summaries

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
 }