5.4. Dynamical Green’s function

Using $\mathcal{H}\mathrm{\Phi }$, we can calculate a dynamical Green’s function

(5.16)$$I(z)=⟨{\mathrm{\Phi }}^{\prime }|\frac{1}{\mathcal{H}-z\hat{I}}|{\mathrm{\Phi }}^{\prime }⟩,$$

where $|{\mathrm{\Phi }}^{\prime }⟩=\hat{O}|{\mathrm{\Phi }}_0⟩$ is an excited state and $\hat{O}$ is an excitation operator defined as a single excitation operator

(5.17)$$\sum _{i,{\sigma }_1}{A}_{i{\sigma }_1}{c}_{i{\sigma }_1}(c_{i{\sigma }_1}^{†})$$

or a pair excitation operator

(5.18)$$\sum _{i,j,{\sigma }_1,{\sigma }_2}{A}_{i{\sigma }_{1}j{\sigma }_2}{c}_{i{\sigma }_1}{c}_{j{\sigma }_2}^{†}(c_{i{\sigma }_1}^{†}{c}_{j{\sigma }_2}).$$

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

(5.19)$$\hat{O}=\hat{S}(\mathbf{k})=\sum _j{\hat{S}}_j^z{e}^{i\mathbf{k}\cdot {\mathbf{r}}_{\mathbf{j}}}=\sum _j\frac{1}{2}(c_{j\uparrow }^{†}{c}_{j\uparrow }-c_{j\downarrow }^{†}{c}_{j\downarrow })e^{i\mathbf{k}\cdot {\mathbf{r}}_{\mathbf{j}}}.$$

There are two modes implemented in $\mathcal{H}\mathrm{\Phi }$. 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).