Onset of a Limit Cycle and Universal Three-Body Parameter in Efimov Physics

The Efimov effect is the only experimentally realized universal phenomenon that exhibits the renormalization-group limit cycle with the three-body parameter parametrizing a family of universality classes. Recent experiments in ultracold atoms have unexpectedly revealed that the three-body parameter itself is universal when measured in units of an effective range. By performing an exact functional renormalization-group analysis with various finite-range interaction potentials, we demonstrate that the onset of the renormalization-group flow into the limit cycle is universal, regardless of short-range details, which connects the missing link between the two universalities of the Efimov physics. A close connection between the topological property of the limit cycle and few-body physics is also suggested.

Figure 1

Energy spectrum of three-identical bosons with resonant interaction (not to scale). The abscissa shows the inverse $s$-wave scattering length $a^{-1}$, and the ordinate shows the square root of the energy eigenvalue multipled by its sign. The Efimov states are related to one another by the universal scaling factor of $e^{2\pi /s_0}\simeq 22.7^2$ with $s_0\simeq 1.00624$. The three-body parameters that set the energy scale of the lowest-lying Efimov state are labeled here as $a_{-}$ and ${\kappa }^{*}$. Here, $a_{-}$ denotes the $s$-wave scattering length at the triatomic resonance and ${\kappa }^{*}$ is the binding wave number of the lowest-lying Efimov state at unitarity. They are uniquely related to each other.

Figure 2

Exact RG flows of the three-body coupling constant $g_3$. The abscissa shows the logarithm of the RG scale $\ln (k{r}_{\mathrm{eff}})$ in units of the effective range $r_{\mathrm{eff}}$, and the ordinate shows the three-body coupling constant $g_3$. Different curves correspond to different interaction potentials: Gaussian (circle), van der Waals (diamond), Yukawa (triangle), and square-well (square) potentials. The universal onset point of the limit cycle coincides with the universal value of the three-body parameter ${\kappa }^{*}{r}_{\mathrm{eff}}=0.49$.

Figure 3

Diagrammatic representations of the exact FRG equations for (a) the one-body sector and (b) the two-body sector. The derivative $\widetilde{{\partial }_k}$ acts only on a regulator which is contained in the internal propagator. (c) Decomposition of the four-point 1PI vertex into the total-momentum and the relative-momentum parts.

Figure 4

(a) Decomposition of the six-point 1PI vertex, which describes the three-body scattering process. The double lines and the shaded vertices are the same as in Fig. 3. The curly brackets mean symmetrization with respect to external lines. (b) Definition of the particle-dimer scattering amplitude $T_k$ represented as a square vertex. (c) Integral form of the exact FRG equation for the three-body sector. Integration of the RG equation with respect to the cutoff scale $k$ has been analytically performed.

Figure 5

Schematic illustration of the limit cycle on the space of the three- and four-body coupling constants $g_3$ and $g_4$. A torus of $g_3\text{−}{g}_4$ space can be constructed by enclosing the space periodically. The limit cycle winds twice in the $g_4$ direction as it winds once in the $g_3$ direction. This reflects the fact that each Efimov state is associated with two four-body bound states [37, 38]. We suggest that such a nontrivial topology of the limit cycle may support the robustness of the number of bound states against a continuous change of the Hamiltonian.