diff --git a/src/qmc-loop-algorithm-report.tex b/src/qmc-loop-algorithm-report.tex
index a7064a38522fa01faa15e959a9d0528276fb3722..1c4f3f37d578087d5bbc02dad0d0550271660b98 100644
--- a/src/qmc-loop-algorithm-report.tex
+++ b/src/qmc-loop-algorithm-report.tex
@@ -162,7 +162,7 @@
\die{} = \num{0.63} > \num{0.45} \Rightarrow \text{decline}
\end{align*}
}
-\only<5>{\Centering\includegraphics[width=0.56\linewidth]{bttf.jpg}}
+\only<5| handout:0>{\Centering\includegraphics[width=0.56\linewidth]{bttf.jpg}}
\only<1-4,6->{
\visible<7->{
Suggest to flip third site \(\vec{\sigma}_3 \overset{?}{=} \uparrow\uparrow\uparrow\uparrow\)
@@ -253,14 +253,48 @@ Always accept the new configuration
\blfootnote{\cite{Gubernatis2016}}
\blfootnote{\cite{gehrbachelor}}
\end{frame}
+\begin{frame}{Interim Summary}
+\begin{minipage}{0.48\linewidth}
+\begin{block}{Problem}
+ \begin{itemize}
+ \item Exponential configuration space
+ \item Most configurations have \(\approx 0\) contribution
+ \end{itemize}
+\end{block}
+\begin{block}{Monte Carlo}
+ \begin{itemize}
+ \item Create Markov Chain with detailed-balance algorithm
+ \begin{itemize}
+ \item Metropolis
+ \item Heat-Bath
+ \end{itemize}
+ \item Calculate observable for each configuration in chain
+ \item Average of observable is expectation value
+ \item Use blocking analysis to get independent values
+ \end{itemize}
+\end{block}
+\end{minipage}
+\hfill
+\begin{minipage}{0.48\linewidth}
+\begin{block}{Metropolis}
+ \begin{itemize}
+ \item Suggest a change to the last configuration \(C \to C'\)
+ \item Accept it with certain probability
+ \item If declined that means the old configuration \(C\) was re-sampled
+ \end{itemize}
+\end{block}
+\begin{block}{Heat-Bath}
+ \begin{itemize}
+ \item Local change in energy, by setting local values in the configuration
+ \item Set local value according to local Boltzmann distribution
+ \end{itemize}
+\end{block}
+\end{minipage}
+\end{frame}
\section{Quantum Monte Carlo}
-\subsection{XXZ Quantum Spin Chain}
-\subsection{Trotter Decomposition}
-\subsection{Updating Schemes}
-\subsection{Loop Updates}
-\subsection{Sign Problem}
\begin{frame}{Quantum Monte Carlo}
\begin{minipage}{0.45\linewidth}
+\subsection{XXZ Quantum Spin Chain}
\begin{block}{XXZ Quantum Spin Chain}
\begin{align*}
H &= J_x \sum_i (S_i^xS_{i+1}^x + S_i^yS_{i+1}^y) + J_z\sum_i S_i^z S_{i+1}^z \\
@@ -295,6 +329,7 @@ Always accept the new configuration
\end{minipage}
\blfootnote{\cite{Assaad}}
\end{frame}
+\subsection{Trotter Decomposition}
\begin{frame}{Trotter Decomposition}
\begin{minipage}{0.5\linewidth}
\begin{small}
@@ -327,6 +362,7 @@ Always accept the new configuration
\blfootnote{\cite{bchformula}}
\blfootnote{\cite{Sandvik2010}}
\end{frame}
+\subsection{Updating Schemes}
\begin{frame}{Updating Schemes}
\begin{minipage}{0.48\linewidth}
\begin{block}{Local Update}
@@ -342,12 +378,13 @@ Always accept the new configuration
\end{minipage}
\hfill
\begin{minipage}{0.48\linewidth}
-\begin{block}{Single Loop Update}
+\begin{block}{Local Loop Update}
\includegraphics[width=0.98\linewidth]{singleloopupdate.pdf}
\end{block}
\end{minipage}
\blfootnote{\cite{Assaad}}
\end{frame}
+\subsection{Loop Updates}
\begin{frame}{Loop Updates}
\begin{minipage}{0.48\linewidth}
\includegraphics[width=\linewidth]{graphmap.pdf}
@@ -364,6 +401,7 @@ Always accept the new configuration
\end{minipage}
\blfootnote{\cite{Assaad}}
\end{frame}
+\subsection{Sign Problem}
\begin{frame}{Sign Problem}
\begin{minipage}{0.48\linewidth}
\begin{block}{Partition Sum}
@@ -381,7 +419,7 @@ Always accept the new configuration
\begin{block}{New Expectation Value}
\begin{align*}
\expval{O}
- &= \tr\left(e^{-\beta H}O\right) \\
+ &= \frac{1}{Z}\tr\left(e^{-\beta H}O\right) \\
&\overset{!}{=} \frac{\sum_Cw_CO_C}{\sum_Cw_C}
= \frac{\sum_C\abs{w_C}\text{sign}_CO_C}{\sum_C\abs{w_C}\text{sign}_C} \\
&\approx \frac{\sum_{i=1}^{N} \text{sign}_{C_i}O_{C_i}}{\sum_{i=1}^{N} \text{sign}_{C_i}}
@@ -390,9 +428,61 @@ Always accept the new configuration
\end{minipage}
\blfootnote{\cite{werner}}
\end{frame}
+\begin{frame}{Summary}
+\begin{minipage}{0.48\linewidth}
+\begin{block}{Problem}
+ \begin{itemize}
+ \item Exponential configuration space
+ \item Most configurations have \(\approx 0\) contribution
+ \end{itemize}
+\end{block}
+\begin{block}{Monte Carlo}
+ \begin{itemize}
+ \item Create Markov Chain with detailed-balance algorithm
+ \begin{itemize}
+ \item Metropolis
+ \item Heat-Bath
+ \end{itemize}
+ \item Calculate observable for each configuration in chain
+ \item Average of observable is expectation value
+ \item Use blocking analysis to get independent values
+ \end{itemize}
+\end{block}
+\end{minipage}
+\hfill
+\begin{minipage}{0.48\linewidth}
+\begin{block}{Quantum Monte Carlo}
+\begin{itemize}
+ \item Rewrite partition sum as sum of weights
+ \begin{align*}
+ Z = \tr(e^{-\beta H}) \overset{!}{=} \sum_C w_C
+ \end{align*}
+ \begin{itemize}
+ \item Maybe apply path integral
+ \begin{small}
+ \begin{align*}
+ Z = \sum_{C=(\vec{\sigma}_1,\hdots,\vec{\sigma}_N)}\bra{\vec{\sigma}_1}e^{-\Delta\tau H}\ket{\vec{\sigma}_2}\cdots\bra{\vec{\sigma}_N}e^{-\Delta\tau H}\ket{\vec{\sigma}_1}
+ \end{align*}
+ \end{small}
+ \item Apply Trotter decomposition if possible
+ \end{itemize}
+ \item Find corresponding formula for observables
+ \begin{align*}
+ \expval{O} = \frac{1}{Z}\tr(e^{-\beta H}O) \overset{!}{=} \frac{\sum_C w_CO_C}{\sum_Cw_C}
+ \end{align*}
+ \item Sample with classical Monte Carlo using \(\abs{w_C}\)
+ \item Pray to god that the sign problem is not too big
+\end{itemize}
+\end{block}
+\end{minipage}
+\end{frame}
+\begin{frame}{The End}
+ \Centering
+ \Large{Thank you for your attention}
+\end{frame}
{
- \nocite{*}
+ %\nocite{*}
\printbibliography
}