diff --git a/Makefile b/Makefile
index 9e7a88d62742ee2440354c87d0eae59e10ef009d..4b7f76f1ffe68ab385445633e632174ce11adeff 100644
--- a/Makefile
+++ b/Makefile
@@ -12,7 +12,7 @@ build/$(name).pdf: src/$(name).tex $(refs) fau-beamer/styles/beamerthemefau.sty
clean:
rm -rf build
- find src/ ! -name src ! -name $(name).tex ! -name *.bib ! -exec rm -rf {} +
+ find src/ -type f ! -name src ! -name $(name).tex ! -name *.bib ! -exec rm -rf {} +
show: build/$(name).pdf
xdg-open build/$(name).pdf
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diff --git a/src/qmc-loop-algorithm-report.tex b/src/qmc-loop-algorithm-report.tex
index 7e93fce3e69ad49b5e3aae8a2ab551b562586497..aeea767682149cecac19509e5e087783049cf61b 100644
--- a/src/qmc-loop-algorithm-report.tex
+++ b/src/qmc-loop-algorithm-report.tex
@@ -1,6 +1,7 @@
\ifx\pdfminorversion\undefined\else\pdfminorversion=4\fi
-\documentclass[aspectratio=169,t]{beamer}
-\usepackage[institute=Nat,aspectratio=169]{styles/beamerthemefau}
+\documentclass{beamer}
+\usepackage[institute=Nat]{styles/beamerthemefau}
+\usefonttheme{professionalfonts}
\usepackage[UKenglish]{babel}
\usepackage[utf8]{inputenc}
@@ -19,30 +20,165 @@
\date{2022-12-12}
\title{Quantum Monte Carlo and the Loop Algorithm}
+\subtitle{Physics Seminar}
\author{Stefan Gehr}
+\institute[FAU]{Friedrich-Alexander Universität Erlangen-Nürnberg}
\begin{document}
-\begin{frame}
- \titlepage
-\end{frame}
+\begin{trueplainframe}
+\titlepage
+\end{trueplainframe}
+
\begin{frame}{Table of contents}
\tableofcontents
\end{frame}
-\section{Monte Carlo Basics}
-\begin{frame}{Monte Carlo Basics}
-\begin{minipage}{0.40\linewidth}
-\begin{align*}
- a^2 + b^2 = c^2
-\end{align*}
+\section{Classic Monte Carlo}
+\begin{frame}{Metropolis Algorithm}
+\begin{minipage}{0.45\linewidth}
+ Classic 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
+ \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)
+ \end{align*}
+ assures configuration \(\vec{\sigma}_i\) is sampled with correct weight \(w_{\vec{\sigma}_i}\).
+\end{minipage}
+\hfill
+\begin{minipage}{0.45\linewidth}
+ Metropolis:
+ \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}}
+ \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
+ \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*}
+\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*}
+ 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*}
+\end{frame}
+\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{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 \(\vec{\sigma}_0 = (\uparrow, \uparrow, \downarrow, \uparrow) = (1, 1, -1, 1)\). \(M = ((\uparrow, \uparrow, \downarrow, \uparrow))\) \\
+ 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*}
+\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}
+\end{frame}
+\section{Quantum Monte Carlo}
+\begin{frame}{Quantum Monte Carlo}
+\begin{minipage}{0.45\linewidth}
+ 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
+ \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})
+ = \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}
+ = \frac{J_z}{4}\ket{\uparrow\uparrow} \qquad
+ H_{\text{two sites}} \ket{\downarrow\downarrow}
+ = \frac{J_z}{4}\ket{\downarrow\downarrow}
+ .\end{align*}
\end{minipage}
\hfill
-\begin{minipage}{0.55\linewidth}
-\begin{align*}
- \sqrt{a} + \sqrt{b} \ne \sqrt{c}
-\end{align*}
+\begin{minipage}{0.45\linewidth}
+ \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)
+ \end{align*}
\end{minipage}
\end{frame}
+\begin{frame}{Quantum Monte Carlo}
+\begin{minipage}{0.45\linewidth}
+ Trotter decomposition
+ \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*}
+ \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}}
+ + \mathcal{O}(\Delta\tau^2)
+ .\end{align*}
+\end{minipage}
+\hfill
+\begin{minipage}{0.45\linewidth}
+ \includegraphics[width=0.8\linewidth]{worldline.pdf}
+\end{minipage}
+\end{frame}
+
{
\nocite{*}
\printbibliography
diff --git a/src/references.bib b/src/references.bib
index 046271bf23fcf32784609a2c0db4e13da32e2824..327c7c22651452551e1157c53f6d24d8f45ccd3e 100644
--- a/src/references.bib
+++ b/src/references.bib
@@ -16,3 +16,31 @@
note = {Personal note}
}
+@unpublished{werner,
+ author = {Philipp Werner},
+ title = {Continuous-Time Impurity Solvers (Lecture Notes)},
+ institution = {Autumn-School Hands-on LDA=DMFT},
+ year = {2011},
+ url = {https://www.cond-mat.de/events/correl11/manuscript/Werner.pdf},
+ note = {Class handout}
+}
+
+@incollection{Assaad,
+ doi = {10.1007/978-3-540-74686-7_10},
+ url = {https://doi.org/10.1007/978-3-540-74686-7_10},
+ publisher = {Springer Berlin Heidelberg},
+ pages = {277--356},
+ author = {F.F. Assaad and H.G. Evertz},
+ title = {World-line and Determinantal Quantum Monte Carlo Methods for Spins, Phonons and Electrons},
+ booktitle = {Computational Many-Particle Physics}
+}
+
+@inproceedings{Sandvik2010,
+ doi = {10.1063/1.3518900},
+ url = {https://doi.org/10.1063/1.3518900},
+ year = {2010},
+ publisher = {{AIP}},
+ author = {Anders W. Sandvik and Adolfo Avella and Ferdinando Mancini},
+ title = {Computational Studies of Quantum Spin Systems},
+ booktitle = {{AIP} Conference Proceedings}
+}