Universal trimers with p-wave interactions and the faux-Efimov effect

TL;DR

This work identifies and characterizes a family of $p$-wave universal trimers with symmetry $L^{\Pi}=1^{\pm}$ across several fermionic configurations, including two-component and three-equal-mass systems. It employs the hyperspherical adiabatic method to compute Born-Oppenheimer potentials $W_\nu(R)$, including the diagonal Born-Huang correction $Q_{\nu\nu}(R)$ and nonadiabatic couplings, and extracts the long-range exponents $l_{e,\nu}$ to determine threshold behavior. Universality is tested by modeling two-body interactions with Lennard-Jones potentials to tune $a_s$ and $V_p$ across multiple poles, revealing robust $L^{\Pi}=1^{+}$ trimers and, in certain regimes, a faux-Efimov channel for $L^{\Pi}=1^{-}$ that vanishes when diagonal corrections are included. The results have implications for three-body recombination and lifetimes in ultracold fermionic mixtures and provide a framework for observing universal trimers and associated resonances in experiments. The analysis highlights how the asymptotic $R^{-2}$ coefficient governs threshold laws and shapes the three-body continuum in these systems.

Abstract

An unusual class of $p$-wave universal trimers with symmetry $L^Π=1^{\pm}$ is identified, for both a two-component fermionic trimer with $s$- and $p$-wave scattering length close to unitarity and for a one-component fermionic trimer at $p$-wave unitarity. Moreover, fermionic trimers made of atoms with two internal spin components are found for $L^Π=1^{\pm}$, when the $p$-wave interaction between spin-up and spin-down fermions is close to unitarity and/or when the interaction between two spin-up fermions is close to the $p$-wave unitary limit. The universality of these $p$-wave universal trimers is tested here by considering van der Waals interactions in a Lennard-Jones potential with different numbers of two-body bound states; our calculations also determine the value of the scattering volume or length where the trimer state hits zero energy and can be observed as a recombination resonance. The faux-Efimov effect appears with trimer symmetry $L^Π=1^{-}$ when the two fermion interactions are close to $p$-wave unitarity and the lowest $1/R^2$ coefficient gets modified, thereby altering the usual Wigner threshold law for inelastic processes involving 3-body continuum channels.

Figures (8)

• Figure 1: (color online). Shown are the lowest effective adiabatic potential curves versus hyperradius for several different $p$-wave scattering volumes ($V_{p}$) in the van der Waals length unit $r_{\text{vdW}}$, for the system $\downarrow\uparrow\uparrow$ with the interaction between spin-up and spin-down fermion set at the $s$-wave unitary limit with $L^{\Pi}=1^-$. As $\lvert V_p \rvert$ gets larger, the potential curve should approach the three-body threshold whose value closes the $l_e=0$ in the adiabatic potential energy curve. The inset plots the behavior of depth of effective adiabatic potential curves versus the various scattering volume $V_p$.
• Figure 2: (color online). (a) Three-body Born-Oppenheimer potential curves are presented for two spin-up and one spin-down fermion ($\uparrow\downarrow\uparrow$) with symmetry $L^{\Pi}=1^{-}$. The interaction between spin-up and spin-down fermions is set at the $s$-wave unitary limit with no deep dimers in different spin states, and the two spin-up fermionic interaction is set at the $2^{\text{nd}}$$p$-wave pole, which has deep $p$-, $f$- and $h$-wave bound states. The letter labeling the potential curves converging to a negative energy represents the angular momentum quantum number $l$ of the dimer, and the number labels the relative angular momentum between the third atom and the dimer $l'$. The inset shows the detail of Born-Oppenheimer potential curves in the short range. (b) Shown is the derivative of the sum of eigenphase shifts as a function of energy which is proportional to the time delay and corresponds to a resonance that is a universal $p$-wave trimer; in this case, and the resonance peak is located around $E=-0.129\, E_{\text{vdW}}$.
• Figure 3: (color online). (a) Three-body Born-Oppenheimer potential curves for two spin-up and one spin-down fermion ($\uparrow\downarrow\uparrow$) with total angular momentum and parity $L^{\Pi}=1^{-}$. The interaction between spin-up and spin-down fermions is set at the $s$-wave unitary limit with no deep opposite spin dimers, and the two spin-up fermion interaction is set at the $3^{\text{rd}}$$p$-wave pole, which has deep $p$-, $f$-, $h$- and $j$-wave bound states. The letter represents the angular momentum quantum number $l$ of the dimer, and the number labels the angular momentum $l'$ between the third atom and the dimer. The inset shows expanded details of the Born-Oppenheimer potential curves in the short range. (b) The derivative of the sum of eigenphase shift is plotted as a function of energy near the universal $p$-wave trimer, and the peak is located around $E=-0.129\, E_{\text{vdW}}$. The resonance decay width in van der Waals energy units is also indicated on the figure.
• Figure 4: (color online). The lowest effective adiabatic potential curves are shown versus the hyperradius for several different $p$-wave scattering volumes ($V_p$) in van der Waals units of length and energy, for the system $\downarrow\uparrow\uparrow$ that has weak interaction between the two spin-up fermions for symmetry $L^{\Pi}=1^+$. As $\lvert V_p \rvert$ gets larger, the potential curve approaches the three-body threshold with a centrifugal barrier strength close to the value implied by the reduced value $l_e=1$ in the adiabatic potential energy curve. The inset shows the evolution of the depth of the effective adiabatic potential curves near their minima.
• Figure 5: (color online). Shown are $p$-wave universal trimer state energies for the system of one spin-down and two spin-up fermions ($\uparrow\downarrow\uparrow$), plotted versus the inverse of the $p$-wave scattering length ($a_p \equiv V_{p}^{1/3}$). The $V_p(a_p)$ and $V'_p(a'_p)$ represent respectively the $p$-wave scattering volume (or length) between unequal spin fermion and same spin fermions. Circles (solid blue) show the two-body $p$-wave bound state. Squares (solid gold) are for fixed $V'_p=-2\, r_{\text{vdW}}^{3}$ and with $V_p$ tuned. Inverted-triangles are with $V'_p=V_p$ being tuned. Diamonds (solid green) are for a fixed $V'_p=\infty\, r_{\text{vdW}}^{3}$ and with $V_p$ tuned. Triangles (solid red) are for fixed $V_p=\infty\, r_{\text{vdW}}^{3}$ and with $V'_p$ tuned. The open (red) circles are for three spin-up fermion and all interactions $V_p$ tuned simultaneously. (a), trimer symmetry $L^{\Pi}=1^+$. (b), trimer symmetry $L^{\Pi}=1^-$