gftool.lattice.triangular.gf_z

gftool.lattice.triangular.gf_z(z, half_bandwidth)

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

Note, that the spectrum is asymmetric and in \([-2D/3, 4D/3]\), where \(D\) is the half-bandwidth. The Green’s function is evaluated as complete elliptic integral of first kind, see [horiguchi1972].

Parameters

zcomplex np.ndarray or complex

Green’s function is evaluated at complex frequency z.

half_bandwidthfloat

Half-bandwidth of the DOS of the triangular lattice. The half_bandwidth corresponds to the nearest neighbor hopping \(t=4D/9\).

Returns

gf_zcomplex np.ndarray or complex

Value of the triangular lattice Green’s function

References

horiguchi1972

Horiguchi, T., 1972. Lattice Green’s Functions for the Triangular and Honeycomb Lattices. Journal of Mathematical Physics 13, 1411–1419. https://doi.org/10.1063/1.1666155

Examples

>>> ww = np.linspace(-1.5, 1.5, num=500, dtype=complex) + 1e-64j
>>> gf_ww = gt.lattice.triangular.gf_z(ww, half_bandwidth=1)

>>> import matplotlib.pyplot as plt
>>> _ = plt.axhline(0, color='black', linewidth=0.8)
>>> _ = plt.axvline(-2/3, color='black', linewidth=0.8)
>>> _ = plt.axvline(+4/3, color='black', linewidth=0.8)
>>> _ = plt.plot(ww.real, gf_ww.real, label=r"$\Re G$")
>>> _ = plt.plot(ww.real, gf_ww.imag, '--', label=r"$\Im G$")
>>> _ = plt.ylabel(r"$G*D$")
>>> _ = plt.xlabel(r"$\omega/D$")
>>> _ = plt.xlim(left=ww.real.min(), right=ww.real.max())
>>> _ = plt.legend()
>>> plt.show()

(png, pdf)