gftool.lattice.lieb.dos_mp

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

Multi-precision DOS of non-interacting 2D lieb lattice.

The delta-peak at eps=0 is ommited and must be treated seperately! Without it, the DOS integrates to 2/3.

Besides the delta-peak, the DOS diverges at eps=±half_bandwidth/2**0.5.

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

\[\begin{split}∫^0 dϵ DOS(ϵ)[f(ϵ) - f(-D/\sqrt{2})] + ∫_0 dϵ DOS(ϵ)[f(ϵ) - f(+D/\sqrt{3})] \\ + [f(-D/\sqrt{2}) + f(0) + f(+D/\sqrt{2})]/3\end{split}\]

where \(D\) is the half_bandwidth, or symmetrize the integral.

The Green’s function and therefore the DOS of the 2D Lieb lattice can be expressed in terms of the 2D square lattice gftool.lattice.square.dos, see [kogan2021].

Parameters:

epsmpmath.mpf or mpf_like

DOS is evaluated at points eps.

half_bandwidthmpmath.mpf or mpf_like

Half-bandwidth of the DOS, DOS(| eps | > half_bandwidth) = 0. The half_bandwidth corresponds to the nearest neighbor hopping t=D * 2**1.5.

Returns:

mpmath.mpf

The value of the DOS.

See also

gftool.lattice.lieb.dos

Vectorized version suitable for array evaluations.

gftool.lattice.square.dos_mp

References

[kogan2021]

Kogan, E., Gumbs, G., 2020. Green’s Functions and DOS for Some 2D Lattices. Graphene 10, 1–12. https://doi.org/10.4236/graphene.2021.101001

Examples

Calculated integrals

>>> from mpmath import mp
>>> mp.identify(mp.quad(gt.lattice.lieb.dos_mp, [-1, -2**-0.5, 0, 2**-0.5, 1]))
'(2/3)'

>>> eps = np.linspace(-1.5, 1.5, num=501)
>>> dos_mp = [gt.lattice.lieb.dos_mp(ee, half_bandwidth=1) for ee in eps]

>>> import matplotlib.pyplot as plt
>>> for pos in (-2**-0.5, 0, +2**-0.5):
...     _ = plt.axvline(pos, color='black', linewidth=0.8)
>>> _ = plt.plot(eps, dos_mp)
>>> _ = 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)