gftool.bethe_gf_z

gftool.bethe_gf_z(z, half_bandwidth)

Local Green’s function of Bethe lattice for infinite coordination number.

\[G(z) = 2(z - s\sqrt{z^2 - D^2})/D^2\]

where \(D\) is the half bandwidth and \(s=sgn[ℑ{ξ}]\). See [georges1996].

Parameters:

zcomplex array_like or complex

Green’s function is evaluated at complex frequency z.

half_bandwidthfloat

Half-bandwidth of the DOS of the Bethe lattice. The half_bandwidth corresponds to the nearest neighbor hopping t=D/2.

Returns:

complex np.ndarray or complex

Value of the Bethe Green’s function.

References

[georges1996]

Georges et al., Rev. Mod. Phys. 68, 13 (1996) https://doi.org/10.1103/RevModPhys.68.13

Examples

>>> ww = np.linspace(-1.5, 1.5, num=500)
>>> gf_ww = gt.lattice.bethe.gf_z(ww, half_bandwidth=1)

>>> import matplotlib.pyplot as plt
>>> _ = plt.plot(ww, gf_ww.real, label=r"$\Re G$")
>>> _ = plt.plot(ww, gf_ww.imag, '--', label=r"$\Im G$")
>>> _ = plt.xlabel(r"$\omega/D$")
>>> _ = plt.ylabel(r"$G*D$")
>>> _ = plt.axhline(0, color='black', linewidth=0.8)
>>> _ = plt.xlim(left=ww.min(), right=ww.max())
>>> _ = plt.legend()
>>> plt.show()

(png, pdf)