| Name | mini-pole JSON |
| Version |
0.2
JSON |
| download |
| home_page | https://github.com/Green-Phys/MiniPole |
| Summary | The Python code provided implements the matrix-valued version of the Minimal Pole Method (MPM) as described in arXiv:2410.14000. |
| upload_time | 2024-12-05 00:34:58 |
| maintainer | None |
| docs_url | None |
| author | Lei Zhang |
| requires_python | >=3.8 |
| license | MIT |
| keywords |
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# 1. MiniPole
The Python code provided implements the matrix-valued version of the Minimal Pole Method (MPM) as described in [arXiv:2410.14000](https://arxiv.org/abs/2410.14000), extending the scalar-valued method introduced in [Phys. Rev. B 110, 035154 (2024)](https://doi.org/10.1103/PhysRevB.110.035154).
The input of the simulation is the Matsubara data $G(i \omega_n)$ sampled on a 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 and $\frac{2n\pi}{\beta}$ for bosons, and $n_{\omega}$ is the total number of sampling points.
## 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=True
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,).
# 2. Examples
The scripts in the *examples* folder demonstrate the usage of MPM and MPM-DLR.
## i) 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. 8 in [arXiv:2410.14000](https://arxiv.org/abs/2410.14000).
### Steps:
a) Download the input data file [Si_dlr.h5](https://drive.google.com/file/d/1_bNvbgOHewiujHYEcf-CCpGxlZP9cRw_/view?usp=drive_link) 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 [arXiv:2410.14000](https://arxiv.org/abs/2410.14000).
- `--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`.
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"description": "# 1. MiniPole\nThe Python code provided implements the matrix-valued version of the Minimal Pole Method (MPM) as described in [arXiv:2410.14000](https://arxiv.org/abs/2410.14000), extending the scalar-valued method introduced in [Phys. Rev. B 110, 035154 (2024)](https://doi.org/10.1103/PhysRevB.110.035154).\n\nThe input of the simulation is the Matsubara data $G(i \\omega_n)$ sampled on a 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 and $\\frac{2n\\pi}{\\beta}$ for bosons, and $n_{\\omega}$ is the total number of sampling points.\n\n## i) The standard MPM is performed using the following command:\n\n**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)**\n \n Parameters\n ----------\n 1. G_w : ndarray\n An (n_w, n_orb, n_orb) or (n_w,) array containing the Matsubara data.\n 2. w : ndarray\n An (n_w,) array containing the corresponding real-valued Matsubara grid.\n 3. n0 : int or str, default=\"auto\"\n If \"auto\", n0 is automatically selected with an additional shift specified by n0_shift.\n If a non-negative integer is provided, n0 is fixed at that value.\n 4. n0_shift : int, default=0\n The shift applied to the automatically determined n0.\n 5. err : float\n Error tolerance for calculations.\n 6. err_type : str, default=\"abs\"\n Specifies the type of error: \"abs\" for absolute error or \"rel\" for relative error.\n 7. M : int, optional\n 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.\n 8. symmetry : bool, default=False\n Determines whether to preserve up-down symmetry.\n 9. G_symmetric : bool, default=False\n If True, the Matsubara data will be symmetrized such that G_{ij}(z) = G_{ji}(z).\n 10. compute_const : bool, default=False\n Determines whether to compute the constant term in G(z) = sum_l Al / (z - xl) + const.\n If False, the constant term is fixed at 0.\n 11. plane : str, optional\n Specifies whether to use the original z-plane or the mapped w-plane to compute pole weights.\n 12. include_n0 : bool, default=True\n Determines whether to include the first n0 input points when weights are calculated in the z-plane.\n 13. k_max : int, default=999\n The maximum number of contour integrals.\n 14. ratio_max : float, default=10\n The maximum ratio of oscillation when automatically choosing n0.\n \n Returns\n -------\n Minimal pole representation of the given data.\n Pole weights are stored in p.pole_weight, a numpy array of shape (M, n_orb, n_orb).\n Shared pole locations are stored in p.pole_location, a numpy array of shape (M,).\n\n## ii) The MPM-DLR algorithm is performed using the following command:\n\n**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)**\n\n Parameters\n ----------\n 1. Al_dlr (numpy.ndarray): DLR coefficients, either of shape (r,) or (r, n_orb, n_orb).\n 2. xl_dlr (numpy.ndarray): DLR grid for the real frequency, an array of shape (r,).\n 3. beta (float): Inverse temperature of the system (1/kT).\n 4. n0 (int): Number of initial points to discard, typically in the range (0, 10).\n 5. nmax (int): Cutoff for the Matsubara frequency when symmetry is False.\n 6. err (float): Error tolerance for calculations.\n 7. err_type (str): Specifies the type of error, \"abs\" for absolute error or \"rel\" for relative error.\n 8. M (int): Specifies the number of poles to be recovered.\n 9. symmetry (bool): Whether to impose up-down symmetry (True or False).\n 10. k_max (int): Number of moments to be calculated.\n 11. Lfactor (float): Ratio of L/N in the ESPRIT algorithm.\n \n Returns\n -------\n Minimal pole representation of the given data.\n Pole weights are stored in p.pole_weight, a numpy array of shape (M, n_orb, n_orb).\n Shared pole locations are stored in p.pole_location, a numpy array of shape (M,).\n\n# 2. Examples\n\nThe scripts in the *examples* folder demonstrate the usage of MPM and MPM-DLR.\n\n## i) MPM-DLR Algorithm\n\nThe *examples/MPM_DLR* folder contains scripts to recover the band structure of Si, as shown in the middle panel of Fig. 8 in [arXiv:2410.14000](https://arxiv.org/abs/2410.14000).\n\n### Steps:\n\na) Download the input data file [Si_dlr.h5](https://drive.google.com/file/d/1_bNvbgOHewiujHYEcf-CCpGxlZP9cRw_/view?usp=drive_link) to the *examples/MPM_DLR/* directory.\n\nb) 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).\n\nc) Plot the band structure by running **python3 plt_band_dlr.py --obs=`<option>`**.\n\n### Note:\n\na) 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\".\n\nb) 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\".\n\nc) Full Parameters for **cal_band_dlr.py**:\n\n - `--obs` (str): Observation type used in the script. Default is `\"S\"`.\n - `--n0` (int): Parameter $n_0$ as described in [arXiv:2410.14000](https://arxiv.org/abs/2410.14000).\n - `--err` (float): Error tolerance for computations. Default is `1.e-10`.\n - `--symmetry` (bool): Specifies whether to preserve up-down symmetry in calculations.\n\nd) Full Parameters for **plt_band_dlr.py**:\n\n - `--obs` (str): Observation type used in the script. Default is `\"S\"`.\n - `--w_min` (float): Lower bound of the real frequency in eV. Default is `-12`.\n - `--w_max` (float): Upper bound of the real frequency in eV. Default is `12`.\n - `--n_w` (int): Number of frequencies between `w_min` and `w_max`. Default is `200`.\n - `--eta` (float): Broadening parameter. Default is `0.005`.\n",
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