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\begin{document}

\begin{center}
{\bf Homework 4: Effective Action Formulation, due on april 30}
\end{center}
\vskip 0.5cm
Note: Additional information on Quantum Montecarlo and optional
excercises can be found at http://www.physics.rutgers.edu/~udo/qmc.html

 
\vskip 0.5cm
In this homework we will apply the effective action reasoning, that leads to
the density functional theory, to a simpler model, namely the classical 
Ising system. 

We want to study
\ben
Z[J] = \sum_{\{S_i\}} e^{- (\sum_{ij}J_{ij} S_i S_j + \sum_i J_i S_i )}
=e^{-W[J]}.
\een

Note that average magnetization is defined as
\ben
\frac{\d W[J]}{\d J_i} = \la S_i \ra = m_i,
\een
and the Legendre transform 
\ben
\G[m] = W[J] - \sum_i J_i m_i.
\een
In the last equation $J_i$ is understood as a function of $m_i$, i.e,
$J = J(m)$ via equation (2).

Our goal is to provide simple approximations for $\G[m]$, following the 
ideas discussed in class (inversion method) :

\begin{enumerate}

\item
Multiply $J_{ij}$ by $\a$, where $\a$ is a parameter which will be set equal 
to one, like the charge of the electron in DFT.
\newline
Evaluate $\G_0[m]$, using equation (3), when $\a = 0$.

\item
Now expand $\G[m]$ in a series of $\a$,
\[
\G[m] = \G_0[m] + \a \G_1[m] + \frac{\a^2}{2} \G_2[m] + \cdots
\]
Evaluate exactly the first two non trivial orders and discuss the physical 
meaning.


\end{enumerate}



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