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\begin{document}
\heading{E5660: Ising model of absorption sites} \auname{Arthur
Shulkin}

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\Dn

Question

M absorption sites are arranged along a ring.

The sites are fermionic, i.e number of particles absorbed at site
$i$ is $n_{i}=0,1$.

Between every two absorption sites a spin $\sigma_{i}=\pm1$ is located.
The ring is surrounded by gas in temperature $T$ and chemical potential
$\mu$.

The absorption energy is $\epsilon>0$ if the two adjacent spins are
in the same direction, and $-\epsilon<0$ otherwise.
\begin{enumerate}
\item Write an expression for the energy $E[\sigma_{i},n_{i}]$ of a given
conguration.
\item Calculate the partition function $\mathbb{Z}(\beta,\mu)$ using the
transfer matrix method, write what is $T_{\sigma_{i},\sigma_{i+1}}$in
this problem.
\item Find Helmholtz function $F(T,\mu)$ assuming $M\gg1$.
\item Write an expression for the number of adsorbed particles $N=\underset{i}{\sum}<n_{i}>$
as a function of $\beta,\mu$.
\item Write an expression for the correlation length $\xi$ that characterizes
the spins in the system.
\end{enumerate}
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\Dn

Solution

1. The energy

\begin{equation}
E[\sigma_{i},n_{i}]=-\epsilon n_{i}\sigma_{i}\sigma_{i+1}-\mu n_{i}\end{equation}


2. The partition function

\begin{equation}
\mathbb{Z}=\underset{\sigma_{i},n_{i}}{\sum}e^{-\beta E[\sigma_{i},n_{i}]}=\underset{\sigma_{i}=\pm1}{\sum}1+e^{-\beta(-\epsilon\sigma_{i}\sigma_{i+1}-\mu)}\end{equation}


defining the transfer matrix

\begin{equation}
T_{\sigma_{i},\sigma_{i+1}}=1+e^{\beta\epsilon\sigma_{i}\sigma_{i+1}+\beta\mu}\end{equation}


\begin{equation}
T_{\sigma_{i},\sigma_{i+1}}=\left[\begin{matrix}1+e^{\beta\epsilon+\beta\mu} & 1+e^{-\beta\epsilon+\beta\mu}\\
1+e^{-\beta\epsilon+\beta\mu} & 1+e^{\beta\epsilon+\beta\mu}\end{matrix}\right]\end{equation}


Notice that $T_{\sigma_{i},\sigma_{i+1}}$is of the form of $\left[\begin{matrix}a & b\\
b & a\end{matrix}\right]$, it is straightforward to find its eigenvalues

\begin{equation}
\lambda_{\pm}=(1+e^{\beta\epsilon+\beta\mu})\pm(1+e^{-\beta\epsilon+\beta\mu})\end{equation}


\begin{equation}
\lambda_{+}=2+2e^{\beta\mu}\cosh(\beta\epsilon)\quad\lambda_{-}=2e^{\beta\mu}\sinh(\beta\epsilon)\end{equation}


one can write the partition function using the transfer matrix 

\[
\mathbb{Z}=\underset{\sigma_{1}}{\sum}\cdots\underset{\sigma_{M}}{\sum}<\sigma_{1}|T|\sigma_{2}>\cdots<\sigma_{M}|T|\sigma_{1}>=\]


\begin{equation}
<\sigma_{1}|T^{M}|\sigma_{1}>=\lambda_{+}^{M}+\lambda_{-}^{M}\sim\lambda_{+}^{M}\end{equation}


3. The Helmholtz function

\begin{equation}
F=-\frac{1}{\beta}\ln(\mathbb{Z})=-\frac{1}{\beta}[M\ln(2+2e^{\beta\mu}\cosh(\beta\epsilon))]\end{equation}


4. The number of absorbed particles

\begin{equation}
N=-\frac{\partial F}{\partial\mu}=M\frac{\cosh(\beta\epsilon)}{e^{-\beta\mu}+\cosh(\beta\epsilon)}\end{equation}


5. The correlation length

\begin{equation}
e^{-\frac{1}{\xi}}=\frac{\lambda_{-}}{\lambda_{+}}=\frac{\sinh(\beta\epsilon)}{e^{-\beta\mu}+\cosh(\beta\epsilon)}\end{equation}

\end{document}
