Issue with H + P at finite T: does not agree with zero T for T = 0.0000001

[Chiffafox]

Stepan ran calculations for H + P at finite but ultralow T, such that there should effectively be no change compared to the zero T calculation. This is because at ultra low T the coupling strengths in the fictitious Hilbert space are vanishingly small.

However, the two approaches give majorly different bandplots. By comparison, making the same check with either H or P on their own shows no difference between ultralow T and zero T bandplots.

[matthewcarbone]

@Chiffafox I actually was able to test this today. Here's some results, which would appear to imply this problem is resolved:

Code used to generate the images:

#!/usr/bin/env python3

import numpy as np

import sys

sys.path.append("..")

from ggce import Model, System, DenseSolver  # noqa


if __name__ == "__main__":

    k = 0.5 * np.pi
    w = np.linspace(-2, 5, 50)

    # Check the true T=0 case
    model = Model.from_parameters(hopping=0.1, temperature=0.0)
    model.add_("Holstein", 1.25, 2, 2, dimensionless_coupling_strength=2.5)
    model.add_("Peierls", 2.25, 2, 2, dimensionless_coupling_strength=2.5)
    solver = DenseSolver(System(model))
    results = solver.spectrum(k, w, eta=0.005, pbar=False).squeeze()
    np.savetxt("T0.txt", np.array([w, -results.imag / np.pi]).T)

    # Check the true T=epsilon case
    model = Model.from_parameters(hopping=0.1, temperature=1e-6)
    model.add_(
        "Holstein",
        1.25,
        2,
        2,
        phonon_extent_tfd=2,
        phonon_number_tfd=3,
        dimensionless_coupling_strength=2.5,
    )
    model.add_(
        "Peierls",
        2.25,
        2,
        2,
        phonon_extent_tfd=2,
        phonon_number_tfd=3,
        dimensionless_coupling_strength=2.5,
    )
    solver = DenseSolver(System(model))
    results = solver.spectrum(k, w, eta=0.005, pbar=False).squeeze()
    np.savetxt("Tepsilon.txt", np.array([w, -results.imag / np.pi]).T)

Do you believe this is sufficient to put this to bed? If not please let me know, I can run some other tests in some other parameter regimes, perhaps.

[matthewcarbone]

@Chiffafox this is now safely resolved, right? 😁

[Chiffafox]

Yessir! And we have memorialized it as a test. Should I link it to that PR somehow?

[matthewcarbone]

Sure go for it, then feel free to close the issue! Thanks!

[Chiffafox]

With @x94carbone commits above and using the finT test from #45 , we confirm this issue is closed.