PronyAC

This is a Python program for performing Prony Analytic Continuation, as described in Phys. Rev. B 110, 035154 (2024).

🚨 Important: A more stable implementation is now available at Green-Phys/MiniPole, where all future updates will be maintained. Compared to PronyAC, MiniPole can automatically determine $n_0$, uses a more stable ESPRIT algorithm instead of the Prony approximation, and supports multiple orbital cases.

The input of the simulation is the Matsubara data $G(i \omega_n)$ sampled on a uniform grid $\lbrace i\omega_{n_0}, i\omega_{n_0 + \Delta n}, \cdots, i\omega_{n_0 + (N_{\omega}-1) \Delta n} \rbrace$, where $\omega_n=\frac{(2n+1)\pi}{\beta}$ for fermions and $\frac{2n\pi}{\beta}$ for bosons, $n_0 \geq 0$, $\Delta n \geq 1$ and $N_{\omega}$ is the total number of sampling points.

The Prony Analytic Continuation is performed using the following command:

PronyAC(G_w, w, err = None, optimize = False, symmetry = False, pole_real = False, reduce_pole = True, plane = None, k_max = 999, x_range = [-np.inf, np.inf], y_range = [-np.inf, np.inf])

1. G_w is a 1-d array containing the Matsubara data.
2. w is the corresponding sampling grid $\lbrace \omega_n \rbrace$.
3. If err (>=1.e-12) is given, the continuation will be carried out in this tolerance.
4. If err is not given, the tolerance will be chosen to be the last singular value in the exponentially decaying range when optimize is False and will be chosen to be the presumably optimal one when optimize is True. There is no guarantee that setting optimize to be True will always find the optimal solution. So it is suggested to fine-tune err to achieve the best performance.
5. symmetry determines whether to impose the up-down symmetry in the complex plane.
6. pole_real determines whether to restrict the poles exactly on the real axis when symmetry is True.
7. reduce_pole determines whether to discard the poles whose weights are smaller than the error tolerance.
8. If plane is set to be not None, the computation of pole weights will be enforced in either the original plane (when plane is "z") or the mapped plane (when plane is "w").
9. k_max is the maximum number of contour integrals.
10. Only poles located within the rectangle x_range[0] < x < x_rang[1] and y_range[0] < y < y_range[1] are retained.

Note: parameters 6 to 10 are rarely used in most cases. If there is no strong motivation to change them, please leave them as default.

The results are stored in variables pole_weight and pole_location. The instance method check_valid() is provided for checking intermediate steps.

Other classes and functions are also provided to facilitate testings of toy models. The "examples" folder contains original data of Phys. Rev. B 110, 035154 (2024) and corresponding scripts to generate them.