\to 0$ limit reproduces the published first-order-in-$\zeta$ corrections (component 1), with $\lambda_1$ and $\lambda_2$ determined microscopically or from a documented quantitative experimental calibration rather than as phenomenological density-linear parameters (component 3).

References

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2. G. Zhu, "BEC-BCS Crossover with Feshbach Resonance for Three-Hyperfine-Species Model", Ph.D. thesis, University of Illinois at Urbana-Champaign (2012) (companion document with detailed calculations, cited as ref. [15] of the paper above). arXiv:1212.6691. DOI: 10.48550/arXiv.1212.6691. https://arxiv.org/abs/1212.6691
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