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 {\mathcal H}\Phi.

Prerequisite

The prerequisite of this utility is the same as that of mVMC or {\mathcal H}\Phi.

Supported quantities

This utility supports the Fourier transformation of the following quantities:

One-body correlations

\begin{align}
\langle {\hat c}_{{\bf k} \uparrow}^{\dagger} {\hat c}_{{\bf k} \uparrow}\rangle
&\equiv \frac{1}{N_{\rm cell}} \sum_{i j}^{N_{\rm site}} e^{-i {\bf k}\cdot({\bf R}_i - {\bf R}_j)}
\langle {\hat c}_{i \uparrow}^{\dagger} {\hat c}_{j \uparrow}\rangle
\\
\langle {\hat c}_{{\bf k} \downarrow}^{\dagger} {\hat c}_{{\bf k} \downarrow}\rangle
&\equiv \frac{1}{N_{\rm cell}} \sum_{i j}^{N_{\rm site}} e^{-i {\bf k}\cdot({\bf R}_i - {\bf R}_j)}
\langle {\hat c}_{i \downarrow}^{\dagger} {\hat c}_{j \downarrow}\rangle
\end{align}

Density-density correlation

\begin{align}
\langle {\hat \rho}_{\bf k} {\hat \rho}_{\bf k}\rangle
\equiv \frac{1}{N_{\rm cell}^2} \sum_{i j}^{N_{\rm site}} e^{-i {\bf k}\cdot({\bf R}_i - {\bf R}_j)}
\langle {\hat \rho}_{i} {\hat \rho}_{j}\rangle
\end{align}

Spin-Spin correlations

\begin{align}
\langle {\hat S}_{\bf k}^{z} {\hat S}_{\bf k}^{z} \rangle
&\equiv \frac{1}{N_{\rm cell}^2} \sum_{i j}^{N_{\rm site}} e^{-i {\bf k}\cdot({\bf R}_i - {\bf R}_j)}
\langle {\hat S}_{i}^{z} {\hat S}_{j}^{z} \rangle
\\
\langle {\hat S}_{\bf k}^{+} {\hat S}_{\bf k}^{-} \rangle
&\equiv \frac{1}{N_{\rm cell}^2} \sum_{i j}^{N_{\rm site}} e^{-i {\bf k}\cdot({\bf R}_i - {\bf R}_j)}
\langle {\hat S}_{i}^{+} {\hat S}_{j}^{-} \rangle
\\
\langle {\hat {\bf S}}_{\bf k} \cdot {\hat {\bf S}}_{\bf k} \rangle
&\equiv \frac{1}{N_{\rm cell}^2} \sum_{i j}^{N_{\rm site}} e^{-i {\bf k}\cdot({\bf R}_i - {\bf R}_j)}
\langle {\hat {\bf S}}_{i} \cdot {\hat {\bf S}}_{j} \rangle
\end{align}