Three and four identical fermions near the unitary limit

TL;DR

This work analyzes three- and four-identical-fermion systems near $s$-wave and $p$-wave unitary limits using a hyperspherical Born–Oppenheimer approach with explicitly correlated Gaussians. It demonstrates universal long-range tails in the hyperradial potentials, notably a $1/R^3$ term proportional to the scattering length, and derives the associated universal coefficients and effective angular momenta that shape low-energy scattering. In the $s$-wave sector, universality governs continuum states across spin configurations, while in the $p$-wave sector a single tetramer bound state emerges with a linear trimer–tetramer correlation and a near-universal ratio $a_{p,4}^{0^+}/a_{p,3}^{1^-}\approx0.88$, offering a predictive handle on four-body recombination. These results illuminate how few-body fermionic systems near unitarity organize into universal structures, and quantify how short-range details influence bound-state spectra, especially for fermionic tetramers. $

Abstract

This work analyzes the three and four equal-mass fermionic systems near and at the $s$- and $p$-wave unitary limits using hyperspherical methods. The unitary regime addressed here is where the two-body dimer energy is at zero energy. For fermionic systems near the $s$-wave unitary limit, the hyperradial potentials in the $N$-body continuum exhibit a universal long-range $R^{-3}$ behavior governed by the $s$-wave scattering length alone. The implications of this behavior on the low energy phase shift are discussed. At the $p$-wave unitary limit, the four-body system is studied through a qualitative look at the structure of the hyperradial potentials at unitarity for the $L^π=0^{+}$ symmetry. A quantitative analysis shows that there are tetramer states in the lowest hyperradial potentials for these systems. Correlations are made between these tetramers and the corresponding trimers in the two-body fragmentation channels. Universal properties related to the four-body recombination process $\mathrm{A+A+A+A}\leftrightarrow \mathrm{A_3+A}$ are discussed.

Figures (13)

• Figure 1: The lowest few Born--Oppenheimer potential curves for the $(\uparrow\uparrow\downarrow)$ equal mass three--body system are shown for the symmetries $L^{\pi}$=$0^{+}$ in (a), $1^{-}$ in (b) and $2^{+}$ in (c). The two--body interactions between all particles are at the first $s$--wave unitary limit before the formation of an $s$--wave dimer. The horizontal dashed lines represent the non--interacting rescaled potentials with their effective angular momentum quantum numbers $l_{\mathrm{eff}}$ labeled to the right.
• Figure 2: The lowest few Born--Oppenheimer potential curves for the $(\uparrow\uparrow\downarrow\downarrow)$ equal mass four--body system are shown for the symmetries $L^{\pi}$=$0^{+}$ in (a), $1^{-}$ in (b) and $2^{+}$ in (c). The strength of the two--body interaction between all particles are scaled give an infinite $s$--wave scattering length. The horizontal dashed lines represent the non--interacting rescaled potentials with their effective angular momentum quantum numbers $l_{\mathrm{eff}}$ labeled to the right. The hyperradius has been rescaled by the range of the Gaussian interaction $r_0$. For some of the higher degenerate channels, the potentials deviate at small hyperradius due to incomplete basis set convergence.
• Figure 3: The lowest few Born--Oppenheimer potential curves for the $(\uparrow\uparrow\uparrow\downarrow)$ equal mass four--body system are shown for the symmetries $L^{\pi}$=$0^{+}$ in (a), $1^{-}$ in (b) and $2^{+}$ in (c). The strength of the two--body interaction between all particles are scaled give an infinite $s$--wave scattering length. The horizontal dashed lines represent the non--interacting rescaled potentials with their effective angular momentum quantum numbers $l_{\mathrm{eff}}$ labeled to the right. The hyperradius has been rescaled by the range of the Gaussian interaction $r_0$. For some of the higher degenerate channels, the potentials deviate at small hyperradius due to incomplete basis set convergence.
• Figure 4: The lowest few Born--Oppenheimer potential curves for the $(\uparrow\uparrow\downarrow)$ equal mass three--body system are shown for the $1^{-}$ symmetry. The solid curves correspond to potentials that exhibit a reduced value of $l_{\mathrm{eff}}$ at the unitary limit for large hyperradius, whereas the dashed potentials go to the non--interacting potentials at large hyperradius. The strength of the two--body interactions between all particles are rescaled from the non--interacting limit up to the $s$--wave unitary limit ($a_s\rightarrow-\infty$). For each set of curves representing a different channel $\nu$, an increase in scattering length on the negative side corresponds to a curve in the set. Reading from highest to lowest, the highest curve is the non--interacting potential, the lowest is the hyperradial potential at the $s$--wave unitary limit, and a potential in between is for a finite scattering length. The hyperradius has been rescaled by the range of the Gaussian interaction $r_0$. The structure of the potential curves for different systems and symmetries are qualitatively similar to the potentials shown here, thus are not shown.
• Figure 5: The lowest few Born--Oppenheimer potential curves for the $(\uparrow\uparrow\downarrow)$ equal mass three--body system for the $0^{+}$ symmetry in (a), the $(\uparrow\uparrow\downarrow\downarrow)$ equal mass four--body system for the $1^{-}$ symmetry in (b) and the $(\uparrow\uparrow\uparrow\downarrow)$ equal mass four--body system for the $2^{+}$ symmetry in (c). The strength of the two--body interactions between all particles are rescaled from the non--interacting limit up to the $s$--wave unitary limit ($a_s\rightarrow-\infty$). The solid curves are for the adiabatic potential and the dashed curves include the second--derivative non--adiabatic correction. The hyperradius has been rescaled by the range of the Gaussian interaction $r_0$. The curves for the other symmetries are qualitatively similar and not shown here.