gftool.lattice.onedim.dos_mp

gftool.lattice.onedim.dos_mp(eps, half_bandwidth=1)

Multi-precision DOS of non-interacting 1D lattice.

Diverges at the band-edges abs(eps) = half_bandwidth.

This function is particularity suited to calculate integrals of the form \(∫dϵ DOS(ϵ)f(ϵ)\). If you have problems with the convergence, consider removing singularities, e.g. split the integral

\[∫^0 dϵ DOS(ϵ)[f(ϵ) - f(-D)] + ∫_0 dϵ DOS(ϵ)[f(ϵ) - f(+D)] + [f(-D) + f(+D)]/2\]

or symmetrize the integral.

Parameters:

epsmpmath.mpf or mpf_like

DOS is evaluated at points eps.

half_bandwidthmpmath.mpf or mpf_likeloat

Half-bandwidth of the DOS, DOS(| eps | > half_bandwidth) = 0. The half_bandwidth corresponds to the nearest neighbor hopping t=D/2.

Returns:

mpmath.mpf

The value of the DOS.

See also

gftool.lattice.onedim.dos

Vectorized version suitable for array evaluations.

References

[economou2006]

Economou, E. N. Green’s Functions in Quantum Physics. Springer, 2006.

Examples

Calculate integrals (the 1D DOS needs higher accuracy for accurate results):

>>> from mpmath import mp
>>> with mp.workdps(35, normalize_output=True):
...     norm = mp.quad(gt.lattice.onedim.dos_mp, [-1, +1])
>>> norm
mpf('1.0')

>>> eps = np.linspace(-1.1, 1.1, num=501)
>>> dos_mp = [gt.lattice.onedim.dos_mp(ee, half_bandwidth=1) for ee in eps]

>>> import matplotlib.pyplot as plt
>>> _ = plt.plot(eps, dos_mp)
>>> for pos in (-1, 0, +1):
...     _ = plt.axvline(pos, color='black', linewidth=0.8)
>>> _ = plt.xlabel(r"$\epsilon/D$")
>>> _ = plt.ylabel(r"DOS * $D$")
>>> _ = plt.ylim(bottom=0)
>>> _ = plt.xlim(left=eps.min(), right=eps.max())
>>> plt.show()

(png, pdf)