7.1. Overview¶
This document is the manual for the utility to perform the Fourier transformation of the correlation function in the site representation generated by mVMC or HPhi++.
7.1.1. Prerequisite¶
The prerequisite of this utility is the same as that of mVMC or HPhi++.
7.1.2. Supported quantities¶
This utility supports the Fourier transformation of the following quantities:
One-body correlations
\begin{align}
\langle {\hat c}_{{\bf k} \alpha \uparrow}^{\dagger} {\hat c}_{{\bf k} \beta \uparrow}\rangle
&\equiv \sum_{\bf R}^{N_{\bf R}} e^{-i {\bf k}\cdot{\bf R}}
\langle {\hat c}_{{\bf 0} \alpha \uparrow}^{\dagger} {\hat c}_{{\bf R} \beta \uparrow}\rangle
\\
\langle {\hat c}_{{\bf k} \alpha \downarrow}^{\dagger} {\hat c}_{{\bf k} \beta \downarrow}\rangle
&\equiv \sum_{\bf R}^{N_{\bf R}} e^{-i {\bf k}\cdot {\bf R}}
\langle {\hat c}_{{\bf 0} \alpha \downarrow}^{\dagger} {\hat c}_{{\bf R} \beta \downarrow}\rangle
\end{align}
Density-density correlation
\[\begin{align}
\langle {\hat \rho}_{{\bf k}\alpha} {\hat \rho}_{{\bf k}\beta}\rangle
\equiv \frac{1}{N_{\bf R}} \sum_{\bf R}^{N_{\bf R}} e^{-i {\bf k}\cdot{\bf R}}
\langle ({\hat \rho}_{{\bf 0}\alpha} - \langle {\hat \rho}_{{\bf 0}\alpha} \rangle)
({\hat \rho}_{{\bf R}\beta} - \langle {\hat \rho}_{{\bf R}\beta} \rangle) \rangle
\end{align}\]
Spin-Spin correlations
\begin{align}
\langle {\hat S}_{{\bf k}\alpha}^{z} {\hat S}_{{\bf k}\beta}^{z} \rangle
&\equiv \frac{1}{N_{\bf R}} \sum_{\bf R}^{N_{\bf R}} e^{-i {\bf k}\cdot{\bf R}}
\langle {\hat S}_{{\bf 0}\alpha}^{z} {\hat S}_{{\bf R}\beta}^{z} \rangle
\\
\langle {\hat S}_{{\bf k}\alpha}^{+} {\hat S}_{{\bf k}\beta}^{-} \rangle
&\equiv \frac{1}{N_{\bf R}} \sum_{\bf R}^{N_{\bf R}} e^{-i {\bf k}\cdot{\bf R}}
\langle {\hat S}_{{\bf 0}\alpha}^{+} {\hat S}_{{\bf R}\beta}^{-} \rangle
\\
\langle {\hat {\bf S}}_{{\bf k}\alpha} \cdot {\hat {\bf S}}_{{\bf k}\beta} \rangle
&\equiv \frac{1}{N_{\bf R}} \sum_{\bf R}^{N_{\bf R}} e^{-i {\bf k}\cdot{\bf R}}
\langle {\hat {\bf S}}_{{\bf 0}\alpha} \cdot {\hat {\bf S}}_{{\bf R}\beta} \rangle
\end{align}
