gftool.lattice.onedim.hilbert_transform¶
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gftool.lattice.onedim.hilbert_transform(xi, half_bandwidth)[source]¶ Hilbert transform of non-interacting DOS of the 1D 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 1D lattice
- Returns
- hilbert_transformcomplex np.ndarray or complex
Hilbert transform of xi.
See also
Notes
Relation between nearest neighbor hopping t and half-bandwidth D
\[2t = D\]