Nonuniversal bound states of two identical heavy fermions and one light particle

We study the behavior of the bound-state energy of a system consisting of two identical heavy fermions of mass $M$ and a light particle of mass $m$. The heavy fermions interact with the light particle through a short-range two-body potential with positive $s$-wave scattering length $a_s$. We impose a short-range boundary condition on the logarithmic derivative of the hyperradial wave function and show that, in the regime where Efimov states are absent, a nonuniversal three-body state cuts through the universal three-body states previously described by Kartavtsev and Malykh [O. I. Kartavtsev and A. V. Malykh, J. Phys. B 40, 1429 (2007)]. The presence of the nonuniversal state alters the behavior of the universal states in certain regions of the parameter space. We show that the existence of the nonuniversal state is predicted accurately by a simple quantum defect theory model that utilizes hyperspherical coordinates. An empirical two-state model is employed to quantify the coupling of the nonuniversal state to the universal states.

Figure 1

Hyperangular eigenvalue $s_0\left(R_0\right)$ as a function of the mass ratio $\kappa$ for $R_0/a_s=0.0001$. The dashed lines labeled I and II mark the mass ratios ${\kappa }_1$ and ${\kappa }_2$, at which the universal states with energies $E_{\text{u},1}$ and $E_{\text{u},2}$ first become bound.

Figure 2

Dotted, dashed, and dash-dotted lines show the scaled three-body energy $E/E_{\text{2b}}$ as a function of $\delta$ using $R_0/a_s=0.0001$, $0.0003$, and $0.001$, respectively, for (a) $\kappa =8.6$ and (b) $\kappa =10$. The solid horizontal lines show the energy $E_{\text{u},1}$ of the universal state and the solid vertical lines show the energy $E_{\text{nu}}$ of the nonuniversal state in the two-state model (see Sec. 4) for $R_0/a_s=0.001$.

Figure 3

Solid lines show the scaled three-body energies $E/E_{\text{2b}}$ as a function of $\delta$ for $\kappa =13$ and $R_0/a_s=0.0001$. The three-body resonance is located at ${\delta }_c\approx 0.15$.

Figure 4

The solid line shows the wave function $F_0\left(R\right)$ for $\kappa =10$, $R_0/a_s=0.0001$, $E/E_{\text{2b}}=-1.4$, and $\delta =-0.33$. As can be seen in Fig. 2b, this angle is close to ${\delta }_c$. For comparison, the inset shows the universal wave function for $\delta =1.53$ (i.e., away from the three-body resonance).

Figure 5

Critical short-range phase ${\delta }_c\left(R_0\right)$ as a function of $\kappa$. The solid, dotted, and dashed lines show the QDT prediction for $R_0/a_s=0.0001,0.0003$, and $0.001$, respectively. The circles show ${\delta }_c\left(R_0\right)$ obtained by analyzing the numerical solutions of the hyperradial Schrödinger equation (5) for $R_0/a_s=0.0001$. The inset shows a blowup of the large-$\kappa$ region.

Figure 6

Thick solid, dotted, and dashed lines show the quantity $(E-E_{\text{u},1})/E_{\text{2b}}$ as a function of $\delta -{\delta }_c$ for $R_0/a_s=0.0001$ and $\kappa =10,12.4$, and 13, respectively. The circles show the quantity $(E-E_{\text{u},2})/E_{\text{2b}}$ for $R_0/a_s=0.0001$ and $\kappa =13$. The thin dashed lines show the results of the two-state model.

Figure 7

Circles, diamonds, and squares show the off-diagonal coupling $\beta /E_{\text{2b}}$ used in the two-state model (11) as a function of $\kappa$ for $R_0/a_s=0.0001,0.0003$, and $0.001$, respectively, on a logarithmic scale. We used $E_{\text{u}}=E_{\text{u},1}$ in the two-state model. The dependence of $\beta /E_{\text{2b}}$ on the value of $R_0$ increases with increasing $\kappa$.