Gaussian Parametrization of Efimov Levels: Remnants of Discrete Scale Invariance

Abstract

The Efimov window, or universal window, is a particular energy region in which few-body systems are loosely bound. Many characteristics of those systems are universal; their properties are largely independent of the particular interaction that produces the binding. This suggests the use of simple potential models characterized by few parameters to describe dynamical properties. Essentially, the potential parameters are set in order to reproduce a small number of data, and then the potential strength can be varied to explore the window. Among others, the Gaussian potential has been used to characterize the universal window. From calculations of the ground state and of the first excited state of three identical bosons using a Gaussian potential along the window, we determine what is called the gaussian-level function. This function is independent of the particular Gaussian potential used in this procedure. Moreover, it contains finite-range corrections which are of particular importance when theoretical predictions are compared to experimental data.

Appendix: Calculation of the Virtual-State Energy

In this Appendix we describe the method, introduced in Ref. [14], we used to calculate the virtual state in the case of negative scattering length. The S-matrix is approximated by a Padé function. The virtual state is the pole, in the positive imaginary k-plane, of the approximated S-matrix.

1.1 Padé Approximant for the S-matrix

The s-wave S-matrix reads

$$\begin{aligned} S(k) = \mathrm {exp}[i2\delta (k)]\,, \end{aligned}$$

(12)

where $\delta (k)$ is the phase-shifts. For certain values of the momentum the S-matrix has some simple poles whose physical meaning are bound states, virtual states or resonances. The approximation of the S-matrix with a rational function

$$\begin{aligned} S(k) \approx \frac{a_0 + a_1k + \dots + a_m k^m}{b_0 + b_1k + \dots + b_n k^n}\,, \end{aligned}$$

(13)

is called Padé approximant of order (n, m). The parameters in Eq. (13) are tuned in order to reproduce the S-matrix and all of its derivatives. Since the derivatives of the S-matrix are not always known, fitting the parameters in Eq. (13) to get Eq. (12) it is not directly possible. However, the number of parameters in Eq. (13) can be reduced thanks to the following properties of the S-matrix:

• Asymptotic behavior Both at high and zero energies the S-matrix tends to one; this implies that $m=n$, $a_n = b_n$, and $a_0 = b_0$. Dividing both the numerator and the denominator by $a_0$, Eq. (13) simplifies to

  $$\begin{aligned} S(k) \approx \frac{1 + \sum _{i=1}^{n-1} \alpha _i k^i + \alpha _n k^n}{1 + \sum _{i=1}^{n-1} \beta _i k^i + \alpha _n k^n}\,, \end{aligned}$$

  (14)

  where $\alpha _i = a_i/a_0$ and $\beta _i = b_i/a_0$.

• Symmetries of the incoming/outgoing amplitudes The scattering matrix can be expressed as the ratio between the incoming and outgoing Jost function [21]

  $$\begin{aligned} S(k)= \frac{f^{(out)} (k)}{f^{(in)} (k)}\,, \end{aligned}$$

  (15)

  that satisfy the symmetries $f^{(in)} (-k) = f^{(out)} (k)$, and $f^{(in)}(k^*) = f^{(out)*}(k)$, allowing to rewrite Eq. (15) as

  $$\begin{aligned} S(k) = \frac{f^{(in)} (-k)}{f^{(in)} (k)}\,. \end{aligned}$$

  (16)

  Using the equivalence of the two expressions, the parametrization Eq. (14) simplifies to

  $$\begin{aligned} S(k) \approx \frac{1 + \sum _{i=1}^{n} \alpha _i k^i}{1 + \sum _{i=1}^{n} (-1)^i \alpha _i k^i}. \end{aligned}$$

  (17)

To summarize, Eq. (17) is the Padé approximant of the S-matrix.

Fig. 4

Poles of the S-matrix in the complex planes for different order of the Padé’s approximation. The spurious poles change the positions in the complex plane while the physical one keeps the same position

1.2 The Algorithm

The parameters $\alpha _i$ of Eq. (17) are obtained with the following algorithm:

• 1. We choose the degree n of the polynomials in the rational approximation (17) and then n different values for the momentum k. Usually a grid of $k_i$ is chosen inside the range $[k_{min}, k_{max}]$.

• 2. For each $k_i$ chosen in the previous step we calculate the phase-shifts (this task is accomplished by solving numerically the Schrödinger equation with the potential Eq. (3)). Given $\delta (k_i)$ we calculate the $S(k_i)$ using Eq. (12) at each $k_i$.

• 3. Equation (17) can be rewritten as

  $$\begin{aligned} \sum _{i=1}^{n} [1 + (-1)^{i+1} S(k)] \alpha _i k^i = S(k) - 1\,, \end{aligned}$$

  (18)

  or equivalently

  $$\begin{aligned} \sum _{j=1}^n A_{ij} \alpha _i = B_i \,, \end{aligned}$$

  (19)

  where $A_{ij} = [1 + (-1)^{j+1} S(k_i)] k_i^j$ and $B_i = S(k_i) - 1$. In this way, the $\alpha _i$ are the just the solutions of a linear system.

• 4. To get the poles in (17) we set the polynomials of the denominator to zero

  $$\begin{aligned} P_n(k) = 1 + \sum _{i=1}^{n} (-1)^i \alpha _i k^i = 0 \end{aligned}$$

  (20)

1.3 Spurious Poles and the Degree of the Padé Approximation

We expect Padé approximant to be a good representation of the S-matrix because the two functions agree in a finite number of points inside the interval $[k_{min}, k_{max}]$. If the agreement were on the full interval, the functions would have been the same thanks to the coincidence theorem [14]. However, as they only coincide in a finite number of points, the theorem does not hold. What has been observed [14] is that the higher the degree of the approximation, the more are the poles of the Padé approximant. These poles are mostly spurious and can be easily detected because they move toward infinity as one increases the degree of the approximation. Only the true poles of the S-matrix stay constant.

    For testing purpose, we solve the Schrödinger equation for two particles of mass $\hbar ^2/m = 43.281307~\text {K}a_0^2$ with the Gaussian potential Eq. (3) where $V_0 = -0.93$ K and $r_0 = 10~a_0$, and we explore how the poles of the S-matrix approximation change increasing the order n. As suggested by Eq. (20), the number of poles is the same as the order n of the approximation. In Fig. 4, we observe that by increasing n, some of the poles remain fixed, whereas the others move. The fixed ones are the poles we are interested in, and they are the only ones with a physical meaning (bound states, virtual states, resonances).

    The choice of the grid is made according to the precision we want to get and to the stability of the algorithm. The following aspects are taken into account:

• The range The outcomes of the Padé approximation is then reasonably independent on the size of the range. That size depends on the number of momenta and on the step size.

• The degree of Padé approximation It is expected that a higher number of momenta $k_i$ (corresponding to a higher number of poles and to a higher degree of the Padé approximation) leads to a better approximation. The left panel of Fig. 5 shows that the momentum starts to be stable (up to the 6th decimal place) with about 9 poles and more. For our purpose we chose $n = 20$ and we take the momenta accurate to the 5th decimal place.

• The step size The stability of the algorithm will definitely depend on the step size. It is expected that the smaller is the step size, the better is the approximation; however, if the step size is too small the range $[k_{min}, k_{max}]$ can be too small as well. In the right panel of Fig. 5 we can see that the algorithm is more stable in the range $[0.035~a_0, 0.055~a_0]$. For our purpose we choose the step size equals to 0.05 $a_0$.

Fig. 5

Stabilization of the momentum of the physical pole obtained by the algorithm increasing the degree of the approximation (left panel) and the size step of the grid (right panel)