diff --git a/src/qmc-loop-algorithm-report.tex b/src/qmc-loop-algorithm-report.tex index 1c4f3f37d578087d5bbc02dad0d0550271660b98..6962fbbd05cd0a8f42f1edbed54eb098767ace9a 100644 --- a/src/qmc-loop-algorithm-report.tex +++ b/src/qmc-loop-algorithm-report.tex @@ -401,6 +401,34 @@ Always accept the new configuration \end{minipage} \blfootnote{\cite{Assaad}} \end{frame} +\begin{frame}{Example} +\begin{minipage}{0.48\linewidth} +\begin{block}{Heisenberg Model} +\begin{align*} + H = J\sum_i (S_i^xS_{i+1}^x + S_i^yS_{i+1}^y + S_i^zS_{i+1}^z) +\end{align*} +\(\implies\) Only graphs \(G=1\) and \(G=2\) need to be considered +\end{block} +\begin{block}{System of Equations} + \begin{align*} + e^{\Delta\tau J /4}\cosh(\Delta\tau J /2) &= W_1 = W_{1,1} + W_{1,2} \\ + e^{\Delta\tau J /4}\sinh(\Delta\tau J /2) &= W_2 = W_{2,2} = W_{1,2} \\ + e^{-\Delta\tau J /4} &= W_3 = W_{1,1} = W_{3,1} + .\end{align*} +\end{block} +\end{minipage} +\hfill +\begin{minipage}{0.48\linewidth} +\begin{block}{Horizontal Breakup Probabilities} +\begin{align*} + \vcenter{\hbox{\includegraphics{square.left.pdf}}}, \vcenter{\hbox{\includegraphics{square.right.pdf}}} &\equiv \tanh(\Delta\tau J /2) \\ + \vcenter{\hbox{\includegraphics{square.diag.lr.pdf}}}, \vcenter{\hbox{\includegraphics{square.diag.rl.pdf}}} &\equiv 1 \\ + \vcenter{\hbox{\includegraphics{square.none.pdf}}}, \vcenter{\hbox{\includegraphics{square.left.right.pdf}}} &\equiv 0 +.\end{align*} +\end{block} +\end{minipage} +\blfootnote{Assaad} +\end{frame} \subsection{Sign Problem} \begin{frame}{Sign Problem} \begin{minipage}{0.48\linewidth}