Minimal Pole Method (MPM)

This repository provides a Python implementation of the matrix-valued Minimal Pole Method (MPM) for both Matsubara and real-frequency data.

🔬 For the Analytic Continuation Community

The method is described in Phys. Rev. B 110, 235131 (2024), which extends the scalar-valued approach introduced in Phys. Rev. B 110, 035154 (2024).

The input Matsubara data is $G(i\omega_n)$, sampled on a non-negative uniform grid $\lbrace i\omega_0, i\omega_1, \cdots, i\omega_{n_\omega - 1} \rbrace$, where

• $\omega_n = \frac{(2n+1)\pi}{\beta}$ for fermions
• $\omega_n = \frac{2n\pi}{\beta}$ for bosons
• $n_\omega$ is the total number of sampling points

Relevant classes: MiniPole, MiniPoleDLR

🌱 For the HEOM Community

For applications involving real-frequency data used in Hierarchical Equations of Motion (HEOM), further details are provided in J. Chem. Phys. 162, 214111 (2025).

Relevant classes: MiniPoleRf, MiniPoleRfDPR

1. Installation

Dependencies

• numpy
• scipy
• matplotlib
• kneed

Installation Commands

1. Install the latest (unreleased) version from source

   python3 setup.py install
   

2. Install the latest released version via pip

   pip install mini_pole
   

2. Usage

i) The standard MPM is performed using the following command:

p = MiniPole(G_w, w, n0 = "auto", n0_shift = 0, err = None, err_type = "abs", M = None, symmetry = False, G_symmetric = False, compute_const = False, plane = None, include_n0 = True, k_max = 999, ratio_max = 10)

Parameters
----------
1. G_w : ndarray
    An (n_w, n_orb, n_orb) or (n_w,) array containing the Matsubara data.
2. w : ndarray
    An (n_w,) array containing the corresponding real-valued Matsubara grid.
3. n0 : int or str, default="auto"
    If "auto", n0 is automatically selected with an additional shift specified by n0_shift.
    If a non-negative integer is provided, n0 is fixed at that value.
4. n0_shift : int, default=0
    The shift applied to the automatically determined n0.
5. err : float
    Error tolerance for calculations.
6. err_type : str, default="abs"
    Specifies the type of error: "abs" for absolute error or "rel" for relative error.
7. M : int, optional
    The number of poles in the final result. If not specified, the precision from the first ESPRIT is used to extract poles in the second ESPRIT.
8. symmetry : bool, default=False
    Determines whether to preserve up-down symmetry.
9. G_symmetric : bool, default=False
    If True, the Matsubara data will be symmetrized such that G_{ij}(z) = G_{ji}(z).
10. compute_const : bool, default=False
    Determines whether to compute the constant term in G(z) = sum_l Al / (z - xl) + const.
    If False, the constant term is fixed at 0.
11. plane : str, optional
    Specifies whether to use the original z-plane or the mapped w-plane to compute pole weights.
12. include_n0 : bool, default=False
    Determines whether to include the first n0 input points when weights are calculated in the z-plane.
13. k_max : int, default=999
    The maximum number of contour integrals.
14. ratio_max : float, default=10
    The maximum ratio of oscillation when automatically choosing n0.

Returns
-------
Minimal pole representation of the given data.
Pole weights are stored in `p.pole_weight`, a numpy array of shape (M, n_orb, n_orb).
Shared pole locations are stored in `p.pole_location`, a numpy array of shape (M,).

ii) The MPM-DLR algorithm is performed using the following command:

p = MiniPoleDLR(Al_dlr, xl_dlr, beta, n0, nmax = None, err = None, err_type = "abs", M = None, symmetry = False, k_max=200, Lfactor = 0.4)

Parameters
----------
1. Al_dlr (numpy.ndarray): DLR coefficients, either of shape (r,) or (r, n_orb, n_orb).
2. xl_dlr (numpy.ndarray): DLR grid for the real frequency, an array of shape (r,).
3. beta (float): Inverse temperature of the system (1/kT).
4. n0 (int): Number of initial points to discard, typically in the range (0, 10).
5. nmax (int): Cutoff for the Matsubara frequency when symmetry is False.
6. err (float): Error tolerance for calculations.
7. err_type (str): Specifies the type of error, "abs" for absolute error or "rel" for relative error.
8. M (int): Specifies the number of poles to be recovered.
9. symmetry (bool): Whether to impose up-down symmetry (True or False).
10. k_max (int): Number of moments to be calculated.
11. Lfactor (float): Ratio of L/N in the ESPRIT algorithm.

Returns
-------
Minimal pole representation of the given data.
Pole weights are stored in `p.pole_weight`, a numpy array of shape (M, n_orb, n_orb).
Shared pole locations are stored in `p.pole_location`, a numpy array of shape (M,).

iii) The standard MPM for real-frequency fitting is performed using the following command:

p = MiniPoleRf(G_rf, func_type = "real", interval_type = "infinite", w_min = -10, w_max = 10, wp_max = 1, sing_vals = None, err = None, M = None, compute_const = False, k_max = 999, Lfactor = 0.4)

Parameters
----------
1. G_rf : list
   A list of length n_orb² containing analytic expressions of the real-frequency Green's functions.
2. func_type : str
    Specifies the type of functions in G_rf; either "real" for real-valued or "complex" for complex-valued.
3. interval_type : str
    Specifies the type of real-frequency interval; either "infinite" or "finite".
4. w_min : float
    Lower bound of the finite real-frequency interval.
5. w_max : float
    Upper bound of the finite real-frequency interval.
6. wp_max : float
    Parameter used in the Möbius transform for the infinite real-frequency interval.
7. sing_vals : list
    List of singular values of G_rf.
8. err : float
    Error tolerance used during the approximation process.
9. M : int
    Number of poles in the final result.
10. compute_constant : bool
    Whether to compute the constant term in the approximation.
11. k_max : int
    Maximum number of contour integrals.
12. Lfactor : float
    Ratio L / N used in ESPRIT.

Returns
-------
Minimal pole representation of the real-frequency Green's functions.
Pole weights are stored in `p.pole_weight`, a numpy array of shape (M, n_orb, n_orb).  
Shared pole locations are stored in `p.pole_location`, a numpy array of shape (M,).

iv) The MPM algorithm for real-frequency fitting using a discrete pole representation (e.g., from AAA results) can be executed with the following command:

p = MiniPoleRfDPR(Al_dpr, xl_dpr, interval_type = "infinite", w_min = -10, w_max = 10, wp_max = 1, err = None, err_type = "abs", cutoff_err = None, cutoff_err_type = "abs", M = None, k_max = 999, Lfactor = 0.4, alpha = 1.0, minimal_k = False)

Parameters
----------
1. Al_dpr : numpy.ndarray
    Complex pole weights, either of shape (r,) or (r, n_orb, n_orb).
2. xl_dpr : numpy.ndarray
    Complex pole locations, an array of shape (r,).
3. interval_type : str
    Specifies the type of real-frequency interval; either "infinite" or "finite".
4. w_min : float
    Lower bound of the finite real-frequency interval.
5. w_max : float
    Upper bound of the finite real-frequency interval.
6. wp_max : float
    Parameter used in the Möbius transform for the infinite real-frequency interval.
7. err : float
    Error tolerance used during the approximation process.
8. err_type : str
    Type of error to use; either "abs" for absolute error or "rel" for relative error.
9. cutoff_err : float
    Cutoff value for h_k.
10. cutoff_err_type : str
    Specifies whether the cutoff is based on absolute ("abs") or relative ("rel") error.
11. M : int
    Number of poles in the final result.
12. k_max : int
    Maximum number of contour integrals.
13. Lfactor : float
    Ratio L / N used in ESPRIT.
14. alpha : float
    Scaling parameter inside the unit disk to accelerate convergence.
15. minimal_k : bool
    Whether to use a minimal number of h_k based on the size of `xl_dpr`.

Returns
-------
Minimal pole representation of the given data.
Pole weights are stored in `p.pole_weight`, a numpy array of shape (M, n_orb, n_orb).  
Shared pole locations are stored in `p.pole_location`, a numpy array of shape (M,).

3. Examples

The scripts in the examples folder demonstrate the usage of MPM, MPM-DLR and MPM-RF.

i) MPM Algorithm

The examples/MPM folder includes a Jupyter notebook that demonstrates how to use MiniPole to recover synthetic spectral functions. You can modify the lambda expression in the GreenFunc class to recover a different spectrum, but please remember to update the lower and upper bounds (x_min and x_max) of the spectrum accordingly. Additional details will be provided in the future.

ii) MPM-DLR Algorithm

The examples/MPM_DLR folder contains scripts to recover the band structure of Si, as shown in the middle panel of Fig. 9 in Phys. Rev. B 110, 235131 (2024).

Steps:

a) Download the input data file Si_dlr.h5 to the examples/MPM_DLR/ directory.

b) Obtain the recovered poles by running python3 cal_band_dlr.py --obs=<option>, where <option> can be "S" (self-energy), "Gii" (scalar-valued Green's function), or "G" (matrix-valued Green's function).

c) Plot the band structure by running python3 plt_band_dlr.py --obs=<option>.

Note:

a) Reference runtime on a single core of a laptop (using the M1 Max Apple chip as an example): 13 seconds for "Gii" and 160 seconds for both "G" and "S".

b) Parallel computation is supported in cal_band_dlr.py to speed up the process on multiple cores. Use the following command: mpirun -n <num_cores> python3 cal_band_dlr.py --obs=<option>, where <num_cores> is the number of cores and <option> is "S," "Gii," or "G".

c) Full Parameters for cal_band_dlr.py:

• --obs (str): Observation type used in the script. Default is "S".
• --n0 (int): Parameter $n_0$ as described in Phys. Rev. B 110, 235131 (2024).
• --err (float): Error tolerance for computations. Default is 1.e-10.
• --symmetry (bool): Specifies whether to preserve up-down symmetry in calculations.

d) Full Parameters for plt_band_dlr.py:

• --obs (str): Observation type used in the script. Default is "S".
• --w_min (float): Lower bound of the real frequency in eV. Default is -12.
• --w_max (float): Upper bound of the real frequency in eV. Default is 12.
• --n_w (int): Number of frequencies between w_min and w_max. Default is 200.
• --eta (float): Broadening parameter. Default is 0.005.

iii) MPM Algorithm for real-frequency fitting

The examples/MPM_RF folder contains Jupyter notebooks that demonstrate how to use MiniPoleRf to obtain poles for both typical spectral functions and a sub-Ohmic bath.