5.2. Full Diagonalization method

5.2.1. Overview

We generate the matrix of \(\hat{\mathcal H }\) by using the real space configuration \(| \psi_j \rangle\)(\(j=1\cdots d_{\rm H}\), where \(d_{\rm H}\) is the dimension of the Hilbert space): \({\mathcal H }_{ij}= \langle \psi_i | \hat {\mathcal H } | \psi_j \rangle\). By diagonalizing this matrix, we can obtain all the eigenvalues \(E_{i}\) and eigenvectors \(|\Phi_i\rangle\) (\(i=1 \cdots d_{\rm H}\)). In the diagonalization, we use a LAPACK routine, such as dsyev or zheev. We also calculate and output the expectation values \(A_i \equiv \langle \Phi_i | {\hat A} | \Phi_i\rangle\). These values are used for the finite-temperature calculations.

5.2.2. Finite-temperature calculations

From \(A_i \equiv \langle \Phi_i | {\hat A} | \Phi_i\rangle\), we calculate the finite-temperature properties by using the relation

(5.8)\[\langle {\hat A}\rangle=\frac{\sum_{i=1}^N A_i {\rm e}^{-\beta E_i}}{\sum_{i=1}^N{\rm e}^{-\beta E_i}}.\]

The calculation should be performed by using the own postscripts.