Three and four identical fermions near the unitary limit

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 $L^{\pi }=0^{+}$ 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 $A+A+A+A\leftrightarrow A_3+A$ are discussed.

Figure 1

The lowest few Born-Oppenheimer potential curves for the $(\uparrow \uparrow \downarrow )$ equal-mass three-body system are shown for the symmetries (a) $L^{\pi }=0^{+}$, (b) $L^{\pi }=1^{-}$, and (c) $L^{\pi }=2^{+}$. Each curve represents a different hyperradial potential with channel index $\nu$ that increases from the lowest to highest curve. 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 noninteracting 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 $r_0$ and shown on a logarithmic scale. For some of the higher degenerate channels, the potentials deviate at small hyperradius due to incomplete basis set convergence.

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 (a) $L^{\pi }=0^{+}$, (b) $L^{\pi }=1^{-}$, and (c) $L^{\pi }=2^{+}$. Each curve represents a different hyperradial potential with channel index $\nu$ that increases from the lowest to highest curve. The strength of the two-body interaction between all particles is scaled give an infinite $s$-wave scattering length. The horizontal dashed lines represent the noninteracting 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 (a) $L^{\pi }=0^{+}$, (b) $L^{\pi }=1^{-}$, and (c) $L^{\pi }=2^{+}$. Each curve represents a different hyperradial potential with channel index $\nu$ that increases from the lowest to highest curve. The strength of the two-body interaction between all particles is scaled give an infinite $s$-wave scattering length. The horizontal dashed lines represent the noninteracting 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 noninteracting potentials at large hyperradius. The strength of the two-body interactions between all particles is rescaled from the noninteracting limit up to the $s$-wave unitary limit ($a_s\to -\infty$). 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. From highest to lowest, the highest curve is the noninteracting 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 is qualitatively similar to the potentials shown here, and thus are not shown.

Figure 5

The lowest few Born-Oppenheimer potential curves for (a) the $(\uparrow \uparrow \downarrow )$ equal-mass three-body system for the $0^{+}$ symmetry, (b) the $(\uparrow \uparrow \downarrow \downarrow )$ equal-mass four-body system for the $1^{-}$ symmetry, and (c) the $(\uparrow \uparrow \uparrow \downarrow )$ equal-mass four-body system for the $2^{+}$ symmetry. The strength of the two-body interactions between all particles is rescaled from the noninteracting limit up to the $s$-wave unitary limit ($a_s\to -\infty$). From highest to lowest, the highest curve is the noninteracting case, the lowest curve is the $s$-wave unitary case, and a curve in between is for a finite $s$-wave scattering length. The solid curves are for the adiabatic potential and the dashed curves include the second-derivative nonadiabatic 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.

Figure 6

The long-range coefficient $C_{3,1}/r_0$ versus $a_s/r_0$ in the lowest Born-Oppenheimer potential for the (a) $(\uparrow \uparrow \downarrow )$, (b) $(\uparrow \uparrow \downarrow \downarrow )$, and (c) $(\uparrow \uparrow \uparrow \downarrow )$ equal-mass systems for the symmetries $L^{\pi }$=$0^{+}$ (circles), $1^{-}$ (squares), and $2^{+}$ (diamonds). For values of the $s$-wave scattering length near the unitary limit ($|a_s/r_0|>10$), $C_{3,\nu }$ grows in proportion to $a_s$, in accordance with Eq. (7). These results are for a single Gaussian two-body interaction of width $r_0$. The data for the other hyperspherical channels yield qualitatively similar results to the lowest channel in each symmetry, and thus are not shown here.

Figure 7

A comparison of the long-range coefficient $C_{3,1}$ versus $a_s$ in the lowest Born-Oppenheimer potential for the $(\uparrow \uparrow \downarrow )$ equal-mass three-body system shown for three different two-body interactions given in Eqs. (5). The parameters for the interactions are given in Table 1 for $s$ waves. The coefficient $C_{3,1}$ and the hyperradius $R$ are rescaled by the range of the single Gaussian interaction $r_0$. The set of three curves is labeled by the ratio of $a_s/r_0$ shown on the right-hand side. Universality kicks in for large hyperradius, as indicated by the different interactions converging to one curve for each scattering length.

Figure 8

Single-channel elastic phase shifts for collisions in the lowest hyperspherical continuum channel. Each curve is computed using a different two-body $s$-wave scattering length, where the ratio $a_s/r_0$ is given for a Gaussian interaction and displayed on the right. The low-energy behavior is governed by the characteristics of the long-range form of the hyperradial potential [see Eq. (7)]. The $1/R^3$ term in the long-range form gives rise to a linear dependence on wave number $k$ of the phase shift, in accordance with the Wigner threshold law. This linear dependence, derived using the Born approximation in Eq. (8), is shown as the black dashed lines.

Figure 9

A Tjon plot of the correlation between the universal three-body binding energy in the $L^{\pi }=1^{-}$ and the lowest four-body binding energy in the $L^{\pi }=0^{+}$ symmetry. Each point type corresponds to a different two-body interaction, given in the legend and labeled by the equation number. The parameters for the interactions are given in Table 1 for $p$ waves. The axes are rescaled by the trimer energy at the unitary limit to place all of the data sets on the same scale.

Figure 10

Lowest hyperradial potentials for the three- and four-identical fermionic systems in the spin-polarized state at the $p$-wave unitary limit. The lower inset plot shows these potentials on a smaller energy scale to highlight the asymptotic behavior of the lowest four-body potential at large hyperradius. The dashed lines show the expected asymptotic behavior as the potential approaches the trimer $+$ free particle threshold. The upper inset plot shows the three-body potential rescaled by $R^2$ with and without the second-derivative nonadiabatic coupling. This highlights the long-range behavior, specifically showing the negative, nonzero coefficient of $R^{-2}$ without the diagonal coupling added, indicating Efimov physics. However, this is not a true Efimov feature, as this coefficient vanishes after including the nonadiabatic coupling.

Figure 11

The hyperradial potentials for the four-fermion system in the $(\uparrow \uparrow \uparrow \uparrow )$ spin configuration. Each curve represents a different hyperradial potential with channel index $\nu$ that increases from the lowest to highest curve. The two-body interaction between $(\uparrow \uparrow )$ spins is set to the first $p$-wave unitary limit. For this set of two-body interactions, there is one two-body breakup threshold below the four-body continuum, labeled by the fragmentation of the spins and the angular momentum of the trimer.

Figure 12

The Born-Oppenheimer potentials for the four-fermion system in the $(\uparrow \uparrow \downarrow \downarrow )$ spin configuration. Each curve represents a different hyperradial potential with channel index $\nu$ that increases from the lowest to highest curve. The two-body interaction between $(\uparrow \uparrow )$ spins is set to the first $p$-wave unitary limit, while the $(\uparrow \downarrow )$ is set to the first $s$-wave unitary limit. For this set of two-body interactions, there is one two-body breakup threshold below the four-body continuum threshold, labeled by the fragmentation of the spins and the angular momentum of the trimer.

Figure 13

The hyperradial potentials for the four-fermion system in the $(\uparrow \uparrow \uparrow \downarrow )$ spin configuration. Each curve represents a different hyperradial potential with channel index $\nu$ that increases from the lowest to highest curve. The two-body interaction between $(\uparrow \uparrow )$ spins is set to the first $p$-wave unitary limit, while the $(\uparrow \downarrow )$ is set to the first $s$-wave unitary limit. For this set of two-body interactions, there are two two-body breakup thresholds below the four-body continuum, labeled by the fragmentation of the spins and the angular momentum of the trimer. The solid curves are the Born-Oppenheimer potentials and the dashed curves are the effective hyperradial potentials that include the diagonal nonadiabatic second-derivative coupling.