Evolution of Efimov states

The Efimov phenomenon manifests itself as an emergent discrete scaling symmetry in the quantum three-body problem. In the unitarity limit, it leads to an infinite tower of three-body bound states with energies forming a geometric sequence. In this work, we study the evolution of these so-called Efimov states using relativistic scattering theory. We identify them as poles of the three-particle $S$ matrix in the complex energy plane, and we study how they transform from virtual states through bound states to resonances when we change the interaction strength. We dial the scattering parameters toward the unitarity limit and observe the emergence of the universal scaling of energies and couplings—a behavior known from the nonrelativistic case. We additionally find that Efimov resonances follow unusual, cyclic trajectories accumulating at the three-body threshold and then disappear at some values of the two-body scattering length. We propose a partial resolution to this “missing states” problem.

Figure 1

Left panel: ratios of subsequent binding energies vs $ma$ ($x$-axis). The horizontal axis for $\mathrm{\Delta }{E}_1/\mathrm{\Delta }{E}_2$ (dashed line) is rescaled by $\lambda$. We show the Efimov ratio, ${\lambda }^2$, as a horizontal, gray line. Right panel: normalized residues of the first three bound-state poles of the ${\mathcal{M}}_{\phi b}$ amplitude as functions of $ma$. The bottom red graph corresponds to the ground state. The upper two curves correspond to excited states (indistinguishable within the numerical precision). The scattering lengths for these residues are rescaled by appropriate powers of $\lambda$, as indicated in the legend.

Figure 2

Residues of the ${\mathcal{M}}_3$ amplitude at the first three trimer poles for $ma=-{10}^6$. Dashed black lines represent the analytic prediction of Ref. [90].

Figure 3

Trajectories of the first three trimer poles in the $(\kappa /m,1/ma)$ plane where $\kappa =\mathrm{sgn}\left(\mathrm{Re}\mathrm{\Delta }E\right)\sqrt{\left|m\mathrm{Re}\mathrm{\Delta }E\right|}$ (states are ordered by their distance from the $\phi b$ threshold). The $3\phi$ and $\phi b$ thresholds are shown explicitly as gray and orange lines. Solid lines denote physical bound states, while dashed ones denote either virtual bound states on the unphysical $\phi b$ sheet or resonances on the nearest $3\phi$ sheet. Stars denote the emergence of a virtual state from the logarithmic cut on the second $\phi b$ sheet. Circles denote the evolution of this virtual state onto a real bound state. Squares denote the further evolution of the state to three-body resonance. Insets show the behavior of trimers near the three-body threshold.

Figure 4

Trajectories of the first three resonances on the nearest unphysical Riemann sheet of the complex $\mathrm{\Delta }E$ plane. Energies of the second and third trimers are rescaled by ${\lambda }^2$ and ${\lambda }^4$, respectively. At large $\left|ma\right|$ and close to the threshold, all trajectories exhibit discrete scaling symmetry. As $\left|ma\right|$ decreases, the scaling symmetry breaks down, although for the second and third resonant states the discrepancy between the trajectories remains small.

Figure 5

Riemann surfaces of the three-body amplitude in $\mathrm{\Delta }E$. The trajectories of the trimers are shown, along with three bound-state positions for some $a$. On the nearest unphysical sheet, the second and third trimers exhibit the cyclic behavior as shown in Fig. 4. Mirror poles are found by continuing up to sheet $-1$. We postulate that the higher trimers come from the further unphysical sheets ($\ge +2$), where their cyclic behavior repeats.

Figure 6

Evolution of the “quirky” pole in the upper half-plane of the $+1$ Riemann sheet. The pole approaches the threshold from complex infinity. As $\left|ma\right|$ decreases, its trajectory forms a fractal pattern with rescaling quotient converging to ${\lambda }^2$.

Figure 7

Imaginary part of ${\mathcal{M}}_2\left(k^{\prime }\right)$ in the complex $k^{\prime }$ plane for $ma=-6$ and $s/m^2=9.01-0.01i$. Black lines represent branch cuts, and blue dots are branch points. Three example integration contours are shown. On the left panel, ${\mathcal{C}}_1$ (yellow, straight line) is the original integration interval of Eq. (3) defining the solution, ${\mathcal{M}}_3(p,k)$, on the physical Riemann sheet of the complex $E^2$ plane. To continue the three-body amplitude through the three-body cut, one deforms ${\mathcal{C}}_1\to {\mathcal{C}}_2$. Amplitude ${\mathcal{M}}_2$ in the integration kernel must be evaluated on its second sheet when $k^{\prime }$ belongs to the green, dashed piece of ${\mathcal{C}}_2$. On the right panel, we show an example contour, ${\mathcal{C}}_3$, defining the ladder amplitude on the $+2$ unphysical Riemann sheet of the $E^2$ plane.

Figure 8

Imaginary part of the amplitude $m^{2}d(p,k)$ in the complex $E^2$ plane for $ma=-8.1$ and ${\varepsilon }_p={\varepsilon }_k=2m^2$. The left panel represents the solution on sheet 0, while the right panel is obtained via the analytic continuation to the nearest unphysical sheet. The “quirky” trimer pole is visible in the right panel, in the upper left corner of the frame.

Figure 9

Inverse $-1/F_3^{\infty }$ as a function of $\mathrm{\Delta }E$ for $ma={10}^6$ (periodic, red curves). Pole positions of ${\mathcal{M}}_3$ are determined by crossing with ${\mathcal{K}}_3$. Two typical forms of ${\mathcal{K}}_3$ are presented: a constant ${\mathcal{K}}_3^{\left(0\right)}/m=2.5\times {10}^5$ represented by a horizontal, blue line, and the so-called isotropic approximation represented by a curved orange graph [105, 106]. For the latter: ${\mathcal{K}}_3^{\left(0\right)}/m={10}^5, {\mathcal{K}}_3^{\left(1\right)}/m=4.5\times {10}^9$, and $\mathrm{\Delta }=(E^2-{\left(E_{\text{thr}}^{\left(3\phi \right)}\right)}^2)/{\left(E_{\text{thr}}^{\left(3\phi \right)}\right)}^2$. Dots represent binding energies in the limit ${\mathcal{K}}_3=0$ for which ${\mathcal{M}}_3=\mathcal{D}$.

Figure 10

The $ma$ dependence of the normalized trimer residues. Real (left panel) and imaginary (right panel) part of the bound-state residue is shown as a red, solid line. Residues of resonance poles are presented as dashed lines without bands. The first resonance is to the right of the dashed, vertical line, while the second is to the left. The $+2$ sheet pole, obtained through the extrapolation procedure, is shown with an error band estimate. Around $ma=-8.71$ the poles approach the threshold, and all residues converge to the same value, around $-0.061$.