gftool.lattice.rectangular.hilbert_transform
- gftool.lattice.rectangular.hilbert_transform(xi, half_bandwidth, scale)[source]
Hilbert transform of non-interacting DOS of the rectangular lattice.
The Hilbert transform is defined
\[\tilde{D}(ξ) = ∫_{-∞}^{∞}dϵ \frac{DOS(ϵ)}{ξ − ϵ}\]The lattice Hilbert transform is the same as the non-interacting Green’s function.
- Parameters:
- xicomplex np.ndarray or complex
Point at which the Hilbert transform is evaluated.
- half_bandwidthfloat
Half-bandwidth of the DOS of the 2D rectangular lattice. The half_bandwidth corresponds to the nearest neighbor hopping \(t=D/2/(scale+1)\).
- scalefloat
Relative scale of the different hoppings \(t_1=scale*t_2\). scale=1 corresponds to the square lattice.
- Returns:
- complex np.ndarray or complex
Hilbert transform at xi.
See also
Notes
Relation between nearest neighbor hopping t, scale γ and half-bandwidth D
\[2(γ+1)t = D\]