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 \({\mathcal H}\Phi\).

7.1.1. Prerequisite

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

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

(7.2)\[\begin{align} \langle {\hat \rho}_{{\bf k}\alpha} {\hat \rho}_{{\bf k}\beta}\rangle \equiv \frac{1}{N_{\bf R}} \sum_{\bf R}^{N_{\bf R}} e^{-i {\bf k}\cdot{\bf R}} \langle ({\hat \rho}_{{\bf 0}\alpha} - \langle {\hat \rho}_{{\bf 0}\alpha} \rangle) ({\hat \rho}_{{\bf R}\beta} - \langle {\hat \rho}_{{\bf R}\beta} \rangle) \rangle \end{align}\]

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}