adapol: Adaptive Pole Fitting for Quantum Many-Body Physics

adapol ("add-a-pole") is a Python package for constructing compact pole approximations of Matsubara functions,

$$ G(\mathrm{i}\nu_n) \approx \sum_{k=1}^{M} \frac{R_k}{\mathrm{i}\nu_n - p_k}, $$

with real poles $p_k$ and scalar or matrix-valued residues $R_k$, using the AAA rational approximation algorithm and nonlinear optimization. Given Matsubara data, or an existing pole expansion (for example, a discretized spectral density or a discrete Lehmann representation), adapol finds an accurate approximation with a specified maximum number of poles, or as few poles as possible. A typical application is hybridization fitting: constructing a compact bath representation of a given hybridization function.

Installation

pip install adapol

The only dependencies are numpy and scipy.

Usage

adapol provides three main functions:

• approx_freq_aaa(F, Z, ...) fits frequency data F, sampled at (typically Matsubara) points Z, with a sum of simple poles, using the AAA algorithm. The number of poles is controlled by a pole budget max_n_poles and/or a AAA error tolerance aaa_tol.
• approx_sop_fast(poles, residues, beta, ...) approximates a given sum of poles by a (hopefully) smaller one in a single AAA pass. The number of poles is again controlled by max_n_poles and/or aaa_tol, and an optional nonlinear_optimization step refines the pole locations.
• approx_sop_tol(poles, residues, tol, beta, ...) finds the smallest sum of poles whose actual error (in $L^2(\tau)$ and $l^2(i \omega_n)$) is below the tolerance tol.

Examples

Three example notebooks demonstrate the usage of these functions in detail. We recommend reading them in the following order.

• semicircle.ipynb — fitting data with a continuous spectrum (semicircular density): the stopping criteria, the nonlinear optimization option, and the error metric.
• discrete.ipynb — fitting multi-orbital data with a discrete spectrum, including an experiment on how the required number of poles scales with the number of orbitals.
• hubbarddimer.ipynb — analytic continuation benchmark for the Hubbard dimer.

Documentation

The reference documentation for the three functions also describes in detail how to use them, as well as information on the algorithms they implement. The same information is contained in the docstrings, e.g. help(adapol.approx_freq_aaa).

Citation

If you use this package in your research, please include a reference to this GitHub repository, and cite the following references:

1. Huang, Zhen, Emanuel Gull, and Lin Lin. "Robust analytic continuation of Green's functions via projection, pole estimation, and semidefinite relaxation," Phys. Rev. B 107, 075151 (2023).
2. Huang, Zhen, Denis Golež, Hugo U. R. Strand, and Jason Kaye. "Automated evaluation of imaginary time strong coupling diagrams by sum-of-exponentials hybridization fitting," SciPost Phys. 19 (5), 121 (2025).

License

adapol is distributed under the GNU General Public License v3.0 (see LICENSE).