gftool.square_gf_z

gftool.square_gf_z(z, half_bandwidth)

Local Green’s function of the 2D square lattice.

\[G(z) = \frac{2}{πz} ∫^{π/2}_{0} \frac{dϕ}{\sqrt{1 - (D/z)^2 \cos^2ϕ}}\]

where \(D\) is the half bandwidth and the integral is the complete elliptic integral of first kind. See [economou2006].

Parameters:

zcomplex np.ndarray or complex

Green’s function is evaluated at complex frequency z.

half_bandwidthfloat

Half-bandwidth of the DOS of the square lattice. The half_bandwidth corresponds to the nearest neighbor hopping t=D/4.

Returns:

complex np.ndarray or complex

Value of the square lattice Green’s function.

References

[economou2006]

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

Examples

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

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

(png, pdf)