Universality in a one-dimensional three-body system

We study a heavy-heavy-light three-body system confined to one space dimension. Both binding energies and corresponding wave functions are obtained for (i) the zero-range and (ii) two finite-range attractive heavy-light interaction potentials. In the case of the zero-range potential, we apply the method of Skorniakov and Ter-Martirosian to explore the accuracy of the Born-Oppenheimer approach. For the finite-range potentials, we solve the Schrödinger equation numerically using a pseudospectral method. We demonstrate that when the two-body ground-state energy approaches zero, the three-body bound states display a universal behavior, independent of the shape of the interaction potential.

Figure 1

Jacobi coordinates $x$ and $y$ for the three particles confined to 1D.

Figure 2

Formation of three-body bound states explained by the two BO potentials $u_{\pm }=u_{\pm }\left(Y\right)$, Eq. (17), as a function of the relative distance $Y$ between the two heavy particles. Only the lower curve, corresponding to $u_{+}=u_{+}\left(Y\right)$, represents an attractive potential for the heavy particles and thus supports three-body bound states.

Figure 3

Mass ratio $M/m$ determining the number $n_{\max }$ of three-body bound states (blue line) obtained numerically from Eq. (19) within the BO approximation together with the semiclassical lower bound (orange line), Eq. (22). Increasing $M/m$ leads to more three-body bound states.

Figure 4

Lowest four (a)–(d) three-body wave functions ${\tilde{\psi }}_n^{★}={\tilde{\psi }}_n^{★}(X,Y)$ corresponding to $n=0,1,2,3$ obtained by the integral equation of Skorniakov and Ter-Martirosian for a mass ratio $M/m=20$, together with corresponding contour plots below. In addition, we depict the lines of interaction, that is $X\pm Y/2=0$, as black lines inside the contour plots. Due to the nondifferentiability of the delta potential, the wave functions show a kink when crossing those lines perpendicularly. All states share the same symmetry with respect to the line $X=0$, whereas the wave functions alter from even to odd with respect to the line $Y=0$. The transformation $Y\to -Y$ represents the exchange of the two heavy particles and hence determines whether they are of bosonic (even case) or of fermionic (odd case) character. The symmetry with respect to the line $X=0$ can be understood within the BO picture. Indeed, the lower-lying light-particle BO wave function ${\phi }_{+}={\phi }_{+}\left(X\right|Y)$, given by Eq. (18) and leading to three-body bound states, is even. Higher excited states have increased size, as indicated by the different scales in the plots.

Figure 5

Accuracy of the BO ground-state energies when compared to the corresponding STM values as a function of the mass ratio $M/m$. Here we depict (a) the energies ${\epsilon }_0$ of the lowest three-body bound state, obtained via the approximate BO approach (blue dots), Eq. (19), and the exact STM integral equation (yellow diamonds), Eq. (28). The BO approach underestimates the bound-state energies, hence all blue dots lie below their yellow partners. With increasing mass ratio $M/m$ the blue dots approach the yellow ones, that is the BO approximation becomes more accurate for larger $M/m$. This behavior is further confirmed (b) by the monotonically decreasing relative error $\delta {\epsilon }_0$ (black squares), defined by Eq. (31).

Figure 6

Comparison between BO approximation and exact STM approach based on the relative deviation in the three-body energy spectrum (top) together with fidelities between bound-state wave functions (bottom). For the four lowest ($n=0,1,2,3$) bound states, we display the relative deviation $\delta {\epsilon }_n$, Eq. (31), of the binding energies (top) and the fidelity ${\mathcal{F}}_n$, Eq. (32), of the corresponding wave functions (bottom) as a function of the mass ratio $M/m$. We distinguish the cases of two bosonic (a), (c) and two fermionic (b), (d) heavy particles. The relative deviation $\delta {\epsilon }_n$ decreases monotonically from almost $20\%$ for $M/m=1$ to less than $2\%$ for $M/m=25$. Higher excited states always show a higher deviation in the binding energy. For all states, the fidelity is increasing monotonically with increasing $M/m$, starting from 0.988 for $M/m=1$ up to 0.999 for $M/m=25$. However, no obvious dependence on $n$ is visible for $M/m\le 25$. For $n=2,3$ the fidelities near mass ratios where the corresponding three-body bound state appears are not sufficiently converged and hence omitted in the picture.

Figure 7

Improved bound-state energies ${\overline{\epsilon }}_n^{\left(\mathrm{BO}\right)}$ defined by Eq. (33) (red symbols), ${\epsilon }_n^{\left(\mathrm{BO}\right)}$ (blue symbols), and ${\epsilon }_n^{★}$ (yellow symbols) as a function of $M/m$. The energies are depicted by dots for the ground state ($n=0$), and by diamonds for the first ($n=1$) excited state. The deviation between ${\overline{\epsilon }}_n^{\left(\mathrm{BO}\right)}$ and ${\epsilon }_n^{★}$ is reduced by up to an order of magnitude compared to the result in zero order between ${\epsilon }_n^{\left(BO\right)}$ and ${\epsilon }_n^{★}$, presented in Fig. 6. For the first excited state, the red diamonds lie on top of the yellow ones, indicating that the values including the diagonal correction are almost indistinguishable from the exact ones.

Figure 8

Asymptotic behavior of the two-body ground-state energy ${\mathcal{E}}_{\mathrm{g}}^{\left(2\right)}$ as a function of the potential depth $\left|v_0\right|$. The energy of a weakly bound ground state, that is $\left|{\mathcal{E}}_{\mathrm{g}}^{\left(2\right)}\right|\ll 1$, shows a quadratic dependence on $\left|v_0\right|$ for different interaction potentials, illustrated for the case of a Gaussian-shaped potential $f_{\text{G}}$, Eq. (35), and for a potential $f_{\text{L}}$ being characterized by the cube of a Lorentzian, Eq. (36), by empty blue and filled red circles, respectively. The dashed red and blue lines are given by Eqs. (38) and (39) accordingly, and confirm this dependency.

Figure 9

Universal behavior of the 1D three-body system illustrated by the scaled three-body binding energy ${\epsilon }_n$, Eq. (6) (top), and the fidelity ${\mathfrak{F}}_n$, Eq. (41) (middle and bottom), as a function of the two-body binding energy ${\mathcal{E}}_{\mathrm{g}}^{\left(2\right)}$ for three different interaction potentials: $f_{\mathrm{G}}$, Eq. (35) (empty blue symbols), $f_{\mathrm{L}}$, Eq. (36) (filled red symbols), and $f_{\delta }$, Eq. (12) (gray lines). We display the first four energies (top) and fidelities (middle and bottom) for (a), (c) bosonic ($n=0,2$) and (b), (d) fermionic ($n=1,3$) heavy particles for the mass ratio $M/m=20$. Close to a resonance of the heavy-light system, that is for ${\mathcal{E}}_{\mathrm{g}}^{\left(2\right)}\to 0$, each energy ${\epsilon }_n$ and fidelity ${\mathfrak{F}}_n$ with $n=0,1,2,3$ approaches the value determined by the contact interaction $f_{\delta }$. Thus we observe for all presented states a universal behavior, independent of the shape $f$ of the interaction potential.