Numerical results for "Non-linear responses and three-particle correlators in correlated electron systems exemplified by the Anderson impurity model"

This dataset contains the numerical data and plot scripts to reproduce the figures from the paper "Non-linear responses and three-particle correlators in correlated electron systems exemplified by the Anderson impurity model" submitted for publication to Physical Review B and already available as preprint on arXiv. As the title suggests it consists of the data necessary to compute linear and non-linear response functions. Most results were calculated with w2dynamics a continuous-time quantum Monte Carlo (QMC) solver, but some were obtained via exact diagonalization (ED). In addition to the numerical data for the figures, the dataset also includes auxiliary data files as well as parameter files and submission scripts for the w2dynamics calculations in an attempt to increase reproducibility.

ED results

ED results are only available for the Anderson impurity model (AIM) with a single bath site where the on-site Coulomb interaction U = 1 and the inverse temperature β = 20. They are found in anderson_impurity_model/single_bath_site/exact/magnetic/ together with two simple gnuplot scripts for easy visualization. The .dat files are text files with columns separated by white space and blocks separated by single blank lines. The header line specifies the quantities in each column but the notation deviates from the paper in some instances:

  • mu: shifted chemical potential μ; the paper uses the one-particle energy ε = -μ - U/2
  • chi_ch: charge response function χ_{ch} = density response function χ_d
  • chi_sp: spin response function χ_{sp} = magnetic response function χ_m

The data file with suffix _phsym contains the results for the particle–hole symmetric case, i.e. bath site energy ε_1 = 0 with different hybridization strengths V = {0, 0.05, 0.10, 0.15, 0.20, 0.25} in the different data blocks. The other data file contains the results for V = 0.2 with different bath site energies ε_1 = {0, 0.05, 0.10, 0.15, 0.20, 0.25} in the different data blocks.

QMC results

As mentioned above, the QMC results were obtained with w2dynamics and make up most of the dataset. They are available for an Anderson impurity model (AIM) with a single bath site or a constant density of states (DOS) and a Hubbard model (HM) on a square lattice. The directory structure is rather straight forward, grouping the results by model type, model parameters, measured w2dynamcis quantity, and finally a number to distinguish different runs. The notation for model parameters, the w2dynamics quantities as well as the different file types are briefly explained below. For more information on things like the measured quantities, data or parameter files please check out the documentation of w2dynamics aka its README and wiki.

Notation for model parameters used in directory and filenames:

  • n: total density n
  • u: on-site Coulomb interaction U
  • beta: inverse temperature β
  • d: half bandwidth D of the constant DOS
  • v: hybridization strength V
  • e: energy level ε_1 of the single bath site
  • mu: chemical potential μ; the paper uses the one-particle energy ε = -μ instead

Brief explanation of w2dynamics quantities:

  • dmft: a series of iterative steps converging towards a self-consistent DMFT solution; required for computing other quantities of the HM
  • g: 1-particle Green's function G
  • p2: fully contracted 2-particle Green's function P2(ω) = ∑_ν ∑_ν' G^2(ν, ν', ω)
  • p3: partially contracted 2-particle Green's function P3(ν', ω) = ∑_ν G^2(ν, ν', ω)
  • nn: 2-particle correlator X^2
  • raman: 3-particle correlator X^3

Brief explanation of file types and formats:

  • Data files: The w2dynamics results are stored in HDF5 files. In addition to the numerical data they contain all necessary input data to reproduce the simulation run. Their name starts with the type of w2dynamics measurement, then some parameters and end with a timestamp.
  • Parameter files: The input files for w2dynamics are usually called Parameters.in and provided for easier reproducibility of the results.
  • Hybridization files: The text files deltaiw and deltatau are required when solving an AIM. They define the hybridization function Δ in Matsubara frequencies and imaginary time. The code for generating the files can be found in anderson_impurity_model/generate_delta_files.py.
  • Hk file: The file hubbard_model/2d_square_lattice/hubbard_2d_48_48_1.hk contains the non-interacting (tight-binding) part of the Hamiltonian H(k) for a square lattice with a single band, nearest-neighbor hopping t = 1 and 48 × 48 k points.
  • Submission scripts: All computations were done on the VSC-4. The job submission scripts are usually called submit.sh and provided for easier reproducibility of the results.
  • Mu-scan scripts: They generate parameter files and submission scripts for different values of the chemical potential and are usually called _mu_scan.sh.
  • PDF files: Many directories contain PDF files that visualize parts of the data with various plots. They are, however, not publication grade plots.

Plot scripts

The plot_scripts folder contains all Python files necessary to generate the figures of the paper. They are provided for transparency and easier reproducibility. Note however, that the Python code uses the w2dynamics notation that differs from the paper for many quantities (see the QMC results section for a brief explanation).

  • style.py: defines font sizes, colors, common dimensions, and other settings for the plot style of the figures
  • utility.py: defines some helpful utility functions as well as the variable DATA_ROOT_DIRECTORY. The latter is used by all plot functions to find the data files and must point to the directory containing the folders hubbard_model and anderson_impurity_model. If, after downloading the dataset, you move the plot scripts or data directories, you must change DATA_ROOT_DIRECTORY accordingly.
  • line_plots.py: contains code for the line plots, i.e., figures 3, 5, 6, 7, and 11.
  • color_plots.py: contains code for the color plots, i.e., figures 4, 8, 9, 10, and 12.
  • atomic_limit.py: defines nice and simple wrapper functions for the atomic limit computations implemented in _3p.py and _3p_mu_scan.py.
  • _3p.py and _3p_mu_scan.py: contain the implementation details for the atomic limit computations.

The plot scripts require Python 3.7 and the following dependencies:

The versions in parentheses are the ones that were used when generating the figures for the paper. The last three packages are not available via conda or in the Python Package Index. Please check their respective README files for how to install them.

To generate all figures, cd into the plot_scripts directory of the dataset and execute:

python line_plots.py
python color_plots.py

The two Python scripts also allow interactive execution similar to Jupyter notebooks. Magic comments starting with # %% divide the code into cells that can be run separately. This is supported in many common Python IDEs like VS Code, PyCharm, and Spyder.

Important
The generated figures still count as part of the article and therefore fall under APS copyright. Please check out the Terms and Conditions and/or the APS Copyright FAQ for your rights as a third party.

License

All Python files are licensed under an MIT License (see LICENSE-MIT), the rest of this dataset is licensed under a Creative Commons Attribution 4.0 International License.