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\noindent {\bf Assignment 4:
The Metal Insulator Transition in Clean Systems}
(corrected version) 
\bigskip

This the last  assignment. 
It is designed to give you 
a  better understanding of the tricky parts of the Fourrier
transformation in a  simple example of the solution of
a many body problem. 
%Understand really well analytically what
%you expect at each step  before writting and runnin the code! 

The problem is to discretize and solve the discrete version of the
following system of DMFT equations for $G(i\omega_{n}), \omega_{n} =
(2n+1)\pi T,n = -\infty \ldots \infty$.

\begin{enumerate}
\item $G_{o}(i\omega_{n}) = [i\omega_{n} - t^{2}G(i\omega_{n})]^{-1}$
\item $G_{o}(\tau) = T \sum^{\infty}_{n=-\infty}
e^{-i\omega_{n}\tau}G_{o}(i\omega_{n}) ) $
\item $\sum(\tau) = -U^{2} G_{o}(\tau)^{2} G_{o} (-\tau)$
\item $\sum(i\omega_{r}) = \frac{1}{2} \int^{\beta}_{-\beta}
e^{i\omega_{n}\tau} \sum(\tau) d\tau$
\item $G(i\omega_{n}) = \frac{1}{[G^{-1}_{o}(i\omega_{n}) -
\Sigma(i\omega_{n})]}$
\end{enumerate}
T U and t are parameters whose physical meaning are temperature,
interaction strength and hopping integral.  $G(i\omega_{n}) \propto
i\omega_{n}$ means insulating behavior $G(i\omega_{n}) \simeq i(sig
\omega_{n})$ as $\omega_{n}\rightarrow 0$ means metallic behavior.

You should regard 1 - 5 as a toy model system having the minimal type
of non linearities necessary to model a complex physical phenomena, i.e.
Mott transition.
%the same way that the Lorentz sytem of Assignment 2 described a
%minimal truncation of the Navier Stokes equations describing heat
%convection.

We want to explore numerically  the full solution of (1) - (5).  

However first
solve the trivial cases U = 0, and t = 0  
analytically.
Discuss the limiting behavior of the self energy $\Sigma$ as
the frequency goes to zero.

Then choose a discretization in $\tau$ space $\tau_{i}$, and a
cuttoff in frequency space, play with N  the number of slices of
$[-\beta , \beta]$.  See how the results change when $U$ is changed.

You are given the information that $G(o^{+}) = - \frac{1}{2}, \ G(o^{-})
= \frac{1}{2}$, so that $G(\tau)$ has a discontinuity at 0.
Furthermore since the frequencies that enter (2) are of the form
$(2n+1)\pi T, G(\tau + \beta) = - G(\tau)$.  Also $G(i\omega_{n}) =
-G(-i\omega_{n})$.  

The strategy for solving the system 1-5 is to first discretize it and
then solve it by iteration.  Steps (1) (3) and (5) are trivial to
implement.  Step 2 involves a Fourier transform of a function with a
long time tail, transform $[G_{o}(i\omega_{n})
- \frac{1}{i\omega_{n}}]$ numerically and $\frac{1}{i\omega_{r}}$
analytically $T \sum_{n} \frac{e^{i\omega_{n}\tau}}{i\omega_{n}} =
\{^{\frac{1}{2} \ \ \beta > \tau > o} _{\frac{1}{2} \ \  -\beta
< \tau < o}$ 
(Check!) and observe how Fourier  trades a long $\frac{1}{i\omega}$
tail for a discontinuity.

Step 4 involves a Fourrier transform of a function with a known
discontinuity.  You first interpolate (linear interpolation is enough)
and then Fourier transform the interpolation.  Make sure the
interpolating function has the right discontinuity and the right
boundary values $\sum(-\beta + o)$ and $\sum(\beta - o)$.  $(\sum(\tau
+ \beta) = - \sum(\tau))$.  Now you see how Fourier converts
discontinuities into long $\frac{1}{i\omega}$ tails.  Now explore the
high temperature 
regime set $t = \frac{1}{2}, \ T = .02$.  Start with the U=o solution as
an initial guess and solve the system 1-5 for U = .5, 1, l.5, 2, 2.5,
3 then begin with the t=o solution as an initial guess and solve
(1-5), for U = 6, 5, 4, 3.  [When you increase U or decrease U, use
the previously obtained solution as an initial guess.]  Display the
$G(i\omega_{n}) \ vs \ i\omega_{n}$ for different U's in a graph.  Is the
evolution as a 
function of U smooth?  Now repeat the same steps at much lower
temperatures T = .02 paying close attention to the neighborhood of the
point U=3.  Is the evolution smooth?











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