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

Universal behavior has been found inside the window of Efimov physics for systems with $N=4,5,6$ particles. Efimov physics refers to the emergence of a number of three-body states in systems of identical bosons interacting via a short-range interaction becoming infinite at the verge of binding two particles. These Efimov states display a discrete scale invariance symmetry, with the scaling factor independent of the microscopic interaction. Their energies in the limit of zero-range interaction can be parametrized, as a function of the scattering length, by a universal function. We have found, using the form of finite-range scaling introduced by A. Kievsky and M. Gattobigio [Phys. Rev A 87, 052719 (2013)], that the same universal function can be used to parametrize the ground and excited energy of $N\le 6$ systems inside the Efimov-physics window. Moreover, we show that the same finite-scale analysis reconciles experimental measurements of three-body binding energies with the universal theory.

Figure 1

The experimental data on ${}^7$Li [10], in the form of ratio between the three-body binding energy $E_3^1$ and the two-body binding energy $E_2$, as a function of $1/({\kappa }_1^3{a}_B+{\Gamma }_1^3)$ with the values of the three-body parameters given in the text. The solid curve represents the prediction of the universal function, Eq. (2). In the upper abscissa the magnetic field from Ref. [26] is given.

Figure 2

The TBG and PT ground-state $N$-body binding-energy $E_N^0$ in units of $E_2$ as a function of (a) $1/{\kappa }_0^N{a}_B$, and (b) $1/({\kappa }_0^N{a}_B+{\Gamma }_0^N$). $E_2$ is the two-body binding energy for $a>0$, and the two-body virtual state energy for $a<0$. In panel (b) the ground-state energies in terms of the scaled variable collapse onto the zero-range curve (solid line).

Figure 3

The TBG and PT excited-state $N$-body binding-energy $E_N^1$ in units of $E_2$ as a function of (a) $1/{\kappa }_1^N{a}_B$ and (b) $1/({\kappa }_1^N{a}_B+{\Gamma }_1^N$). $E_2$ is the two-body binding energy for $a>0$ and the two-body virtual state for $a<0$. In panel (b) the excited-state energies in terms of the scaled variable collapse onto the zero-range curve (solid line).

Figure 4

The TBG finite-range scaling parameters ${\Gamma }_m^N$ as a function of the angle $\xi$ derived from Eq. (6). The symbols are the same as in Figs. 2 and 3.

Figure 5

The TBG and PT ground- and excited-state $N$-body binding energies $E_N^m$ in units of $E_2$ as a function of $1/({\kappa }_m^N{a}_B+{\Gamma }_m^N)$. $E_2$ is the two-body binding energy for $a>0$, and the two-body virtual state for $a<0$. The symbols are the same as in Figs. 2 and 3 of the paper. All the data collapse on the three-body universal curve (solid line) calculated in the scaling limit.

Figure 6

The ground-state N-body binding energy $E_N^0$ from Ref. [41] in units of ${\hslash }^2/m{a}^2$ as a function of $1/({\kappa }_0^{N}a+{\Gamma }_0^N)$. The solid curve represent the universal zero-range curve.