
\sect{One dimensional XY model}

%Baruch's  A26

Polymer in two dimensions:  Configurations of a polymer are described
by a set of vectors ${\bf t}_i$ of length $a$ in two dimensions (for i = 1,...,N), or alternatively
by the angles $\phi_i$ between successive vectors, as indicated in the figure below. The energy of a configuration $\{\phi_i\}$ is
\[{\cal H}=-\kappa \sum_{i=1}^{N-1}{\bf t}_i\cdot{\bf t_{i+1}}=-\kappa a^2 \sum_{i=1}^{N-1}\cos\phi_i   \]

\begin{itemize}
\item[(a)] Show that the correlations $\langle {\bf t}_n\cdot{\bf t}_m\rangle$ decay exponentially with distance and obtain an expression for the "persistence length" $a\xi$; you can leave the answer in terms of simple integrals.

Hint: Show  ${\bf t}_n\cdot{\bf t}_m=a^2\re \, \{\eexp{i\sum_{j=n}^{m-1}\phi_j}\}$.
\item[(b)] The end-to-end distance  ${\bf R}$ is defined as illustrated in the figure. Calculate $\langle R^2\rangle$ in the limit $N\gg 1$.  
\end{itemize}

\begin{center}
\includegraphics[scale=0.7]{A26.eps}
\end{center}


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