DQMC/scheme1.tex

\documentclass[tikz]{standalone}
\usepackage{physics}
\usetikzlibrary{arrows.meta}
\begin{document}

\begin{tikzpicture}[]

	% frame
	\draw[-,thick] (0,0) -- (5.5,0);
	\draw[-,dash pattern=on 20pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 20pt,thick] (5,0) -- (7,0);
	\draw[->,thick] (6.5,0) -- (8,0) node[anchor=west]{$\text{time} = N \tau $};
	\draw[->,thick] (0,0)  node[anchor=north ]{$0$} -- (0,4) node[anchor=north east]{states};
	
	% vertical lines
	\draw[-] (1,0) node[anchor=north]{$1$} -- (1,4);
	\draw[-] (2,0) node[anchor=north]{$2$} -- (2,4);
	\draw[-] (3,0) node[anchor=north]{$3$} -- (3,4);
	\draw[-] (4,0) node[anchor=north]{$4$} -- (4,4);
	\draw[-] (5,0) node[anchor=north]{$5$} -- (5,4);
	\draw[-] (7,0) node[anchor=north]{$N = \infty$} -- (7,4);
	
	% initial and final states
	\draw[] (0,2) node[anchor=east]{$\ket{i_0}$};
	\draw[] (7,3) node[anchor=west]{$\ket{i}$};
	\draw[] (8,2.5) node[anchor=west]{$\displaystyle \ket{\Phi_0} = \sum_i c_i \ket{i}$};
	\draw[] (8,1.5) node[anchor=west]{$\displaystyle  c_i \sim \sum_{\substack{\text{all paths} \\ \text{ending at $\ket{i}$}}} \prod_{k}^{\infty} T_{i_k i_{k+1}}$};
	
	% paths
	\draw[thick,magenta] (0,2) node{$\bullet$} -- (1,3) node{$\bullet$} -- (2,2) node{$\bullet$} -- (3,3) node{$\bullet$} -- (4,3.5) node{$\bullet$} -- (5,3.75) node{$\bullet$};% -- (7,3) node{$\bullet$};
	\draw[thick,red] (0,2) node{$\bullet$} -- (1,2) node{$\bullet$} -- (2,2) node{$\bullet$} -- (3,1.5) node{$\bullet$} -- (4,3) node{$\bullet$} -- (5,3) node{$\bullet$};% -- (7,3) node{$\bullet$};
	\draw[thick,green] (0,2) node{$\bullet$} -- (1,1) node{$\bullet$} -- (2,2) node{$\bullet$} -- (3,2.5) node{$\bullet$} -- (4,2.5) node{$\bullet$} -- (5,2) node{$\bullet$};% -- (7,3) node{$\bullet$};
	\draw[thick,blue] (0,2) node{$\bullet$} -- (1,0.25) node{$\bullet$} -- (2,1) node{$\bullet$} -- (3,1) node{$\bullet$} -- (4,2) node{$\bullet$} -- (5,1) node{$\bullet$};% -- (7,3) node{$\bullet$};

	\draw[thick,magenta,dash pattern=on 22pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 22pt] (5,3.75) node{$\bullet$} -- (7,3) node{$\bullet$};
	\draw[thick,red,dash pattern=on 20pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 20pt] (5,3) node{$\bullet$} -- (7,3) node{$\bullet$};
	\draw[thick,green,dash pattern=on 25pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 25pt] (5,2) node{$\bullet$} -- (7,3) node{$\bullet$};
	\draw[thick,blue,dash pattern=on 30pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 30pt] (5,1) node{$\bullet$} -- (7,3) node{$\bullet$};

\end{tikzpicture}

\end{document}