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\author{Jeffrey Kuan}
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\title{Matrix Product Ansatz}
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\begin{document}
\begin{abstract}
I like Taylor Swift.
\end{abstract}
\maketitle
%\part{Introduction}
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\section{About this webpage}
This webpage was created with \href{https://ximera.osu.edu/}{Ximera}, an
interactive textbook platform hosted by Ohio State University. The Ximera Project is funded 2024-2026 (with no other external funding) by a
\$2,125,000 \href{https://www.ed.gov/grants-and-programs/grants-higher-education/improvement-postsecondary-education/open-textbooks-pilot-program}{Open Textbooks Pilot Program} grant from the federal Department of Education.
\section{Introduction}
The \underline{Matrix Product Ansatz} is a method for finding
stationary measures in an interacting particle system with open
boundary conditions. Let the generator be written in the form
\[
B_1 + \sum_{l=1}^l w_{l,l+1} + \bar{B}_L
\]
where we use the subscript notation. In other words,
\(w\) is a local two--site generator, while \(B\) acts on
the left--most lattice site, and \(\bar{B}\) acts on the right--most
lattice site.
Denote a particle configuration on \(L\) sites
by
\[
\mathcal{C} = (\tau_1,\ldots,\tau_L),
\]
where each \(tau_i\) are particle variables equally
either 0 or 1, indicating the number of particles. The Ansatz
takes the form
\[
\mathcal{S}(\mathcal{C}) = \frac{1}{Z_L} \langle W | \prod_{i=1}^L ((1-\tau_i)E + \tau_i D) | V \rangle
\]
where \(E\) and \(D\) are elements of a non--commutative algebra, and
the product is ordered from the index \(i=1\) on the left to \(i=L\) on the right.
In \cite{KS97}, it was proven that there is a solution of the form
\[
w\left[ \binom{E}{D} \otimes \binom{E}{D} \right]
\]
\begin{thebibliography}{10}
\bibitem[CRV14]{CRV14} N Crampe, E Ragoucy and M Vanicat, ``Integrable approach to simple exclusion processes with boundaries. Review and progress.''
Journal of Statistical Mechanics: Theory and Experiment, Volume 2014, November 2014
\bibitem[DEHP93]{DEHP93} B Derrida, M R Evans, V Hakim and V Pasquier
Exact solution of a 1D asymmetric exclusion model using a matrix formulation
Journal of Physics A: Mathematical and General, Volume 26, Number 7, 1993.
\bibitem[KS97]{KS97} K. Krebs and S. Sandow, Matrix Product Eigenstates for One-Dimensional Stochastic Models
and Quantum Spin Chains, J. Phys. A30 (1997) 3165 and arXiv:cond-mat/9610029.
\end{thebibliography}
\end{document}