From 768cf9208ffa3dabf72f268d4aa940bcbc1f7d74 Mon Sep 17 00:00:00 2001
From: Stefan Gehr <stefan.kerman.gehr@fau.de>
Date: Sat, 10 Dec 2022 19:19:35 +0100
Subject: [PATCH] more progress

---
 figures/BoFoQoOo_gausshist_polarisation.pdf |   Bin 0 -> 100249 bytes
 figures/BoFoQoOo_rplot.pdf                  |   Bin 0 -> 137894 bytes
 figures/die.png                             |   Bin 0 -> 6047 bytes
 figures/localupdate.pdf                     |   Bin 0 -> 2612 bytes
 figures/singleloopupdate.pdf                |   Bin 0 -> 471944 bytes
 figures/singleloopupdate.svg                | 18358 ++++++++++++++++++
 figures/sixvertex-map.pdf                   |   Bin 0 -> 93697 bytes
 figures/sixvertex-map.svg                   |  5589 ++++++
 figures/sixvertex-worldline.pdf             |   Bin 0 -> 360277 bytes
 figures/sixvertex-worldline.svg             | 12730 ++++++++++++
 figures/worldline.pdf                       |   Bin 13108 -> 13760 bytes
 src/qmc-loop-algorithm-report.tex           |   322 +-
 12 files changed, 36904 insertions(+), 95 deletions(-)
 create mode 100644 figures/BoFoQoOo_gausshist_polarisation.pdf
 create mode 100644 figures/BoFoQoOo_rplot.pdf
 create mode 100644 figures/die.png
 create mode 100644 figures/localupdate.pdf
 create mode 100644 figures/singleloopupdate.pdf
 create mode 100644 figures/singleloopupdate.svg
 create mode 100644 figures/sixvertex-map.pdf
 create mode 100644 figures/sixvertex-map.svg
 create mode 100644 figures/sixvertex-worldline.pdf
 create mode 100644 figures/sixvertex-worldline.svg

diff --git a/figures/BoFoQoOo_gausshist_polarisation.pdf b/figures/BoFoQoOo_gausshist_polarisation.pdf
new file mode 100644
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diff --git a/figures/sixvertex-worldline.pdf b/figures/sixvertex-worldline.pdf
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diff --git a/figures/worldline.pdf b/figures/worldline.pdf
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diff --git a/src/qmc-loop-algorithm-report.tex b/src/qmc-loop-algorithm-report.tex
index aeea767..213d034 100644
--- a/src/qmc-loop-algorithm-report.tex
+++ b/src/qmc-loop-algorithm-report.tex
@@ -19,165 +19,297 @@
 \addbibresource{references.bib}
 
 \date{2022-12-12}
-\title{Quantum Monte Carlo and the Loop Algorithm}
-\subtitle{Physics Seminar}
+\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}
 
+\newcommand{\die}{\vcenter{\hbox{\includegraphics[width=1em]{die.png}}}}
+
 \begin{document}
 \begin{trueplainframe}
 \titlepage
 \end{trueplainframe}
 
-\begin{frame}{Table of contents}
+\begin{frame}{Outline}
 	\tableofcontents
 \end{frame}
 
-\section{Classic Monte Carlo}
-\begin{frame}{Metropolis Algorithm}
-\begin{minipage}{0.45\linewidth}
-	Classic Ising model
+\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*}
-	We want to create a Markov chain of configurations \((\vec{\sigma}_1, \vec{\sigma}_2, \dots)\) where
-	\(N_{\vec{\sigma}_i} \propto e^{-\beta E(\vec{\sigma}_i)}\) \\
-	Detailed Balance condition
+\end{block}
+	%}
+\begin{block}{Markov Chain}
 	\begin{align*}
-			w_{\vec{\sigma}_i}\,	p(\vec{\sigma}_i \to \vec{\sigma}_j)
-		=	w_{\vec{\sigma}_j}\,	p(\vec{\sigma}_j \to \vec{\sigma}_i)
+		\vec{\sigma}_1\to \vec{\sigma}_2 \to \vec{\sigma}_3 \to \cdots
 	\end{align*}
-	assures configuration \(\vec{\sigma}_i\) is sampled with correct weight \(w_{\vec{\sigma}_i}\).
+	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}
 \end{minipage}
 \hfill
+\begin{minipage}{0.48\linewidth}
+\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
+	\(w_{\vec{\sigma}_i}\approx 0\) \\
+	\(\Rightarrow\) Only sample the important configurations
+	% TODO Tikz circle in circle
+\end{block}
+\begin{block}{Detailed Balance condition}
+	\begin{align*}
+			w_{\vec{\sigma}}\,	p(\vec{\sigma} \to \vec{\sigma}')
+		=	w_{\vec{\sigma}'}\,	p(\vec{\sigma}' \to \vec{\sigma})
+	\end{align*}
+	Assures configuration \(\vec{\sigma}\) is sampled with correct weight \(w_{\vec{\sigma}}\)
+\end{block}
+\begin{block}{Ergodicity}
+	All configurations should be reachable after finitely many steps
+\end{block}
+\end{minipage}
+\end{frame}
+\begin{frame}{Metropolis Algorithm}
 \begin{minipage}{0.45\linewidth}
-	Metropolis:
+\begin{block}{Split the Probability}
 	\begin{align*}
-		p(\vec{\sigma}_i \to  \sigma_j) = p_{\text{prop}}(\vec{\sigma}_i \to  \sigma_j)p_{\text{acc}}(\vec{\sigma}_i \to  \sigma_j) \\
-		\Rightarrow \frac{p_{\text{acc}}(\vec{\sigma}_i\to \vec{\sigma}_j)}{p_{\text{acc}}(\vec{\sigma}_j\to \vec{\sigma}_i)}
-		= \frac{p_{\text{prop}}(\vec{\sigma}_j\to \vec{\sigma}_i)w_{\vec{\sigma}_j}}{p_{\text{prop}}(\vec{\sigma}_i\to \vec{\sigma}_j)w_{\vec{\sigma}_i}}
+		p(\vec{\sigma} \to  \vec{\sigma}') = p_{\text{prop}}(\vec{\sigma} \to  \vec{\sigma}')p_{\text{acc}}(\vec{\sigma} \to  \vec{\sigma}') \\
+		\Rightarrow \frac{p_{\text{acc}}(\vec{\sigma}\to \vec{\sigma}')}{p_{\text{acc}}(\vec{\sigma}'\to \vec{\sigma})}
+		= \frac{p_{\text{prop}}(\vec{\sigma}'\to \vec{\sigma})w_{\vec{\sigma}'}}{p_{\text{prop}}(\vec{\sigma}\to \vec{\sigma}')w_{\vec{\sigma}}}
 	\end{align*}
-	Choose e.g. \(p_{\text{prop}}(\vec{\sigma}_i\to \vec{\sigma}_j) = \frac{1}{N}\) (flip one of \(N\) sites)
-	Accept the new configuration \(\vec{\sigma}_j\) with probability
+	Choose e.g. \(p_{\text{prop}}(\vec{\sigma}\to \vec{\sigma}') = \frac{1}{N}\) (flip one of \(N\) sites)
+	Accept the new configuration \(\vec{\sigma}'\) with probability
 	\begin{align*}
-		\min\left(1,\frac{p_{\text{acc}}(\vec{\sigma}_i\to \vec{\sigma}_j)}{p_{\text{acc}}(\vec{\sigma}_j\to \vec{\sigma}_i)}\right)
-		= \min\left(1,e^{-\beta\, [H(\vec{\sigma}_j) - H(\vec{\sigma}_i)]}\right)
-	.\end{align*}
+		\min\left(1,\frac{p_{\text{acc}}(\vec{\sigma}\to \vec{\sigma}')}{p_{\text{acc}}(\vec{\sigma}'\to \vec{\sigma})}\right)
+		= \min\left(1,e^{-\beta\, [H(\vec{\sigma}') - H(\vec{\sigma})]}\right)
+	\end{align*}
+\end{block}
+\end{minipage}
+\hfill
+\begin{minipage}{0.45\linewidth}
+\visible<2->{
+\begin{block}{Markov Chain}
+\begin{align*}
+	\visible<2->{&\vec{\sigma}_1 \to \cdots \to \vec{\sigma}_A} \\
+	\visible<3->{&\vec{\sigma}_A \overset{\text{change}}{\implies} \vec{\sigma}_B} \\
+	\visible<4->{
+	&\begin{cases}
+		\visible<4->{\die{} \le \min\left(1,\frac{p_{\text{acc}}(\vec{\sigma}_A\to \vec{\sigma}_B)}{p_{\text{acc}}(\vec{\sigma}_B\to \vec{\sigma}_A)}\right) \\
+					 \implies \vec{\sigma}_1 \to \cdots \to \vec{\sigma}_A \to \vec{\sigma}_B} \\
+		\visible<5->{\die{} > \min\left(1,\frac{p_{\text{acc}}(\vec{\sigma}_A\to \vec{\sigma}_j)}{p_{\text{acc}}(\vec{\sigma}_j\to \vec{\sigma}_A)}\right) \\
+					 \implies \vec{\sigma}_1 \to \cdots \to \vec{\sigma}_A \to \vec{\sigma}_A}
+	\end{cases}}
+\end{align*}
+\end{block}
+}
 \end{minipage}
 \end{frame}
-\begin{frame}{Example with Numbers}
-	\(N = 4, J = 1, \mu = 2, \beta = 1\) \\
-	Start with random configuration \(\vec{\sigma}_0 = (\uparrow, \uparrow, \downarrow, \uparrow) = (1, 1, -1, 1)\).
-	Markov chain \(M = (\vec{\sigma}_0) = ((\uparrow, \uparrow, \downarrow, \uparrow))\).
-	\begin{align*}
-		H((\uparrow, \uparrow, \downarrow, \uparrow))
-		= -1\left(\uparrow \uparrow + \uparrow \downarrow + \downarrow \uparrow + \uparrow \uparrow\right) -2 \left(\uparrow + \uparrow + \downarrow + \uparrow\right) = -2\times 2 = -4
-	.\end{align*}
-	Suggest to flip first site \(\vec{\sigma}_1 \overset{?}{=} (\downarrow,\uparrow,\downarrow,\uparrow)\)
-	\begin{align*}
-		H((\downarrow, \uparrow, \downarrow, \uparrow))
-		= -1\left(\downarrow \uparrow + \uparrow \downarrow + \downarrow \uparrow + \uparrow \downarrow\right) -2 \left(\downarrow + \uparrow + \downarrow + \uparrow\right) = -1 \times (-4) = 4
-	.\end{align*}
+\begin{frame}{Hands-On Example}
+\begin{minipage}{0.48\linewidth}
+\visible<1->{
+	\(N = 4, J = 1, \mu = 2, \beta = \num{0.1}\) \\
+	Start with random configuration \(\vec{\sigma}_1 = \uparrow \uparrow \downarrow \uparrow = (1, 1, -1, 1)\).
+}
+\visible<2->{
+	\begin{align*}
+		H(\uparrow\uparrow\downarrow\uparrow) = -4
+	\end{align*}
+}
+\visible<3->{
+	Suggest to flip first site \(\vec{\sigma}_2 \overset{?}{=} \downarrow\uparrow\downarrow\uparrow\)
+	\begin{align*}
+		H(\downarrow\uparrow\downarrow\uparrow) = 4
+	\end{align*}
+}
+\visible<4->{
 	Accept with probability
 	\begin{align*}
-		\min\left(1, e^{-1 \left[4 - (-4)\right]}\right) = e^{-8} \approx \num{0.000335} \\
-		\text{random number }r = \num{0.2} > \num{0.000335} \Rightarrow \text{decline} \Rightarrow \vec{\sigma}_1 = \vec{\sigma}_0 = (\uparrow, \uparrow, \downarrow, \uparrow) \\
-		M = (\vec{\sigma}_0, \vec{\sigma}_1) = ((\uparrow, \uparrow, \downarrow, \uparrow), (\uparrow, \uparrow, \downarrow, \uparrow))
-	.\end{align*}
-	Suggest to flip third site \(\vec{\sigma}_2 \overset{?}{=} (\uparrow, \uparrow, \uparrow, \uparrow)\)
-	\begin{align*}
-		H((\uparrow, \uparrow, \uparrow, \uparrow))
-		= -1\left(\uparrow \uparrow + \uparrow \uparrow + \uparrow \uparrow + \uparrow \uparrow\right) -2 \left(\uparrow + \uparrow + \uparrow + \uparrow\right) = -1 \times 4 - 2 \times 4 = -12 \\
-		\min\left(1, e^{-1 \left[-12 - (-4)\right]}\right) = \min(1, e^{16}) = 1 \ge r \Rightarrow \text{accept} \\
-		\Rightarrow M = (\vec{\sigma}_0, \vec{\sigma}_1, \vec{\sigma}_2) = ((\uparrow, \uparrow, \downarrow, \uparrow), (\uparrow, \uparrow, \downarrow, \uparrow), (\uparrow, \uparrow, \uparrow, \uparrow))
-	.\end{align*}
+		\min\left(1, e^{-\num{0.1} \times \left[4 - (-4)\right]}\right) = e^{-\num{0.8}} \approx \num{0.45} \\
+		\die{} = \num{0.63} > \num{0.45} \Rightarrow \text{decline} \Rightarrow \vec{\sigma}_1 = \vec{\sigma}_0 = \uparrow\uparrow\downarrow\uparrow
+	\end{align*}
+}
+\visible<6->{
+	Suggest to flip third site \(\vec{\sigma}_3 \overset{?}{=} \uparrow\uparrow\uparrow\uparrow\)
+	\begin{align*}
+		H(\uparrow\uparrow\uparrow\uparrow) = -12 \\
+		\min\left(1, e^{-\num{0.1}\times \left[-12 - (-4)\right]}\right) = \min(1, e^{\num{1.6}}) = 1 \\
+		\die{} \le 1 \Rightarrow \text{accept} \implies \vec{\sigma}_2 = \uparrow\uparrow\uparrow\uparrow
+	\end{align*}
+}
+\end{minipage}
+\hfill
+\begin{minipage}{0.48\linewidth}
+\begin{block}{Markov Chain}
+\begin{align*}
+	\visible<1->{&\uparrow\uparrow\downarrow\uparrow} \\
+	\visible<5->{&\quad\Downarrow \\
+	&\uparrow\uparrow\downarrow\uparrow \to \uparrow\uparrow\downarrow\uparrow \\
+	}
+	\visible<7->{&\quad\Downarrow \\
+	&\uparrow\uparrow\downarrow\uparrow \to \uparrow\uparrow\downarrow\uparrow
+	\to \uparrow\uparrow\uparrow\uparrow
+	}
+\end{align*}
+\end{block}
+\end{minipage}
 \end{frame}
+\subsection{Heat-Bath Algorithm}
 \begin{frame}{Heat-Bath Algorithm}
-	Sample the proposed change with correct probability \(\Rightarrow\) change gets always accepted. \\
-	The local energy of site \(i\) is
+\begin{minipage}{0.45\linewidth}
+\begin{block}{Idea}
+\[ p(\vec{\sigma} \to \vec{\sigma}') = p(\vec{\sigma}') \]
+Always accept the new configuration
+\end{block}
+\begin{block}{Local Energy}
 	\begin{align*}
 		E_i(\sigma_i) = -J \left(\sigma_{i-1}\sigma_i + \sigma_i\sigma_{i+1}\right) -\mu\sigma_i
-	.\end{align*}
-	The possible values are \(\sigma_i \in \{\uparrow, \downarrow\} = \{+1,-1\}\). We set site \(i\) to \(\uparrow\) with probability
-	 \begin{align*}
-		p(\uparrow) = \frac{e^{-\beta E_i(\uparrow)}}{e^{-\beta E_i(\uparrow)} + e^{-\beta E_i(\downarrow)}}
-	.\end{align*}
-	\(N = 4, J = 1, \mu = 2, \beta = 1\) \\
-	Start with random configuration \(\vec{\sigma}_0 = (\uparrow, \uparrow, \downarrow, \uparrow) = (1, 1, -1, 1)\). \(M = ((\uparrow, \uparrow, \downarrow, \uparrow))\) \\
+	\end{align*}
+\end{block}
+\begin{block}{Probability}
+	\begin{align*}
+		p_i(\uparrow) = p_i(\downarrow \,\to \, \uparrow) = p_i(\uparrow \,\to  \,\uparrow)
+		= \frac{e^{-\beta E_i(\uparrow)}}{e^{-\beta E_i(\uparrow)} + e^{-\beta E_i(\downarrow)}}
+	\end{align*}
+\end{block}
+\end{minipage}
+\hfill
+\begin{minipage}{0.45\linewidth}
+\begin{block}{Example}
+	\(N = 4, J = 1, \mu = 2, \beta = \num{0.1}\) \\
+	Start with random configuration
+	\[\uparrow\uparrow\downarrow\uparrow = (1,1,-1,1)\]
 	Set value at site 1
 	\begin{align*}
-		E_1(\uparrow) &= -1\left(\uparrow\uparrow + \uparrow\uparrow\right) - 2\uparrow = -1 \times 2 - 2 \times 1 = -4 \\
-		E_1(\downarrow) &= -1\left(\uparrow\downarrow + \downarrow\uparrow\right) - 2\downarrow = -1 \times (-2) - 2 \times (-1) = 4 \\
-		p(\uparrow) &= \frac{e^{-1 \times (-4)}}{e^{-1\times (-4)} + e^{-1 \times 4}} \approx \num{0.99966} \\
-		\text{random number }r &= 0.84 < \num{0.99966} \Rightarrow \sigma_1 = \downarrow \,\Rightarrow \vec{\sigma}_1 = (\uparrow, \uparrow, \downarrow, \uparrow) \\
-		M &= ((\uparrow, \uparrow, \downarrow, \uparrow), (\uparrow, \uparrow, \downarrow, \uparrow))
-	.\end{align*}
+		E_1(\uparrow) = -4 \qquad
+		E_1(\downarrow) = 4 \\
+		p(\uparrow) = \frac{e^{-\num{0.1} \times (-4)}}{e^{-\num{0.1}\times (-4)} + e^{-\num{0.1} \times 4}} \approx \num{0.69} \\
+		\die{}
+		\begin{cases}
+			< \num{0.69} \implies \uparrow\uparrow\downarrow\uparrow \to \uparrow\uparrow\downarrow\uparrow \\
+			\ge \num{0.69} \implies \uparrow\uparrow\downarrow\uparrow \to \downarrow\uparrow\downarrow\uparrow \\
+		\end{cases}
+	\end{align*}
+\end{block}
+\end{minipage}
 \end{frame}
 \begin{frame}{Observables}
-\begin{itemize}
-	\item We start saving the Markov chain after a certain amount of sweeps (equilibrium was reached).
-	\item Easily calculate expectation values \(\expval{O} = \frac{1}{Z}\sum_{\vec{\sigma}}w_{\vec{\sigma}}O(\vec{\sigma}) \approx \frac{1}{\abs{M}}\sum_{i=1}^{\abs{M}}O(\vec{\sigma}_i)\)
-	\item For uncertainties use blocking analysis
-\end{itemize}
+\begin{minipage}{0.45\linewidth}
+\begin{enumerate}
+	\item<1-> \(N_{\text{eq}}\) sweeps to reach equilibrium
+	\item<2-> Use equilibrated MC steps for sampling
+	\item<3-> Expectation values \(\expval{O} = \frac{1}{Z}\sum_{\vec{\sigma}}w_{\vec{\sigma}}O(\vec{\sigma}) \approx \frac{1}{N_{\text{samples}}}\sum_{i=1}^{N_{\text{samples}}}O(\vec{\sigma}_i)\)
+	\item<4-> Blocking analysis for correlations
+	\begin{align*}
+		R = \frac{m\sigma_{\overline{O}_{\text{block}}}^2}{\sigma_O^2} \qquad
+		\tau_O = \frac{1}{2}(R_{\text{sat}}-1)
+	\end{align*}
+\end{enumerate}
+\visible<5->{\includegraphics[width=0.8\linewidth]{BoFoQoOo_gausshist_polarisation.pdf}}
+\end{minipage}
+\hfill
+\begin{minipage}{0.45\linewidth}
+	\visible<5->{\includegraphics[width=\linewidth]{BoFoQoOo_rplot.pdf}}
+\end{minipage}
 \end{frame}
 \section{Quantum Monte Carlo}
+\subsection{XXZ Quantum Spin Chain}
+\subsection{Trotter Decomposition}
 \begin{frame}{Quantum Monte Carlo}
 \begin{minipage}{0.45\linewidth}
-	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 \\
-		&= \frac{J_x}{2}\sum_i \left(S_i^+S_{i+1}^- + S_i^-S_{i+1}^+\right) + J_z \sum_i S_i^z S_{i+1}^z \\
-		&\text{with}\quad S^+ = S^x + iS^y, \quad S^- = S^x - iS^y
-	.\end{align*}
-	Look at two sites only
+		&= \frac{J_x}{2}\sum_i \left(S_i^+S_{i+1}^- + S_i^-S_{i+1}^+\right) + J_z \sum_i S_i^z S_{i+1}^z
+	\end{align*}
+	\[S^+ = S^x + iS^y, \quad S^- = S^x - iS^y \]
+\end{block}
+\begin{block}{Simple Case \(N=2\)}
 	\begin{align*}
-		H_{\text{two sites}} = \frac{J_x}{2}(S_1^+S_2^-+S_1^-S_2^+)+J_zS_1^zS_2^z \\
-		H_{\text{two sites}} \frac{1}{\sqrt{2}}(\ket{\uparrow\downarrow}\pm\ket{\downarrow\uparrow})
+		H_{2} &= \frac{J_x}{2}(S_1^+S_2^-+S_1^-S_2^+)+J_zS_1^zS_2^z \\
+		H_{2} &\frac{1}{\sqrt{2}}(\ket{\uparrow\downarrow}\pm\ket{\downarrow\uparrow})
 		= \left(-\frac{J_z}{4}\pm\frac{J_x}{2}\right)\frac{1}{\sqrt{2}}(\ket{\uparrow\downarrow}\pm\ket{\downarrow\uparrow}) \\
-		H_{\text{two sites}} \ket{\uparrow\uparrow}
+		H_{2} &\ket{\uparrow\uparrow}
 		= \frac{J_z}{4}\ket{\uparrow\uparrow} \qquad
-		H_{\text{two sites}} \ket{\downarrow\downarrow}
+		H_{2} \ket{\downarrow\downarrow}
 		= \frac{J_z}{4}\ket{\downarrow\downarrow}
-	.\end{align*}
+	\end{align*}
+\end{block}
 \end{minipage}
 \hfill
 \begin{minipage}{0.45\linewidth}
+\begin{block}{Possible Boltzmann Weights}
 	\begin{align*}
-	\vcenter{\hbox{\includegraphics{square.none.pdf}}} &\equiv \bra{\downarrow\downarrow} e^{-\Delta\tau H_{\text{two sites}}} \ket{\downarrow\downarrow} = e^{\Delta\tau J_z / 4}\\
-	\vcenter{\hbox{\includegraphics{square.left.right.pdf}}} &\equiv \bra{\uparrow\uparrow} e^{-\Delta\tau H_{\text{two sites}}} \ket{\uparrow\uparrow} = e^{\Delta\tau J_z / 4} \\
-	\vcenter{\hbox{\includegraphics{square.left.pdf}}} &\equiv \bra{\uparrow\downarrow} e^{-\Delta\tau H_{\text{two sites}}} \ket{\uparrow\downarrow} = e^{\Delta\tau J_z /4}\cosh(\Delta\tau J_x /2) \\
-	\vcenter{\hbox{\includegraphics{square.right.pdf}}} &\equiv \bra{\downarrow\uparrow} e^{-\Delta\tau H_{\text{two sites}}} \ket{\downarrow\uparrow} = e^{\Delta\tau J_z /4}\cosh(\Delta\tau J_x /2) \\
-	\vcenter{\hbox{\includegraphics{square.diag.rl.pdf}}} &\equiv \bra{\uparrow\downarrow} e^{-\Delta\tau H_{\text{two sites}}} \ket{\downarrow\uparrow} = -e^{\Delta\tau J_z /4}\sinh(\Delta\tau J_x /2) \\
-	\vcenter{\hbox{\includegraphics{square.diag.lr.pdf}}} &\equiv \bra{\downarrow\uparrow} e^{-\Delta\tau H_{\text{two sites}}} \ket{\uparrow\downarrow} = -e^{\Delta\tau J_z /4}\sinh(\Delta\tau J_x /2)
+	\vcenter{\hbox{\includegraphics{square.none.pdf}}} &\equiv \bra{\downarrow\downarrow} e^{-\Delta\tau H_{2}} \ket{\downarrow\downarrow} = e^{\Delta\tau J_z / 4}\\
+	\vcenter{\hbox{\includegraphics{square.left.right.pdf}}} &\equiv \bra{\uparrow\uparrow} e^{-\Delta\tau H_{2}} \ket{\uparrow\uparrow} = e^{\Delta\tau J_z / 4} \\
+	\vcenter{\hbox{\includegraphics{square.left.pdf}}} &\equiv \bra{\uparrow\downarrow} e^{-\Delta\tau H_{2}} \ket{\uparrow\downarrow} = e^{\Delta\tau J_z /4}\cosh(\Delta\tau J_x /2) \\
+	\vcenter{\hbox{\includegraphics{square.right.pdf}}} &\equiv \bra{\downarrow\uparrow} e^{-\Delta\tau H_{2}} \ket{\downarrow\uparrow} = e^{\Delta\tau J_z /4}\cosh(\Delta\tau J_x /2) \\
+	\vcenter{\hbox{\includegraphics{square.diag.rl.pdf}}} &\equiv \bra{\uparrow\downarrow} e^{-\Delta\tau H_{2}} \ket{\downarrow\uparrow} = -e^{\Delta\tau J_z /4}\sinh(\Delta\tau J_x /2) \\
+	\vcenter{\hbox{\includegraphics{square.diag.lr.pdf}}} &\equiv \bra{\downarrow\uparrow} e^{-\Delta\tau H_{2}} \ket{\uparrow\downarrow} = -e^{\Delta\tau J_z /4}\sinh(\Delta\tau J_x /2)
 	\end{align*}
+\end{block}
 \end{minipage}
 \end{frame}
-\begin{frame}{Quantum Monte Carlo}
-\begin{minipage}{0.45\linewidth}
-	Trotter decomposition
+\begin{frame}{Trotter Decomposition}
+\begin{minipage}{0.5\linewidth}
+\begin{block}{Split Hamiltonian}
 	\begin{align*}
-		H &= \underbrace{\frac{J_x}{2}\sum_{\text{odd } i}(S_i^+S_{i+1}^-+S_i^-S_{i+1}^+) + J_z\sum_{\text{odd } i}S_i^zS_{i+1}^z}_{H_1} \\
-		  &+ \underbrace{\frac{J_x}{2}\sum_{\text{even }i}(S_i^+S_{i+1}^-+S_i^-S_{i+1}^+) + J_z\sum_{\text{even }i}S_i^zS_{i+1}^z}_{H_2}
-	.\end{align*}
+		H &= \underbrace{\frac{J_x}{2}\sum_{\text{odd } i}(S_i^+S_{i+1}^-+S_i^-S_{i+1}^+) + J_z\sum_{\text{odd } i}S_i^zS_{i+1}^z}_{H'} \\
+		  &+ \underbrace{\frac{J_x}{2}\sum_{\text{even }i}(S_i^+S_{i+1}^-+S_i^-S_{i+1}^+) + J_z\sum_{\text{even }i}S_i^zS_{i+1}^z}_{H''}
+	\end{align*}
+\end{block}
+\begin{block}{Path Integral}
 	\begin{align*}
 		\tr\left[e^{-\beta H}\right]
 		&= \tr\left[\left(e^{-\Delta\tau H}\right)^m\right] \\
-		&= \tr\left[\left(e^{-\frac{\Delta\tau}{2}H_2}e^{-\Delta\tau H_1}e^{-\frac{\Delta\tau}{2}H_2} + \mathcal{O}(\Delta\tau^3)\right)^m\right] \\
-		&= \tr\left[\left(e^{-\Delta\tau H_1}e^{-\Delta\tau H_2}\right)^m\right] + \mathcal{O}(\Delta\tau^2) \\
-		&= \sum_{\vec{\sigma}_1\cdots\vec{\sigma}_{2m}}\bra{\vec{\sigma}_1}e^{-\Delta\tau H_1}\ketbra{\vec{\sigma}_{2m}}
-		e^{-\Delta\tau H_2}\ket{\vec{\sigma}_{2m-1}} \\
-		&\cdots\bra{\vec{\sigma}_3}e^{-\Delta\tau H_1}\ketbra{\vec{\sigma}_{2}}
-		e^{-\Delta\tau H_2}\ket{\vec{\sigma}_{1}}
+		&= \tr\left[\left(e^{-\frac{\Delta\tau}{2}H''}e^{-\Delta\tau H'}e^{-\frac{\Delta\tau}{2}H''} + \mathcal{O}(\Delta\tau^3)\right)^m\right] \\
+		&= \tr\left[\left(e^{-\Delta\tau H'}e^{-\Delta\tau H''}\right)^m\right] + \mathcal{O}(\Delta\tau^2) \\
+		&= \sum_{\vec{\sigma}_1\cdots\vec{\sigma}_{2m}}\bra{\vec{\sigma}_1}e^{-\Delta\tau H'}\ketbra{\vec{\sigma}_{2m}}
+		e^{-\Delta\tau H''}\ket{\vec{\sigma}_{2m-1}} \\
+		&\cdots\bra{\vec{\sigma}_3}e^{-\Delta\tau H'}\ketbra{\vec{\sigma}_{2}}
+		e^{-\Delta\tau H''}\ket{\vec{\sigma}_{1}}
 		+ \mathcal{O}(\Delta\tau^2)
-	.\end{align*}
+	\end{align*}
+\end{block}
 \end{minipage}
 \hfill
 \begin{minipage}{0.45\linewidth}
 	\includegraphics[width=0.8\linewidth]{worldline.pdf}
 \end{minipage}
 \end{frame}
+\begin{frame}{Updating Schemes}
+\begin{minipage}{0.48\linewidth}
+\begin{block}{Local Update}
+	\Centering{\includegraphics[width=0.6\linewidth]{localupdate.pdf}}
+\end{block}
+\begin{block}{Six Vertex Model}
+	\Centering{
+		\includegraphics[width=0.4\linewidth]{sixvertex-map.pdf}
+		\qquad
+		\includegraphics[width=0.4\linewidth]{sixvertex-worldline.pdf}
+	}
+\end{block}
+\end{minipage}
+\hfill
+\begin{minipage}{0.48\linewidth}
+\begin{block}{Single Loop Update}
+	\includegraphics[width=0.98\linewidth]{singleloopupdate.pdf}
+\end{block}
+\end{minipage}
+\end{frame}
 
 {
 	\nocite{*}
-- 
GitLab