qmc-loop-algorithm-report.tex

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\date{2022-12-12}
\title{Path Integral Quantum Monte Carlo}
\subtitle{Quantum and Classical Algorithms for Quantum Many-Body Systems}
\author{Stefan Gehr}
\institute[FAU]{Friedrich-Alexander Universität Erlangen-Nürnberg}

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\begin{document}
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\begin{frame}{Outline}
	\tableofcontents
\end{frame}

\section{Classical Monte Carlo}
\subsection{Classical Ising Model}
\subsection{Markov Chain}
\subsection{Detailed Balance Condition}
\subsection{Metropolis Algorithm}
\begin{frame}{Monte Carlo Basics}
\begin{minipage}{0.48\linewidth}
	%\only<2>{
	%\visible<2>{
\begin{block}{Classical Ising Model}
	\begin{align*}
		H(\vec{\sigma}) &= - \sum_{i}\left(J\sigma_i\sigma_{i+1} + \mu \sigma_i\right) \\
		\sigma_i &\in \{-1, +1\}\qquad \vec{\sigma} = (\sigma_1, \dots, \sigma_N) \\
		Z &= \sum_{\vec{\sigma}} w_{\vec{\sigma}} = \sum_{\vec{\sigma}}e^{-\beta H(\vec{\sigma})}
	\end{align*}
\end{block}
	%}
\begin{block}{Markov Chain}
	\begin{align*}
		\vec{\sigma}_1\to \vec{\sigma}_2 \to \vec{\sigma}_3 \to \cdots
	\end{align*}
	The number of occurrences of \(\vec{\sigma}\) according to Boltzmann weight
	\[N_{\vec{\sigma}} \propto e^{-\beta H(\vec{\sigma})} =: w_{\vec{\sigma}}\]
\end{block}
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\begin{block}{Exponential Configuration Space}
	\begin{align*}
		C = \{\vec{\sigma} = (\sigma_1,\dots,\sigma_N) \,|\, \sigma_i \in \{-1,+1\}\}
	\end{align*}
	Most configurations \(\vec{\sigma}_i\) are highly unlikely