Examples

Several example scripts are provided in the Github repo of the green-mbtools package, particularly for the post-processing of Green output. Here, we will go over some of the basic cases.

Initialization

For analyzing the output of Green’s function simulation using, e.g. GW, GF2, or self-energy embedding theory, we use the MB_post class from green_mbtools.pesto.mb. The script examples/MB_post_example.py demonstrates the key steps in initializing the post-processing class:

  1. Read the input and output files from a Green simulation, and initialize the many-body post processing class:

    from green_mbtools.pesto import mb

if __name__ == "__main__":
    fname_inp = '/path/to/input.h5'
    fname_sim = '/path/to/sim.h5'
    ir_file = '/path/to/ir.h5'

    mb_obj = mb.initialize_MB_post(sim_path=fname_sim, input_path=fname_inp, ir_file=ir_file)

  2. Access Green’s function, self-energy, and other information

    if __name__ == "__main__":

    # ... initialization

    mu = mb_obj.mu()  # chemical potential
    gtau = mb_obj.gtau()  # Green's function
    sigma_tau = mb_obj.sigma  # self-energy on imaginary-time axis
    dm = mb_obj.dm()  # get density matrix from Green's function

Matsubara Transformation

We are often interested in visualizing the Green’s function and self-energy on the Matsubara frequency or \(i\omega\) axis rather than the imaginary-time or \(\tau\) axis. As the data in Green software package is stored on sparse grids also known as the intermediate representation (IR) grids, we use the information from the IR grid file to execute the \(\tau \leftrightarrow i\omega\) transformation.

Once an MB_post object is initialized, the transformation can be performed simply as:

if __name__ == "__main__":

    # ... initialization

    # tau to omega transform
    sigma_iw = mb_obj.ir.tau_to_w(sigma_tau)
    g_iw = mb_obj.ir.tau_to_w(gtau)
    # back transform
    sigma_tau_back = mb_obj.ir.w_to_tau(sigma_iw)
    gtau_back = mb_obj.ir.w_to_tau(g_iw)

The examples/IR_transform.py script also shows how such a transformation can be performed without initializing the MB_post object for one of the testing data files. The IR transform is supported by the class IR_factory:

if __name__ == "__main__":

    # ... initialization

    # Data files
    sim_file = '../tests/test_data/H2_GW/sim.h5'
    f = h5py.File(sim_file, 'r')
    it = f["iter"][()]
    G_tau = f["iter" + str(it) + "/G_tau/data"][()].view(complex)
    tau_mesh = f["iter" + str(it) + "/G_tau/mesh"][()]
    f.close()

    # Here the H2 GW simulation uses lambda = 1e4
    ir_file = '../tests/test_data/ir_grid/1e4.h5'
    beta = tau_mesh[-1]  # inverse temperature in simulation
    nts = tau_mesh.shape[0]
    my_ir = ir.IR_factory(beta, ir_file)

    # Fourier transform from G(tau) to  G(iw_n)
    G_iw = my_ir.tau_to_w(G_tau)
    # Fourier transform from G(iw_n) to G(tau)
    G_tau_2 = my_ir.w_to_tau(G_iw)

Analytic continuation

One of the primary steps in the post-processing of Green’s function data is analytic continuation (AC) of the results from imaginary time and frequency axis to the real frequency axis. The green-mbtools.pesto module supports several implementations for AC based on theory of:

In addition, an interface to the Maxent AC library is also provided (see examples/maxent.py.

Nevanlinna AC is one of the more robust approaches in our opinion and a direct interface is provided in the MB_post class. Continuing with the above examples, we can directly obtain the spectral function on the real axis using:

if __name__ == "__main__":

    # ... initialization

    # w_min and w_max: min and max real frequency bracket (in a.u.)
    # n_real: number of frequency points to consider -- more poitns = smoother plots
    # eta: Broadening parameter
    n_real = 10001
    w_min = -5.0
    w_max = 5.0
    eta = 0.01
    freqs, Aw = mb_obj.AC_nevanlinna(n_real=n_real, w_min=w_min, w_max=w_max, eta=eta)

Note

Most AC implementations use parallelization using multiprocessing utilities in python. As a result, analytic continuation should always be placed inside the if __name__ == "__main__": block.

Wannier interpolation

For solid-state systems, simulations are generally performed on evenly sampled grid-points in the first Brillouin zone in the reciprocal space. However, the spectral function and band structure are generally desirable along a special, high-symmetry path in the same Brillouin zone. This is facilitated by k-point or Wannier interpolation, which is provided by green_mbtools.pesto.ft library.

Examples for interpolating the mean-field (e.g., DFT, Hartree-Fock) and the correlated calculation (e.g., GW, GF2, SEET) results are provided in examples/winter_mean_field.py and examples/winter_correlated.py, respectively.

Following the above exmple, once the MB_post object is initialized, the interpolation can be performed directly using the function wannier_interpolation.

from ase.spacegroup import crystal
from green_mbtools.pest import mb

if __name__ == "__main__":

    # initialization

    # obtain high-symmetry path for fictitious H2 solid example using ase
    cc = crystal(
        symbols=['H', 'H'],
        basis=[(-0.25, -0.25, -0.25), (0.25, 0.25, 0.25)],
        spacegroup=group,
        cellpar=[a, b, c, alpha, beta, gamma], primitive_cell=True
    )
    path = cc.cell.bandpath('GXMGR', npoints=100)
    kpts_inter = path.kpts

    # Obtain interpolated Green's function, self-energy, tau-mesh, Fock and overlap matrix
    G_tk_int, Sigma_tk_int, tau_mesh, Fk_int, Sk_int = MB.wannier_interpolation(
        kpts_inter, hermi=True, debug=debug
    )

Note

The accuracy of interpolation depends on the density of the original Brillouin zone grid. To reduce the errors, we recommend using PySCF to directly calculate the one-body Hamiltonain and the overlap matrix along the high-symmetry path rather than interpolation. See, e.g., examples/useful_scripts/nvnl_winter_analysis.py.

Other useful scripts

For practical application, a combination of initialization, interpolation and analytic continuation is required. We provide several python scripts to automate these tasks for solid-state and molecular systems. These are available in the examples/useful_scripts directory.