Topological origin of universal few-body clusters in Efimov physics

Efimov physics is renowned for the self-similar spectrum featuring the universal ratio of one eigenenergy to its neighbor. Even more esoteric is the numerically unveiled fact that every Efimov trimer is accompanied by a pair of tetramers. Here we demonstrate that this hierarchy of universal few-body clusters has a topological origin by identifying the numbers of universal three- and four-body bound states with the winding numbers of the renormalization-group limit cycle in theory space. The finding suggests a topological phase transition in mass-imbalanced few-body systems which should be tested experimentally.

Figure 1

(a) Cutoff $k$ dependencies of the three-body and four-body coupling constants $g_3\left(k\right)$ and $g_4\left(k\right)$ at the unitarity limit $1/a=0$. The abscissa shows the logarithm of the cutoff $k$. The blue dots and the purple diamonds show $g_3\left(k\right)/2$ and $g_4\left(k\right)/250$, where the multiplication factors $1/2$ and $1/250$ are introduced to display the two flows simultaneously. (b) RG flow in theory space of the three-body and four-body coupling constants. The abscissa shows $\arctan (g_3/16)$ and the ordinate shows $\arctan (g_4/6250)$, where we use arctan to display diverging coupling constants, and $g_3$ and $g_4$ are multiplied by 1/16 and 1/6250 for the sake of simultaneous display. The brown curve shows the RG limit cycle obtained by eliminating the cutoff $k$ dependence of $g_3\left(k\right)$ and $g_4\left(k\right)$ from (a). If we glue the edges of the figure to form a torus, the limit cycle winds once in the $g_3$ direction and twice in the $g_4$ direction, revealing a topological nature. (c) Schematic illustration of the topological feature of the limit cycle described in (b). (c) is adapted from Fig. 5 of Ref. [38] with modifications.

Figure 2

Diagrammatic expressions for the exact FRG equations for (a) the two-body sector and (b) the three-body sector. The black dot in (a) shows the bare particle exchange interaction introduced in Eq. (1), the solid curve shows the propagator $G_{\psi }$ of a particle, and the double line shows the propagator $G_{\varphi ,k}$ of a dimer introduced in Eq. (7). (c) Integral form of the exact FRG equation for the three-body sector. Decomposition of (d) the three-body 1PI vertex and (e) the four-body 1PI vertex. The curly brackets indicate symmetrization with respect to external lines. (f) Integral form of the exact FRG equation for the four-body sector. (g) Definition of the particle-trimer scattering amplitude $T_k^{\left(4\right)}$ represented as a square vertex.