gftool.lattice.sc.gf_z_mp

gftool.lattice.sc.gf_z_mp(z, half_bandwidth=1)

Multi-precision Green’s function of non-interacting 3D simple cubic lattice.

Has a van Hove singularity (continuous but not differentiable) at z = ±D/3.

Implements equations (1.24 - 1.26) from [delves2001].

Parameters:

zmpmath.mpc or mpc_like

Green’s function is evaluated at complex frequency z.

half_bandwidthmpmath.mpf or mpf_like

Half-bandwidth of the DOS of the simple cubic lattice. The half_bandwidth corresponds to the nearest neighbor hopping \(t=D/6\).

Returns:

mpmath.mpc

Value of the Green’s function at complex energy z.

References

[economou2006]

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

[delves2001]

Delves, R. T. and Joyce, G. S., Ann. Phys. 291, 71 (2001). https://doi.org/10.1006/aphy.2001.6148

Examples

>>> ww = np.linspace(-1.1, 1.1, num=500)
>>> gf_ww = np.array([gt.lattice.sc.gf_z_mp(wi) for wi in ww])

>>> import matplotlib.pyplot as plt
>>> _ = plt.axhline(0, color="black", linewidth=0.8)
>>> _ = plt.axvline(-1/3, color="black", linewidth=0.8)
>>> _ = plt.axvline(+1/3, color="black", linewidth=0.8)
>>> _ = plt.plot(ww.real, gf_ww.astype(complex).real, label=r"$\Re G$")
>>> _ = plt.plot(ww.real, gf_ww.astype(complex).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)