Efimov physics implications at $p$-wave fermionic unitarity

TL;DR

This work investigates Efimov physics in three-fermion systems at $p$-wave unitarity using Lennard–Jones interactions within the adiabatic hyperspherical framework. By solving the fixed-$R$ angular problem and incorporating the diagonal adiabatic correction $Q_{\nu\nu}(R)$, the authors identify $p$-wave unitary channels across multiple spin configurations and symmetries, revealing universal long-range potentials without a true Efimov spectrum. The results demonstrate that $Q_{\nu\nu}(R)$ is essential to obtaining physically meaningful adiabatic potentials, converting apparent Efimov behavior seen in Born–Oppenheimer pictures into non-Efimovian, universal channels with $l_e(l_e+1)$ near integers. Across various two-body poles and potential shapes, the adiabatic curves exhibit robust universality, with implications for threshold laws and recombination in unitary Fermi gases.

Abstract

Efimov physics at $p$-wave unitarity for three equal mass fermions in multiple symmetries interacting via Lennard-Jones potentials is predicted to modify the long range interaction potential energy, but without producing a true Efimov effect. This analysis treats the following total orbital angular momenta and parities, $J^Π=0^{+}, 1^{+}, 1^{-}$ and $2^{-}$, for either three spin-polarized fermions ($\uparrow \uparrow \uparrow $), or two spin-up and one spin-down fermion ($\downarrow \uparrow \uparrow $). Our results for the long range interaction in some of those cases agree with previous work by Werner and Castin and by Blume {\it et al.}, namely in cases where the $s$-wave scattering length goes to infinity. The present results extend those calculated interaction energies to small and intermediate hyperradii comparable to the van der Waals length, and we consider additional unitarity scenarios where the $p$-wave scattering volume approaches infinity. The crucial role of the diagonal hyperradial adiabatic correction term is identified and characterized.

Figures (8)

• Figure 1: (color online). Shown are the lowest adiabatic potential curves versus hyperradius for several different $s$-wave scattering lengths ($a_{s}$) in the van der Waals length unit $r_{\text{vdW}}$, for the system ($\downarrow\uparrow\uparrow$) with $J^{\Pi}=1^{-}$. As $|a_{s}|$ gets larger, the potential curve should approach the red dotted line whose value equals the coefficient of $1/(2\mu R^{2})$ in the adiabatic potential energy curve.
• Figure 2: (color online). Shown are the adiabatic potential curves for the first to sixth channels versus hyperradius for the ($\downarrow\uparrow\uparrow$) system. The interaction between the two fermions in the same spin state has been set at the $p$-wave unitary limit ($V_{p} \rightarrow \infty$), and the interactions between fermions in different spin states have also been set at the $s$-wave unitary limit ($a_{s} \rightarrow \infty$), for the various symmetries $J^{\Pi}$. (a) and (b), the solid (blue) curve corresponds to the $p$-wave unitary channel and the coefficient of $1/(2\mu R^{2})$ is approximately $2$. (c), the solid (blue) curve and dash-dot (green) curve shows $p$-wave universal property and their coefficients of $1/(2\mu R^{2})$ are computed here to have the values $\approx 0.02$ and $\approx 6$, respectively. (d), the lowest adiabatic potential curve (solid curve) represents the $p$-wave unitary channel and the asymptotic $1/(2\mu R^{2})$ coefficient is calculated to have the value $\approx 6$.
• Figure 3: (color online). Shown are the adiabatic potential curves for the first to sixth channels versus hyperradius for the two-component 3-fermion system ($\downarrow\uparrow\uparrow$), with the interaction between fermions in different spin states set at the $p$-wave unitary limit ($V_{p} \rightarrow \infty$) and the interaction between the same spin state fermions set at a weak value with $p$-wave scattering volume close to $V_{p}=-2\, r_{\text{vdW}}^{3}$, for several symmetries $J^{\Pi}$. (a) and (b), the solid (blue) curve corresponds to the $p$-wave unitary channel and the coefficient of $1/(2\mu R^{2})$ is approximately $2$. (c), the solid (blue) curve and dashed (orange) curve exhibit the $p$-wave universal property, and their asymptotic coefficients of $1/(2\mu R^{2})$ are computed here to have the values $\approx 0$ and $\approx 6$, respectively. (d), the lowest adiabatic potential curve (solid curve) represents the $p$-wave unitary channel and the asymptotic $1/(2\mu R^{2})$ coefficient is calculated to have the value $\approx 6$.
• Figure 4: (color online). Shown are the adiabatic potential curves for the first to sixth channels versus hyperradius for the fermions in the same and different spin state both at the $p$-wave unitary limit ($V_{p} \rightarrow \infty$) for the various symmetries $J^{\Pi}$. (a) and (b), the solid (blue) and dashed (orange) curve corresponds to the $p$-wave degenerate unitary channel and the coefficient of $1/(2\mu R^{2})$ is approximately $2$. (c), the solid (blue) curve and dashed (orange) curves show the $1^{\text{st}}$ degenerate $p$-wave universal property and their coefficients of $1/(2\mu R^{2})$ are computed here to have the values $\approx 0.01$. The dash-dotted (green) curve and dotted (red) curve represent the $2^{\text{nd}}$ degenerate $p$-wave unitary channels which coefficients of $1/(2\mu R^{2})$ are closed to $6$. (d), the two lowest adiabatic potential curves (solid and dashed curves) represent the degenerate $p$-wave unitary channels and the $1/(2\mu R^{2})$ coefficient is calculated to have the value $\approx 6$.
• Figure 5: (color online). Shown are the adiabatic potential curves for the first to sixth channels versus hyperradius for the single-component 3-fermion system ($\uparrow \uparrow \uparrow$) with their two-body interactions set at the $p$-wave unitary limit ($V_{p} \rightarrow \infty$) for the various symmetries $J^{\Pi}$. (a) and (b), the solid (blue) curve corresponds $p$-wave unitary channel and the coefficient of $1/(2\mu R^{2})$ is approximately $2$. (c), the solid (blue) and the dashed (orange) curve show the $p$-wave universal property, with asymptotic coefficients of $1/(2\mu R^{2})$ computed here to have the values $\approx 0.01$ and $\approx 6$, respectively. (d), the lowest adiabatic potential curve (solid curve) represents the $p$-wave unitary channel and the $1/(2\mu R^{2})$ coefficient is calculated to have the value $\approx 6$.