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\Large
{Problem set 1}
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\noindent{\bf{1.}}  Reconsider the proof of Wick's theorem discussed in class.  This time, consider a set of 
independent Fermion states (you can just consider one for simplicity, since they are independent) and a 
perturbation of the same form considered in class, except now the operators create or destroy Fermions and 
the conplex numbers $z$ and $z^*$ are replaced by Grassmann numbers.  Following the logic presented in class
and in Coleman's book, derive a Wick's theorem for Fermions.
\smallskip


\noindent{\bf{2.}} See cond-mat/0204351.  This problem is taken from this
paper with little modification (verbatim).
\smallskip

{\em{
We study a spin-S Heisenberg ferromagnet in a d-dimensional hypercubic lattice
with periodic boundary conditions,

\begin{eqnarray}
H=J\sum_{<i,j>}S_{i} \cdot S_{j},
\end{eqnarray}
where the exchange constant $J>0$ and the sum is over all the nearest-neighbor 
pairs on the lattice. In the Schwinger boson representation, a spin operator is 
represented by two types of bosons,

\begin{eqnarray}
S_{i}^{z}=\frac{1}{2}(a_{i\uparrow}^{\dag}a_{i\uparrow}-a_{i\downarrow}^{\dag}a_{i\downarrow}),
\end{eqnarray}

\begin{eqnarray}
S_{i}^{+}=a_{i\uparrow}^{\dag}a_{i\downarrow}, \,\, S_{i}^{-}=a_{i\downarrow}^{\dag}a_{i\uparrow},
\end{eqnarray}
These bosons should satisfy the constraints in their Fock space as follows,
$\sum_{\sigma}a_{i\sigma}^{\dag}a_{i\sigma}=2S$,
with $2S = 1, \, 2, \, ...$ for physical systems.
{\bf{Show that this tranformation is canonical}}.

The hypercubic lattice is divided into two sub-lattices
{\itshape A\/} and {\itshape B\/}. In a nearest neighbor pair $<ij>$,
$i{\in}{\itshape A\/}$  and $j{\in}{\itshape B\/}$. As in the  FM model,
 two Schwinger bosons $a_{i\sigma}$ with $\sigma={\pm}1$, at each site $i$, 
 and a bond field
$A_{ij}^{\dag}=\frac{1}{2}\sum_{\sigma}{\sigma}a^{\dag}_{i\sigma}a^{\dag}_{j-\sigma}$ 
are introduced.
$A_{ij}$ is  anti-symmetric with respect to the interchange of $i$ and $j$.
 A spin rotation by $\pi$ about the y-axis is made at all the
sites $j$ on the sub-lattice {\itshape B\/}, so that
$a_{j\uparrow}{\longrightarrow}a_{j\downarrow}$,  
$a_{j\downarrow}{\longrightarrow}a_{j\uparrow}$. The bond operator is 
then transformed as 
$A_{ij}^{\dag}\longrightarrow
A_{ij}^{\dag}=\frac{1}{2}\sum_{\sigma}a^{\dag}_{i\sigma}a^{\dag}_{j\sigma}$. 
{\bf{Show that this canonical transformation preserves the constraint
 $\sum_{\sigma}a_{i\sigma}^{\dag}a_{i\sigma}=2S=0,1,2,.....$}}

In terms of bosonic operators, the AFM Heisenberg model reads,

\begin{eqnarray}
H=-\frac{J}{2}\sum_{<ij>}(A_{ij}^{\dag}A_{ij}-2S^{2})+ 
\sum_{i}\lambda_i(\sum_{\sigma}a_{i\sigma}^{\dag}a_{i\sigma}-2S).
\end{eqnarray}

A global Lagrangian multiplier $\lambda$ has been introduced to replace 
the constraint on every site, and a real and uniform mean field amplitude 
$<A_{ij}>=<A_{ij}^{\dag}>=A$ are introduced. $A \neq 0$ describes 
short-range AFM correlations. {\bf{Show that }} the mean field Hamiltonian 
can be written as:

\begin{eqnarray}
H_{MF}=E_{0}+{\lambda}\sum_{k,\sigma}a_{k\sigma}^{\dag}a_{k\sigma}-
\frac{JAz}{2}\sum_{k,\sigma}{\gamma_{k}}(a_{k\sigma}^{\dag}a_{-k\sigma}^{\dag}+h.c),
\end{eqnarray}

with $\gamma_{\vec k}=\frac{1}{z}\sum_{\delta}e^{i\vec k \cdot \vec {\delta}}=1-\epsilon_{\vec k}$,  and $E_{0}=\frac{1}{2}NzJS^{2}-2{\lambda}NS+JA^{2}Nz$.

{\bf{Show that }}  $H_{MF}$ is diagonalized by using the standard Bogoliubov transformation, 
 yielding




\begin{eqnarray}
H_{MF}=E_{0}-{\lambda}N+\sum_{k}\omega_{k}(\alpha_{k}^{\dag}\alpha_{k}+\beta_{k}^{\dag}\beta_{k}+1),
\end{eqnarray}

where $\omega_{\vec k}=(\lambda^{2}-(JzA\gamma_{\vec k})^{2})^{1/2}$.

}}


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