Universality and scaling in the $N$-body sector of Efimov physics

In this talk I will illustrate the universal behavior that we have found inside the window of Efimov physics for systems made of $N\le 6$ particles~[1]. We have solved the Schr\"odinger equation of the few-body systems using different potentials, and we have changed the potential parameters in such a way to explore a range of two-body scattering length, $a$, around the unitary limit, $|a| \rightarrow \infty$. The ground- ($E_N^0$) and excited-state ($E^1_N)$ energies have been analyzed by means of a recent-developed method which allows to remove finite-range effects~[2]. In this way we show that the calculated ground- and excited-state energies collapse over the same universal curve obtained in the zero-range three-body systems. Universality and scaling are reminiscent of critical phenomena; in that framework, the critical point is mapped onto a fixed point of the Renormalization Group (RG) where the system displays scale-invariant (SI) symmetry. A consequence of SI symmetry is the scaling of the observables: for different materials, in the same class of universality, a selected observable can be represented as a function of the control parameter and, provided that both the observable and the control parameter are scaled by some material-dependent factor, all representations collapse onto a single universal curve. Efimov physics is a more recent example of universality, but in this case the physics is governed by a limit cycle on the RG flow with the emergence of a discrete scale invariance (DSI). The scaling of the few-body energies can be interpreted as follow: few-body systems (at least up to $N=6$), inside the Efimov window, belong to the same class of universality, which is governed by the limit cycle. These results can be summarized by the following formula \begin{equation} E_N^n/E_2 = \tan^2\xi \\ \qquad \kappa^n_N a_B + \Gamma^n_N = \frac{\mathrm{e}^{-\Delta(\xi)/2s_0}}{\cos\xi}\,. \end{equation} where the function $\Delta(\xi)$ is universal and it is determined by the three-body physics, and $s_0=1.00624$. The parameter $\kappa^n_N$ appears as a scale parameter and the shift $\Gamma_n^N$ is a finite-range scale parameter introduced to take into account finite-range corrections~[2].\\[4pt] [1] M. Gattobigio and A. Kievsk, arXiv:1309.1927 (2013).\\[0pt] [2] A. Kievsky and M. Gattobigio, Phys. Rev. A {\bf 87}, 052719 (2013).