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

gf_zmpmath.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)