5.4. Dynamical Green’s function

Using HPhi++, we can calculate a dynamical Green’s function

\[I(z) = \langle \Phi ' | \frac{1}{ {\mathcal H}- z\hat{I} } | \Phi '\rangle,\]

where \(|\Phi ' \rangle = \hat{O} | \Phi _0 \rangle\) is an excited state and \(\hat{O}\) is an excitation operator defined as a single excitation operator

\[\sum_{i, \sigma_1} A_{i \sigma_1} c_{i \sigma_1} (c_{i\sigma_1}^{\dagger})\]

or a pair excitation operator

\[\sum_{i, j, \sigma_1, \sigma_2} A_{i \sigma_1 j \sigma_2} c_{i \sigma_1}c_{j \sigma_2}^{\dagger} (c_{i\sigma_1}^{\dagger}c_{j\sigma_2}).\]

For example, the dynamical spin susceptibilities can be calculated by defining \(\hat{O}\) as

\[\hat{O} = \hat{S}({\bf k}) = \sum_{j}\hat{S}_j^z e^{i {\bf k} \cdot \bf {r}_j} = \sum_{j}\frac{1}{2} (c_{j\uparrow}^{\dagger}c_{j\uparrow}-c_{j\downarrow}^{\dagger}c_{j\downarrow})e^{i {\bf k} \cdot \bf {r}_j}.\]

There are two modes implemented in \({\cal H}\Phi\). One is the continued fraction expansion method by using Lanczos method  [1] and the other is the shifted Krylov method [2] . See the reference for the details of each algorithm.

[1]	E. Dagotto, Rev. Mod. Phys. 66, 763-840 (1994).

[2]	S.Yamamoto, T. Sogabe, T. Hoshi, S.-L. Zhang, T. Fujiwara, Journal of the Physical Society of Japan 77, 114713 (2008).