Crossover trimers connecting continuous and discrete scaling regimes

For a system of two identical fermions and one distinguishable particle interacting via a short-range potential with a large $s$-wave scattering length, the Efimov trimers and Kartavtsev-Malykh trimers exist in different regimes of mass ratio. The Efimov trimers are known to exhibit a discrete scaling invariance, while the Kartavtsev-Malykh trimers feature a continuous one. We point out that a third type of trimer, “crossover trimers,” exists universally regardless of short-range details of the potential. These crossover trimers have neither discrete nor continuous scaling invariance. We show that the crossover trimers continuously connect the discrete and continuous scaling regimes as the mass ratio and the scattering length are varied. We identify the regions for the Kartavtsev-Malykh trimers, Efimov trimers, crossover trimers, and nonuniversal trimers in terms of the mass ratio and the $s$-wave scattering length by investigating the scaling property and universality (i.e., independence on short-range details) of the trimers.

Figure 1

Schematic of the energy spectrum for (a) Kartavtsev-Malykh (KM) trimers (thick lines) and (b) Efimov trimers (thick curves), where ${\mu }_{FL}=m_F{m}_L/\left(m_L+m_F\right)$, $a_s$ is the $s$-wave scattering length, and $\Lambda$ is the three-body parameter. The rightmost thin yellow (gray) lines show the dimer binding energy.

Figure 2

(a)–(c) Schematics of the energy spectra for different regimes of the mass ratio. Curves are deformed from the real ones for the sake of illustration. The crossover trimer region, Kartavtsev-Malykh (KM) trimer region, Efimov trimer region, and nonuniversal region are distinguished according to their scaling properties (see Sec. 3 for their identification) and are drawn as thick orange (gray) curves in (a)–(c), thick blue (dark gray) curves in (a), thick green (dark gray) curves in (c), and black dotted curves in (a)–(c), respectively. The dashed curves in (a) and (c) are the ones as predicted from the continuous scaling law of the KM trimers and the discrete scaling law of the Efimov trimers, respectively. The dots on the positive-$a_s$ side indicate the fermion-dimer dissociation points at which the trimer dissociates into a fermion and a dimer, and the fermion-dimer $p$-wave scattering volume diverges. In panel (c), only four levels of the Efimov series are shown.

Figure 3

Regions of (a) the ground-state, (b), (c) first-excited, and (d) second-excited trimers as a function of the mass ratio and ${\left(\Lambda a_s\right)}^{-1}$, where $\Lambda$ is the three-body parameter and $a_s$ is the $s$-wave scattering length. For the ground-state and first-excited trimers, there are four regimes: the Efimov, KM, crossover, and nonuniversal. For the second-excited trimer, only the Efimov region appears. A trimer dissociates into a fermion and a dimer at the fermion-dimer $p$-wave resonance (red curve on the positive-$a_s$ side), while it dissociates into three particles at the three-body resonance line (red curve on the negative $a_s$ side). The blue dashed curves and black dotted curves correspond to $q_n=0.40$ [see Eq. (6)] and $r_n=0.40$ [see Eq. (5)], which delimit the Efimov region and KM trimer region, respectively. The black dashed-dotted curves correspond to $s_n=0.90$ [see Eq. (9)], which separates the crossover and nonuniversal (i.e., model-dependent) trimer regions. The color contour is used for the sake of clarity. The nonuniversal region increases with increasing mass ratio, and the Efimov region disappears for the mass ratio well above 50 (see discussions in Sec. 4).

Figure 4

(a), (b) Spectra of trimers with several different mass ratios $\alpha =m_F/m_L$ as measured from the dimer binding energy for (a) the ground-state trimer and (b) the first-excited trimer. The insets show the energy of the trimers as measured from the vacuum, together with the dimer binding energy (red solid curve). (c), (d) Contour plots of $r_n$ defined in Eq. (5) for (c) the ground-state trimer and (d) the first-excited trimer. The red solid curve shows the fermion-dimer dissociation threshold, while the black-dotted curve is the $r_n=0.40$ curve, which delimits the region in which the continuous scaling invariance is well satisfied (i.e., the KM-trimer regime).

Figure 5

(a) Energy spectra of the ground-state, first-excited, and second-excited trimers for $m_F/m_L=20.0$ and $m_F/m_L=30.0$. The binding energy is measured from the three-body continuum. For the first- and second-excited trimers, the radial scaling transformation has been performed. On the positive-$a_s$ side, the fermion-dimer dissociation points are shown for each trimer state. (b) Energy spectra as measured from the dimer binding energy for $m_F/m_L=20.0$ and $m_F/m_L=30.0$. The same radial scaling transformation is performed as in (a).

Figure 6

(a) Binding energy of the ground to fourth-excited trimers at the unitarity limit. The radial scaling transformation has been performed. (b) Ratio of the binding energy between the adjacent trimer levels at the unitarity limit. (c) Ratio of the $s$-wave scattering length at which the trimer dissociates into three particles. (d) Ratio of the $s$-wave scattering length at which a trimer dissociates into a fermion and a dimer. In (b)–(d), the universal scaling ratio obtained from Eq. (3) is shown as black solid curves.

Figure 7

Energy of trimers as measured from the dimer binding energy for (a) the ground-state trimer at $m_F/m_L=10.0$ and (b) the first-excited trimer at $m_F/m_L=13.3$. The energy for the sharp momentum cutoff (red solid curve) and Gaussian momentum cutoff (blue dashed curve) are compared. The energy of the Gaussian cutoff after a rescaling of the three-body parameter is also shown as green dashed-dotted curves. The scaling factors are taken to be 1.85 for (a) and 1.90 for (b). The black dotted lines are the linear fits in the large-$a_s$ region.

Figure 8

(a) Ratio of binding energy between adjacent trimer levels at the unitarity limit. Two models are compared: the STM equation with the sharp momentum cutoff and the Gaussian cutoff. (b) Energy spectra from the ground-state trimer to the fourth-excited trimer at $m_F/m_L=30.0$ for the sharp momentum cutoff (blue dash-dotted curves) and the Gaussian momentum cutoff (green dashed curves). The radial scaling transformation has been performed, so that the energy spectra with the different cutoffs are superimposed onto universal curves. In the inset, the energy spectra before the radial scaling transformation are shown. (c) Ratio of $s$-wave scattering length at which the trimer dissociates into three particles. (d) Ratio of the $s$-wave scattering length at which the trimer dissociates into a fermion and a dimer. In (c) and (d), the universal scaling curves between the two models obtained from Fig. 8a are shown as the black solid curves.

Figure 9

$p$-wave scattering volume $v_p^{\left(\mathrm{AD}\right)}$ of the atom-dimer scattering for ${}^6$Li-${}^{53}$Cr (red solid curve, $m_F/m_L=8.80$), ${}^7$Li-${}^{87}$Sr (green dashed curve, $m_F/m_L=12.39$), and ${}^7$Li-${}^{173}$Yb (blue dotted curve, $m_F/m_L=24.65$), calculated with the STM equation with the sharp momentum cutoff. In the inset, the $s$-wave scattering length of the atom-dimer scattering for each atomic combination is shown. The points show the $p$-wave atom-dimer resonance positions for each atomic combination.