\section{Introduction}

\par\noindent
The scattering of the radio waves as they propagate
through the ISM has long been a clue to small scale
inhomogeneities in the ionized interstellar plasma.
The scattering is seen in the dynamic spectrum of
pulsar signals as random modulations in frequency
and time.  Islands of intensity in the dynamic
spectrum, often referred to as ``scintles'',  are
sometimes modulated by a crisscross pattern.  By
Fourier analyzing the dynamic spectrum, Stinebring
et al.\ (2001) discovered the remarkable phenomenon
of parabolic arcs that underlies the criss-cross
substructure.  In this paper, we analyze parabolic
arcs from archival observations of PSR~B1133+16
originally reported by Gupta, Rickett, \& Lyne (1994).

The two-dimensional Fourier spectrum of the primary
dynamic spectrum is often referred to as the secondary
spectrum. However, as described by Cordes et al. (2006),
it can also be regarded as the ``differential
delay-Doppler'' distribution, whose coordinates are
time delay ($f_{\nu}$) and Doppler frequency (or
fringe rate $f_t$), being the variables conjugate
to radio frequency $\nu$ and time $t$.  The theory
of the arcs is described in that paper, which
henceforth we refer to as CRSC, and is also discussed
by Walker at al.\ (2004).

The basic arc phenomenon is remarkably simple, even
though it lay undiscovered for over 30 years.  When
scattered waves arrive at the observer from two angles
they interfere and cause fringes in intensity.  The
observer is moving relative to this interference
pattern and sees a sinusoidal variation in both time
and frequency, which thus appears as a pair of points
in the secondary spectrum. The coordinates of these
points are the differences in group delay and Doppler
frequency between the two scattered waves. The Doppler shift
depends on the relative velocity between the source and
observer along the (scattered) lines of sight. Hence the
\it difference \rm in Doppler shift of waves scattered
along two different paths through the ISM depends
only on the angles of scattering and the \it transverse \rm
velocity of the observer and pulsar relative to the
ISM. It is the beating of these slightly different
Doppler shifts that makes the fringe vary in time
and so their difference equals the fringe rate $f_t$.
There is a quadratic relation of delay to fringe rate
because, while the delay depends on the square of the
angle of scattering, the Doppler shift depends linearly
on the angles.  When one of the waves is unscattered the
result is a simple parabola $f_{\nu} \propto f_t^2$.

The observational properties
of pulsar scintillation arcs have been described by
Hill et al.\ (2003), who observed arcs over a broad
range of wavelengths and found the curvature
to be remarkably constant and to follow the
expected scaling as wavelength squared.
Arc observations from five pulsars are shown by
CRSC, who summarize the various arc phenomenon.
The results show many deviations from
the simple arc $f_{\nu} \propto f_t^2$.
A particuarly striking form is the
occasional appearance of ``reverse arclets''
which are displaced from the origin and whose curvature
is equal in magnitude, but reversed in sign, to
that of the main (forward) arc.
Hill et al.\ (2005) reported on a set of remarkable
stable reverse arclets from PSR B0834+06.

We present results for the evolution of the arcs
seen in PSR~B1133+16 at 408 MHz in archival
observations spanning 16 months.  On two of the
days observed with short time resolution
the delay-Doppler spectrum showed
several discrete arcs with reversed curvature,
similar to those shown by Hill et al. (2005).
We concentrate on fitting a model to one of those days.
From an assumed brightness distribution we calculate
the expected delay-Doppler spectrum using the screen theory of
CRSC. Then we iteratively adjust the parameters of the
model to fit to the observed spectrum.
Although the fitting is not unique we are able to explain the
reversed arcs and obtain an estimate of the scattered
brightness distribution.  This model
distribution is then compared with that expected
from a Kolmogorov scattering medium.
We finish with a brief discussion of what
structures in the interstellar medium might be
responsible.

The technique we use, which is model-fitting to match
the observed secondary spectrum, differs
from the inversion described by Walker and Stinebring
(2005), who perform the match to the primary dynamic
spectrum.  We discuss the differences between the two
methods in \S\ref{sec:theory}.

\begin{figure*}[htb]
\includegraphics[height=15cm, angle=270.0]{f1.ps}
\caption{\it Top Left: \rm  Dynamic Spectrum of PSR~B1133+16
on 8th September 1984 (day 0) with sampling time of 10 seconds.
\it Top Right: \rm Secondary Spectrum of the same data.
The plot is limited to $\pm2$ cpm in $f_t$, since
no signal is visible in the range 2-3 cpm.
The grey scale is logarithmic from the mean noise
level to black at a level 30dB higher, as shown by the
tablet to the right. The plot
shows a set of reversed arcs grouped together
forming a forward parabolic arc predominantly visible
for negative $f_t$.
\it Bottom Left and Right: \rm  Same displays for 9th
September 1984.  Note that the different time scale
from that above;  both show narrow fringes that drift in the same
general sense.  Both secondary spectra
are similar, but the details of the reverse arcs
have changed over one day.}
\label{fig:days01}
\end{figure*}

\section{Observations and Data Analysis}
\label{sec:obs}

The observations were made with the 76m telescope at
Jodrell Bank between September 1984 and January 1986,
as described by Gupta, Rickett, \& Lyne (1994).
An auto-correlation spectrometer was centered
at 408 MHz with 256 channels of 20 kHz
covering 5 MHz. The spectrometer was gated
synchronously with the pulsar period and
spectra were averaged in windows on and off
the pulse. The averaging time was normally
30 seconds and on two occasions it was shortened
to 10 seconds.  The duration ($T_{\rm obs}$)
of most observations was from one to four hours.
The on and off pulse spectra were corrected for
the one-bit auto-correlation normalized and
subtracted to create a primary dynamic spectrum
of pulse intensity $S(\nu,t)$.

The left hand panels of Figure \ref{fig:days01} shows
$S(\nu,t)$ for 8 and 9 September 1984
(which we call days 0 and 1 of the observing sequence);
they exhibit crisscross structure with a
predominance of scintles that drift toward
higher frequency with time. With the 10 second sampling
one can see fine scale fringes drifting in the same direction.

The right panels are the corresponding secondary spectrum
-- $S_2(f_{\nu},f_{t})$ or  delay-Doppler spectrum, which
is the squared magnitude of the
Fourier Transform of the primary spectrum:
\begin{eqnarray}
S(\nu,t) & \longrightarrow & S^{\dagger}(f_{\nu},f_{t})\\
S_2(f_{\nu},f_{t}) & = & |S^{\dagger}(f_{\nu},f_{t})|^2 \; .
\end{eqnarray}
Here $S^{\dagger}(f_{\nu},f_{t})$ is the two
dimensional discrete Fourier transform of the
sampled primary spectrum $S(\nu,t)$.
With time resolution of 10 seconds and frequency resolution of
20kHz, the Nyquist frequencies are $f_{t,{\rm Nyq}}=3$
cycles min$^{-1}$ (cpm)
and $f_{\nu,{\rm Nyq}} = 25.6$ cycles/MHz (or 25.6 $\mu$sec).
The associated spectral resolutions are
$\delta f_t = 1/T_{\rm obs}$ and $\delta f_{\nu}=0.2 \mu$sec.

The $S_2$ plots show structure
which is strongest near the origin and extends out
to large delays ($f_{\nu}$) in the lower part of the panel
(ie for negative $f_t$). We call this the forward arc,
since it is curved toward the $f_{\nu}$ axis.
It consists of discrete curves, which we call reverse arcs,
since they are curved in the opposite sense.
The apexes of the reverse arcs lie approximately
along the parabola $f_{\nu} \propto f_t^2$.
On close inspection one can see several
nearly parallel forward arcs. The top and
bottom panels of Figure \ref{fig:days01}
are separated by 27 hrs and show very similar behavior
in general, but the individual reverse
arcs have clearly changed substantially.
The main goal of our paper is to investigate
these reverse arcs and to find a model for the
interstellar angular scattering function that could
explain them.

\begin{figure*}
\includegraphics[width=13cm]{f2.ps}
\caption{Selected Secondary Spectra of PSR~B1133+16 from
September 1984 to January 1986 in a logarithmic grey scale
over a 30 dB dynamic range, as shown by the tablet in Figure
\ref{fig:days01}
}
\label{fig:full12}
\end{figure*}

\subsection{Time Evolution of the Arcs}
\label{sec:evol}

After rapidly sampled observations were made
on September 8th and 9th 1984, a slower standard
sampling interval of 30 seconds was adopted
for monitoring the ISS of PSR~B1133+16.
Figure \ref{fig:full12} shows $S_2$ plots from
twelve observations spanning 481 days.
Days 0-31 show the same general character
of a V-shaped ``valley'' with little power
along the delay axis,
flanked by diffuse power often appearing
as a set of discrete reverse arcs, which
can be asymmetric in $f_t$.
Then day 45 shows a simpler forward
arc which fades remarkably over days 96 and 142.
A strong forward arc appears
on day 194 and by day 288 multiple reverse
arcs similar to days 0-31 have returned.
Day 415 is similar to day 194, but day 481
has weak and indistinct arc structure.
These remarkable changes in the appearance of the secondary spectra
are evidence that the line of sight
crossed through particular localized structures,
as we discuss in \S~4.

With 30 second sampling
$f_{t,{\rm Nyq}} = 1$~cpm, and so features
beyond $\pm 1$~cpm can be aliassed into
the figure.  This is evident by comparing
the plots for days 0 and 1 in figures (\ref{fig:full12})
and (\ref{fig:days01}).
For example on day 1 figure (\ref{fig:full12}) shows
a linear feature at $f_t\sim 0.75\pm0.15$ cpm
and $f_{\nu} \sim 18 \mu$sec.  This
is an alias from $f_t\sim 0.75-2.0 = -1.25$ cpm,
visible in the lower right panel of Figure \ref{fig:days01}
at  $f_{\nu} \sim 18 \mu$sec with
its apex near $f_t = -1.4$ cpm.
The figure (\ref{fig:days01}) data were a 23 minute
observation immediately following the 30 min
observation figure (\ref{fig:full12}).
This discrete reverse arc changed very little in 40 minutes.
Though it clearly changed between day 0 and 1.
Days 9 and 16 show fuzzy reverse arcs
for both positive and negative $f_t$, which
are probably confused by aliasing, but give evidence
for an evolution in the asymmetry
as well as in the reverse arc features.

In one of the observations of PSR B1133+16 shown by CRSC
the secondary spectrum has two very well defined discrete forward arcs.
Putney and Stinebring (2006) have found multiple arcs
in six pulsars.  For B1133+16 they found the arc curvatures
to fall near four discrete values, but that the arc
strengths were quite variable.
They suggest that these are due to variable amplitude scattering
at four discrete distances from the Earth.
Consequently, we have estimated the curvature $a$
(defined by $f_{\nu} = a f_t^2$) from
each of our observations scaled to 1 GHz.  The mean of our values
over 1984 September to 1986 January was
$a = 6.7\times 10^{-3}$~sec$^3$ with an rms scatter of $0.6 \times 10^{-3}$~sec$^3$ ,
which is consistent with one of their values at
$(6.3 \pm 0.1)\times 10^{-3}$ sec$^3$.  We do not detect multiple
forward arcs, but note the signal-to-noise ratio in our data
is lower than in most of Stinebring's data.  We also note that the
curvature estimation becomes more difficult in the presence of
reverse arclets, and in such cases we trace the arc by the location of
their apexes.

\section{Modeling of arcs from PSR~B1133+16}
\label{sec:model}

\subsection{Theory}
\label{sec:theory}

We use the theory of arcs in the secondary spectrum
as given by CRSC.  Section 3.3 of that paper
analyzes $S_2$ in terms of a ``brightness function''
$B(\theta_x,\theta_y)$ that represents the
instantaneous strength of contributions to
the intensity at a given observing point
as a function of the angle of arrival
${ \mbox{\boldmath $\theta$} }=(\theta_x,\theta_y)$. The theory
provides insight into the basic interference effects
that give rise to the arc phenomenon,
but it does not include the dispersive delays
introduced by propagation through the
irregular plasma density in the ISM.
In \S\ref{sec:simulation}
we use a full electromagnetic simulation
to include such effects.

Consider a pulsar at distance $D$
whose radiation is scattered by a screen
at a distance $sD$ from the pulsar.
The observer sees a superposition of
interference fringes from each pair of apparent
angles of arrival ${ \mbox{\boldmath $\theta$} }_1$ and ${ \mbox{\boldmath $\theta$} }_2$.
The fringes contribute to a single pixel in
$S_2(f_{\nu},f_{t})$ if the angles obey:
\begin{eqnarray}
f_{\nu} &=&  \left( \frac{D(1-s)}{2\,c\,s }\right)
\left(\theta_2^2 - \theta_1^2\right)
\label{eq:fnu} \\
f_t &=&
  \left(\frac{1}{\lambda\, s}\right)(\theta_{x2} - \theta_{x1})  V_{\rm iss} ,
\label{eq:ft}
\end{eqnarray}
where $c$ is the speed of light and
$\lambda$ is the central wavelength observed.
${\bf V_{\rm iss}}$ is the effective transverse velocity
relative to the scattering region
of the line of sight from pulsar to observer,
which both may be moving (Cordes and Rickett, 1998);
without loss of generality we have chosen the orientation
of  ${\bf V_{\rm iss}}$ to define the direction of $\theta_x$.

Equation (8) of CRSC expresses $S_2(f_{\nu},f_{t})$
as a four-fold integral over all pairs of angles, subject
to the constraints of equations \ref{eq:fnu} and \ref{eq:ft}.  The constraints allow
the result to be reduced to a two-fold integral
as given by CRSC equations (9) or (B4 \& B5).
We used the latter, since it is more convenient
numerically, and we rewrite it here in a slightly different
form as $s_2(q,p)$ which is the delay-Doppler spectrum in terms of
normalized delay and fringe rate:
\begin{equation}
p = f _ \nu {2cs \over D(1-s) {\theta_0}^2},
\qquad q = f_t {s\lambda \over  V_{\rm iss}
\theta_0}
\label{eq:pqdef}
\end{equation}
$\theta_0$ is a normalizing angle, typically chosen to be the
angular radius of the diffractive brightness distribution.

The result is
\begin{eqnarray}
s_2(p,q) & = & {\pi \over q}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}
b(\xi_{+},y_1)b(\xi_{-},y_2)dy_1dy_2  \label{eq:s2int}   \nonumber \\
\xi_{\pm} & = & {p \over 2q} \pm {q \over 2} -
{1 \over 2q}({y_2}^2 - {y_1}^2)
\end{eqnarray}

where $b(x,y)$ is the brightness function in terms of
normalized angular coordinates
$x = {\theta_x \over \theta_0},  y = {\theta_y \over \theta_0}$,
and the constraints in equations (\ref{eq:fnu} \& \ref{eq:ft})
become:
\begin{eqnarray}
p & = & ({x_2}^2 - {x_1}^2) + ({y_2}^2 - {y_1}^2) \nonumber \\
q  =  x_2 - x_1  .
\label{eq:pq}
\end{eqnarray}
There is also a second mirrored relation
obtained by mapping $p\Rightarrow -p, q\Rightarrow -q$.

Hence we can describe an arc as the locus of
points in $p,q$ due to interference of a narrow discrete
component in $b(x,y)$ at $x_p,y_p$ with the remaining
extended continuum in $b$, as is done in section 3.4.3 of CRSC.
A primary forward arc $p \ge q^2$ comes from
an undeviated discrete component in $b$
interfering with the extended continuum in $b$.
This is the condition under weak
scintillation described by CRSC.  The boundary of this primary arc
is $f_{\nu} = a f_t^2$ (or $p= q^2$).
In turn $a$ is related to the physical parameters
via equations \ref{eq:ft} and \ref{eq:fnu}.

In general we substitute $x_p,y_p$ for $x_2,y_2$ in equation
(\ref{eq:pq}) to obtain a relation between
$p$ and $q$ depending on $x_1$ and $y_1$, which
vary over the continuum in $b$. The special case
where $b$ is narrow in one
dimension and extended along a line
$y_1=x_1 \tan{\phi}$, at an angle $\phi$ to the $x$-axis,
allows us to eliminate both $x_1$ and $y_1$ to obtain
the quadratic relation between $p$ and $q$:
\begin{equation}
p = x_p^2+y_p^2 - \sec^2(\phi) \;(q-x_p)^2    .
\label{eq:pqarc}
\end{equation}
This reverse arc is a parabola with an apex at
$p_a= x_p^2+y_p^2, q_a= x_p$
and negative curvature $-\sec^2(\phi)$, which
has unit magnitude if $\phi=0$ (i.e.\ scatter broadening
parallel to the velocity). The corresponding forward arc
with its apex at negative $p$ is obtained
by reversing the signs of $p$ and $q$ in
equation (\ref{eq:pqarc}). Both sets of arcs are
shown in Figure 4 of CRSC.

Of course, in general the scatter broadening function
is not one-dimensional, as assumed above. Further,
we have not discussed the power density of $s_2$ along an arc.
This depends on the relative strength of the discrete
component supposed to exist in $b$ and on the shape
and intensity of its continuum, as described by
the integral (\ref{eq:s2int}).

\subsection{Model for September 8th 1984 (day 0)}

The parallax and proper motion of PSR~B1133+16
were measured by Brisken et al. (2002);
they found its distance $D= 0.35\pm 0.02$ kpc and
its transverse velocity $V_{\rm pm}=631 \pm 36$ km~sec$^{-1}$.
In general the scintillation velocity depends
on the transverse velocities of the pulsar, the Earth and
the scattering medium, but since our pulsar is moving
much faster than the Earth and the medium, we have
$ V_{\rm iss}  = V_{\rm pm} (1-s)$.  Hence the arc curvature,
as given by equation (8) of Stinebring et al. (2001),
becomes:
\begin{eqnarray}
a = \frac{D \lambda^2}{2cV_{\rm pm}^2}\frac{s}{1-s} = 0.0244 \frac{s}{1-s} \; {\rm sec}^3  \; ,
\label{eq:curv}
\end{eqnarray}
at our observing wavelength.
The arc curvature $a = 6.7 \times 10^{-3}$ sec$^{3}$,
given in section 2.1, is scaled to 1 GHz.
When scaled back to our observed frequency of 408 MHz,
we have $a = 0.040 \pm 0.003$ sec$^{3}$.
This results in $s = 0.62$, giving a screen at 133 $\pm$ 7 pc from the Earth.
Putting the mean value into equations \ref{eq:fnu} and \ref{eq:ft}
gives the mappings:
\begin{eqnarray}
f_{t,\rm cpm} = 0.15 \,(\theta_{x2,\rm mas}-\theta_{x1,\rm mas}) \nonumber \\
f_{\nu, \rm \mu sec} =
0.26 \, (\theta_{2\rm mas}^2-\theta_{1\rm mas}^2).
\label{eq:map}
\end{eqnarray}

The secondary spectrum from September 8th 1984 shown in
the upper right panel of Figure \ref{fig:days01}
exhibits a series of reverse arcs extending
to delays of 25 $\mu$sec, which are visible predominantly
at negative $f_t$.  We choose these data to model
quantitatively, because of these unusual
\it asymmetric discrete reverse \rm arcs.
Consider, for example, the faint reverse arc
with its apex at $f_{\nu} = 14 \,\mu$sec and
$f_t = -1.1$ cpm, which is visible
over the approximate range $-1.6 \le f_t \le -0.8$ cpm.

A discrete reverse arc
is formed by interference of a point component
in the scattered brightness function
with an anisotropic extended component. To simplify the analysis
consider the normalized brightness $b(x,y)$ to be narrow
in $y$ and extended in $x$, i.e.\ parallel
to the ISS velocity (i.e. $\phi=0$ in equation(\ref{eq:pqarc})).  Thus we write
$b(x,y) = [a_p \delta(x-x_p) + b_e(x)]\delta(y)$,
where $b_e$ represents the extended component
and we use delta functions to model the narrow
structures.  The interference between waves from
$x_p$ and $x$ gives $q=x_p-x$. Thus, if
$b_e(x)$ extends from $x_{-}$ to $x_{+}$, the
reverse arc will have its apex at $p_a = x_p^2$,
$q_a=x_p$ and be visible over
$x_{-}-x_p \le q \le x_{+}-x_p$.

In this model $y=0$ and the apex lies on the basic arc. Hence
substituting $f_{\nu}$ and $f_t$ from above into equation
(\ref{eq:curv}) gives $s=0.63$, which is consistent
with the value 0.63 obtained above.  Using the mapping
in equation \ref{eq:map},
the observed range $-1.6 \le f_t \le -0.8$ cpm
for the reverse arc, corresponds to
interference of a discrete component at
-7.5 mas interfering with an extended component
from -2.7~mas to 2.7~mas.  In a similar way for each discrete
reverse arc on day 0 we can find a corresponding
pair of components in the scattered brightness
(a point and extended component along $\theta_x$).

Following this idea we model the brightness
distribution (in normalized coordinates)
as a sum of Gaussian functions:
\begin{equation}
b(x,y) = \sum_{i=1}^n A_i e^{- {x-x_{oi} \overwithdelims () \sigma_{xi}}^2 -
{y-y_{oi} \overwithdelims () \sigma_{yi}}^2}
\label{eq:modeleq}
\end{equation}
where $A_i$ is the intensity, $\sigma_{xi}$ describes the width
of $x$, $\sigma_{yi}$ describes the width of $y$, $x_{oi}$
is the offset in $x$, and $y_{oi}$ is the offset in $y$.
We made an initial model in the manner described above
for the most obvious reverse arcs in Figure \ref{fig:days01}.
As more components are added the extended component
eventually  becomes the sum of the discrete components.
Then we tried to optimize the model by
minimizing the sum of the squared errors between
the observed $s_2$ and the model obtained by a numerical integral of
equation (\ref{eq:s2int}).

\subsection{Fitting Process}

We chose to work in normalized delay $p$, fringe rate $q$
and angle $x,y$ and used a normalizing angle $\theta_0 = 3.1$~mas,
with the constants evaluated as discussed above.
The fitting was done over two regions where the arcs
are visible in the top right panel of Figure \ref{fig:days01}:
a lower rectangle  -2--0 cpm by 0--25 $\mu$s
and a smaller upper rectangle  0--1 cpm by 0--5 $\mu$s.

We estimated  the noise background in $S_2$ from the
average noise level in $S_2$ over the regions not included
in the fit, and this constant
was added to the model.
We defined $\chi^2$ in a standard fashion:
\begin{eqnarray}
\chi^2 = \sum^{i,j} {{[S_2(f_{\nu,i},f_{t,j})
- S_{2,{\rm mod},i,j}]^2}\over{\sigma_{i,j}^2}} ,
\end{eqnarray}
where $\sigma_{i,j} = S_{2,{\rm mod},i,j}$. At each point $S_2(f_{\nu,i},f_{t,j})$ is an
estimate of the secondary spectrum, which has a standard deviation
$\sigma_{i,j}$ equal to its mean value (following exponential statistics).
Once the model is close to an optimum value it provides an estimate
of $\sigma_{i,j}$. Thus we used the model as an estimate
for $\sigma_{i,j}$ rather than the observed value.

The fitting was done in a semi-automated fashion.
Starting with separate optimization of the parameters describing each
narrow component responsible for the more obvious discrete arcs,
we minimized $\chi^2$ over a rectangle around each apex.
Table \ref{table:model} lists the coordinates of the apexes of six of the most
obvious reverse arcs.
With five parameters for each Gaussian the number of parameters to fit
grows quickly as components are added.  To simplify we started
with all components centered on the x-axis (ie $y_{oi}=0$).
With the position $x_{oi}$ of a component fixed we explored the
influence of $\sigma_{xi}$ and $\sigma_{yi}$.  For the arcs at large
delays we found that $\sigma_{xi}$ had to be quite narrow
to agree with the observed arc thickness in delay,
but that $\sigma_{yi}$ could be substantially wider.
As a result we did not optimize for the $y$-position
or width, but set them all on the $x$ axis with $\sigma_{yi}=0.3$~mas.
This left three parameters to be optimized for each component
(amplitude, $x$-position and width).  However,
as discussed above the extent in $q$ is governed by the $x$-
position and width of the associated extended component,
thus they too had to be adjusted.

Close inspection of the observed $S_2$ shows narrow
parallel forward arcs crossing the reverse arcs.
As can be seen from Figure 4 of CRSC, these are
extensions of the reverse arc to positive $f_t$ values
which are reflected about the origin.  Thus they show
interference of a discrete component ($x_p <0$) with components of
the scattered brightness extended over a wide
range, extending to $x$ values more negative
than $x_p$.  This emphasizes that in fact
the entire set of arcs is caused by the mutual
interference of all possible pairs of points in $b(x,y)$.

As noted above the observations show parallel
forward arcs (for negative $f_t$), but they also
include some that appear as fainter (whiter) stripes.
Such fainter stripes come from a discrete
minimum in $b(x,y)$, whereas the brighter
arcs are from discrete maxima.  We modelled this
effect by putting dips into the brightness
function, which we achieved by
having two offset broad extended components with a
minimum in $b$ between them. These minima improved
the fit slightly over a model without the minima.
Thus in choosing how many
and where to place new components, we were guided by the
visual appearance of the observed $S_2$
in comparison with the model.

Table \ref{table:parameter} gives a list of the parameters for the 9-component
model for $b(x,y)$ (scaled into milli-arcseconds).
Components 1-6 are relatively narrow
($\sigma_x < 0.3$~mas) and
components 7-9 are relatively wide.
It is the interference of the narrow with the
wide components that is responsible for the 6
most obvious discrete reverse arcs.
Their position and width in Doppler frequency
governs the range in $x$ of the brightness function.

We optimized the model to minimize $\chi^2$
with regard to the three variable
parameters of the 9 components. The model
can be seen in the upper panel of
Figure \ref{fig:ss-model},
to be compared with the upper right panel of
Figure \ref{fig:days01}.
However, the value of $\chi^2$ was about five times
the number of points in the fit, indicating that
the model is far from optimum even though
it was a local minimum. The largest discrepancy was
traced to the region near the $f_t$ axis
(from -1 to -2 cpm), in which the observed power level
is elevated but the model is not.  The higher
level in the observations appears to be due
to effects from broadband variation
of the pulse strength (intrinsic
pulse modulations or impulsive interference).
Consequently, we estimated this extra power
by averaging $S_2$ in (from -2 to -3 cpm)
as a function of delay, and added it to the model.

This refinement caused little change in the best parameters
but reduced the minimum $\chi^2$ to about twice the number
of data points.  Visually, the model
resembles the observations in the
asymmetry versus $f_t$, the presence of
6 reverse arcs, the location of their apexes,
and their width in delay ($f_{\nu}$).
Given the success of the fit from a visual perspective,
we decided not to attempt a better fit by
introducing extra components, since
we would just be refining the model for a particular
snapshot of the scattering function $b$.
Further, since our method of fitting relies on the visual
impression in choosing how many components and their
initial parameters, the result is not unique.

The upper panel of Figure \ref{fig:ss-model} shows the
model secondary spectrum. It corresponds
to the brightness model in Table \ref{table:parameter}
with one modification.  In order to represent the
``speckle'' effect in a snapshot of the scattered brightness
we multiplied the sum of Gaussians by
a random function of $x$ (uniform distribution), before computing
the integral in equation (\ref{eq:s2int}).
The effect is to cause a set of fine nearly parallel
forward arcs, somewhat reminiscent of the
observed fine structure in $S_2$.
The model brightness is plotted logarithmically
in the upper panel of Figure \ref{fig:bright-model}.
Note the asymmetry and anisotropy in its
distribution of  discrete components along the
axis parallel to ${\bf V_{\rm iss}}$.

The effect of rotating the
brightness by an angle $\phi$ to the $x$-axis
increases the magnitude of the curvature
in $p,q$ plane, as can be seen
from equation (\ref{eq:pqarc}).  However,
this could be compensated by a change in the
screen distance parameter $s$. Thus we cannot
determine both $s$ and $\phi$ and leave our results
displayed with $\phi=0$.

We now consider what constraint can be placed
on the $y$-extent of the brightness function ($b(x,y)$).
When $y_p$ and $y_1$ are non-zero, we obtain the
a version of equation (\ref{eq:pqarc}) in which
$y_p^2$ is replaced by $y_p^2-y_1^2$.
Let the extended and point components be centered on the $x$-axis
with widths $\sigma_{y}$ and $\sigma_{yp}$, respectively.
Then these ranges broaden the
thickness of the reverse arc in delay (ie $p$ is broadened
by the larger of $\sigma_{y}^2$ and $\sigma_{yp}^2$.
The widths of the reverse arcs in the upper right
panel of Figure \ref{fig:days01} are about $\pm0.05 \mu$sec.
With the scalings from equation (\ref{eq:map}), this
constrains the rms $y$-widths to be less than about
0.44~mas.  We set them (somewhat arbitrarily) to be 0.3~mas.

\subsection{Discussion of the Model}

Although the parameters we used gave us arc and subarc
structures that resemble the particular PSR~B1133+16 data,
we do not imply that it is the only set nor is
it the best set.  We have produced one model
for the brightness distribution that reproduces
the visual form of the observations and is optimized
within a limited range of the parameters.

From this we conclude that the asymmetrical form of
the observed secondary spectrum and the presence of
narrow discrete reverse arcs can be successfully explained
by an asymmetrical and anisotropic brightness function with
a number of narrow discrete features.
The constraint on the widths in $\theta_y$ is important since,
with an $\theta_x$-width of about $\pm 5$~mas, it constrains
the effective axial ratio of the scattered
brightness function to be greater than about 10.
However, an interesting feature
of the model is that the individual features
can be wider in $\theta_y$ than in $\theta_x$; in other words
they may be anisotropic in the orthogonal direction.

As shown by CRSC, an absence of power along the
delay axis in the secondary spectrum
is caused by anisotropic broadening, enhanced
along the direction of the scintillation velocity.
All of 12 of the secondary spectra plotted
in Figure \ref{fig:full12} show no
power along the delay axis. Hence
the same basic anisotropy in the scattering
persisted over about 500 days, but the detailed
form of the scattered brightness changed
substantially over days and weeks.

\subsection{Screen Simulation}
\label{sec:simulation}

CRSC used the simulation code of  Coles et al.\ (1995)
to simulate secondary spectra of waves scattered
by a dispersive phase changing screen.
In normal usage the method synthesizes
one realization of a stationary random phase screen,
which modulates a plane wave. The emerging complex field
is then propagated a distance $z$ to the observer,
by transforming into wavenumber and multiplying by the appropriate
(Fresnel) filter and back transforming to the spatial domain.
The intensity is saved along a straight line track through
the two-dimensional complex wave field.
The result simulates a time series as an observer moves
relative to a scattered wave field.
A dynamic spectrum is a set of such time series, created as
the radio frequency of the incident wave is incremented.
The secondary spectrum is computed in the same
way as for the data.

CRSC simulated the screen as a single two-dimensional
realization of a stationary random process following a
given wavenumber spectrum.  They showed results for a
Kolmogorov screen. When anisotropy was included, some
results had multiple discrete
forward and reverse arcs.  However, the results
differed from our observations of PSR~B1133+16
in that the discrete arcs were considerably
finer and distributed symmetrically in fringe rate
($f_t$).  The addition of a linear gradient in the
phase across the screen (i.e.\ a simple refraction)
caused the secondary spectrum to be
asymmetric in fringe rate. They showed that the phase
gradient needed to be large enough to shift the scattered image by
at least its own radius.  This is a much larger
gradient than is expected randomly
for a medium with a Kolmogorov spectrum.

\begin{figure}[hbt]
\begin{tabular}{c}
\includegraphics[height=5cm]{f3a.eps} \\
\includegraphics[height=5cm]{f3b.eps}
\end{tabular}
\caption{ \it Top\rm  : Secondary spectrum computed from
the brightness distribution of the upper
panel of Figure \ref{fig:bright-model},
by a numerical integration of equation
(\ref{eq:s2int}). Grey scale
is logarithmic with dynamic range of 36 dB.
The fit was done over the two rectangular regions
shown; outside those regions the noise from the
data is plotted.
\it Bottom\rm  : Secondary spectrum from the
screen simulation for the same day (see text)
with dynamic range of 30 dB.
}
\label{fig:ss-model}
\end{figure}

We modified the screen generation code to
make the scattered brightness like that in our model
(i.e.\ the upper panel of Figure \ref{fig:bright-model}).
This modification required an iteratative process,
as follows.  The field emerging from the phase screen
at position ${\bf s}$ has unit amplitude
and is phase modulated $F({\bf s}) = \exp[j\phi({\bf s})]$.
Its two-dimensionsal Fourier transform $F^{\dagger}({ \mbox{\boldmath $\kappa$} })$
is the  spectrum of the field versus wavenumber ${ \mbox{\boldmath $\kappa$} }$,
which maps simply to the scattered angle as
${ \mbox{\boldmath $\kappa$} } = k { \mbox{\boldmath $\theta$} }$ with $k=2\pi/\lambda$.
Our goal is to find
a screen phase $\phi({\bf s})$, for which the squared
magnitude $|F^{\dagger}|^2$ follows our model for the
scattered brightness $B({ \mbox{\boldmath $\theta$} })$.
We started by specifying the magnitude
$F^{\dagger}=\sqrt{B({ \mbox{\boldmath $\theta$} }/k)}$ and assigned it random phase.
Then back transformed to give $F({\bf s})$, which had
both amplitude and phase modulation. A revised
$F({\bf s})$ was obtained by normalizing each point
to unit magnitude. That revised $F$ was re-transformed
and the resulting $F^{\dagger}$ was normalized
to make its magnitude $\sqrt{B({ \mbox{\boldmath $\theta$} }/k)}$
and keep its phase. When this cycle was repeated
the amplitude modulation in $F({\bf s})$ is decreased.
We iterated until the residual amplitude modulation
no longer decreased ($\sim 100$ cycles).  The phase
of this final $F({\bf s})$ was then extracted as the screen
phase for the normal simulation.  This method is based on the concepts
developed by  Gerchberg and Saxton (1972).
As in the normal simulation this screen phase
was scaled proportional to radio wavelength,
to represent the plasma dispersion.

The lower panel of Figure \ref{fig:ss-model}
gives the resulting secondary spectrum.  It has
the same basic features as the simple model
in the upper panel, but shows more pronounced
substructure in the form of fine arc modulations.
The importance of this result is that the basic
arc forms are unchanged by the inclusion of
dispersion in the screen phase, whereas dispersion is not included
in the physics of equation (\ref{eq:s2int}).  A further result
is that we have obtained a realization of the phase
screen itself.  This is then an estimate of
the phase imposed by the ISM during the observations
on our day 0, and we can ask what it tells about
the density structure in the ISM.

\subsection{Comparison with Reverse Arcs from PSR B0834+08}

Hill, et al. (2005) reported how four discrete
reverse arcs observed from PSR B0834+16 moved along
the main forward parabola over 25 days.
The Doppler frequency of the apex of each arc followed a
linear track against time; furthermore the slopes of all
four tracks were equal to the slope predicted by assuming that each
reverse arc was scattered from a fixed structure
in space scanned by the known pulsar proper motion.
Their results suggest that the reverse arcs appearing on successive days
are not unrelated, but are in fact due to the same physical structures.

We have attempted to test this idea with the reverse arcs
observed on day 0 and 1 of our data (PSR B1133+16), which are
27 hours apart.  For the 6 most
obvious reverse arcs on day 0 we calculated
the shift expected due to the pulsar proper motion in 27 hours
and compared the new locations to those observed on day 1.
The expected angular shift is $\delta \theta_x \sim \delta t V_{\rm pm}/D$,
which we substitute into equation \ref{eq:map}, and tabulate
the $f_t$ and $f_{\nu}$  positions before and after this
proper motion shift (see Table \ref{table:proper motion}).   Comparing the
shifted coordinates with those observed on day 1,
we found some agreement. For example, the easily visible reverse
arc no. 4 from day 0 should should move to $f_t$ = -1.25 cpm
and $f_{\nu}$ = 18.75 $\mu$s which is close to a prominent
arc on day 1 (lower right panel of Figure \ref{fig:days01}).
The comparison is complicated by the fact that day 1 shows a higher density
of reverse arcs making it hard to recognize particular structures.
We conclude that these data are not of sufficiently high
signal to noise ratio to confirm or reject the hypothesis
that the arc positions are simply shifted by the pulsar proper
motion.

\subsection{Relationship to W-S Inversion Algorithm}

We now compare our model fitting with the method of
Walker and Stinebring (2005) (WS) who
estimate the scattered brightness function using an
fitting algorithm to the dynamic spectrum.
This has great appeal since it
avoids the messy task of deciding what components
of the secondary spectrum to include in a model fit.
Their method represents the full electric field as
a sum of ``scattered waves'', which are successively
identified and subtracted in order to match
the observed primary dynamic spectrum to the model.
It is somewhat analogous to the CLEAN algorithm in image reconstruction.

WS consider the scattered electric field $U(\nu,t)$ at frequency $\nu$
and time $t$ as a sum of basis functions
$\exp[i2\pi(\nu f_t + f_{\nu}t)]$,
which they call scattered waves, though the formulation is
not derived from scattering theory.
We note that $\nu$ and $t$ are relative to the center of
the observing band and integration time and that WS use
the symbols $\tau$ for $f_{\nu}$ and $\omega$ for $f_t$.

The dynamic spectrum $S(\nu,t)$ is then modelled by $|U(\nu,t)|^2$
and WS designed a procedure to estimate the phasor
($\tilde{U} (f_{\nu},f_t)$) for each scattered wave to best match the
model with the observed dynamic spectrum.  The results
are given as both amplitude and phase
for each differential delay $f_{\nu}$ and Doppler $f_t$.
They applied the method and displayed the amplitudes
for a particular 327 MHz observation in the
sequence from PSR B0834+06 (Hill et al., 2005), in
which there were several well-defined reverse arclets.
They discuss possible applications of the method but
do not relate the results to the scatttered brightness
$B(\theta_x,\theta_y)$.

However, Asplund et al. (2004) used these same results
and mapped them to the scatttered brightness by assuming that
each component $\tilde{U} (f_{\nu},f_t)$ was the result of interference
between an undeviated (reference) wave and a wave scattered at angle
$(\theta_x,\theta_y)$.  This assumption is valid
under conditions of weak scattering, as shown by CRSC,
but it does not seem justified for the 327 MHz observations
of PSR B0834+06.  As WS note, the effective reference wave
in their algorithm starts out as undeviated, but is modified
as each scattered wave component is identified.
Nevertheless the summing of the ``scattered waves''
does include the mutual interference between the components
of the basis functions as well as their interference with the reference component.

One can show that the secondary spectrum is the square
of the auto-correlation of $\tilde{U} (f_{\nu},f_t)$,
as follows:
\begin{eqnarray}
S_2(f_{\nu},f_{t}) = | \stackrel{F_{\nu}}{\Sigma} \stackrel{F_t}{\Sigma} \;
 \tilde{U}(F_{\nu},F_t) \, \tilde{U*} (F_{\nu}+f_{\nu},F_t+f_t) |^2
\end{eqnarray}
Nevertheless, it seems that more work is needed on the theoretical
basis of the algorithm to consider the circumstances under which
the weak scintillation mapping into the angle domain is properly justified.

While the method we have used is cumbersome to implement,
it is more straightforward in that our computed model
directly sums the waves in the domain of $(\theta_x,\theta_y)$
including their mutual interference.

\section{The Density structure in the Interstellar Plasma}

Most interpretations of radio scintillation
have been formulated as constraints on the wavenumber
spectrum of the plasma density.  The Kolmogorov
spectrum has become a default model, even though
several observations point to more fluctuation power
on  wavelengths longer than the size of the
typical radio scattering disc  (a few AU).

The general concept proposed to explain the discrete
reversed arcs is that  substructure in the scattered brightness
$b$, analogous to speckle in a scattered image,
causes substructure in the secondary spectrum,
that takes on a form of points lying on
intersecting reverse and forward
parabolic arcs, traced by equation (\ref{eq:pqarc})
and its reflection in the origin,
with intensity at each point governed by the product
of the brightness of the two interfering components.
The plots in Figure \ref{fig:bright-model} are estimates of
a single realization of the scattered brightness with a particular
sub-structure characteristic of the observing conditions.
We now consider what these images tell us about the
density structure in the interstellar plasma.

\begin{figure}[hbt]
\begin{tabular}{c}
\includegraphics[height=8cm, angle=270]{f4a_clip.eps} \\
\includegraphics[height=8cm, angle=270]{f4b_clip1.eps}
\end{tabular}
\caption{\it Top\rm :  Logarithmic plot
of the model brightness distribution
obtained by fitting to observation on day 0
in Figure \ref{fig:days01} (dynamic range 50 dB).
\it Bottom\rm : brightness distribution obtained from the
full screen simulation (see text).}
\label{fig:bright-model}
\end{figure}

The first point is that the scattered image is highly elongated
with an axial ratio of 10:1 or more. Further, the axis
of elongation is roughly parallel to the
proper motion direction of the pulsar (within, say, $\pm 20\degr$).
This means that the spatial density structures are elongated
in a direction roughly perpendicular to this direction.

The second point is that the images are not centered on the origin.
We see two possible explanations: either refraction
by a large scale transverse gradient in the column density
of electrons; or highly localized scattering caused by a
region that is centered to one side of the line
towards the pulsar.

Refraction seems likely
since similar refraction has been invoked to explain the
persistent drifting in frequency of scintillation features in
the dynamic spectra of some pulsars (e.g. Shishov, 1974;
Hewish, Wolszczan and Graham, 1985; Gupta, Rickett and Lyne, 1994).
The basic idea is that a refracting region deflects the scattered
(and unscattered) waves which causes a lateral shift
in the scintillation pattern.  Since in a plasma the angle
of refraction is strongly dependent on the observing frequency,
the frequency for a given maximum of intensity will drift in time.
Such a drift will be seen as an asymmetry versus Doppler frequency
in the power distribution of the secondary spectrum, this effect
was confirmed by simulations reported by CRSC.

The alternative explanation is that there are highly localized
regions of very irregular (turbulent?) plasma, that can deflect
(ie scatter or refract) the radio waves into our line of sight.
This implies that thay must be small enough to subtend an angle
at the observer smaller than their maximum angle of deflection.
The angles of arrival of these
waves will be determined by how far the enhanced turbulence is
from the direct line of sight and they will not depend
systematically on frequency.

A third point about the scattered images is that they
are ``blotchy'', but the blotches have a lower filling factor
than would be expected for the usual speckle
in an angular spectrum.  (see the simulations shown by CRSC,
in which there is a multitude of reverse arcs
due to speckle with a high filling factor from a Kolmogorov medium).
We conclude that the blotchy nature of our scattered image is not compatible with a
uniform Kolmogorov random medium.  Our results do not, on their
own, allow us to determine the nature of the medium.  However, the
fascinating measurements by Hill et al. (2005) for pulsar
PSR B0834+06 provide an important constraint.  They found that
the frequency-scaling of the positions of the reverse arcs
was not compatible with frequency dependent refraction and
implies the alternative of highly localized regions of
enhanced scattering as the cause of each reverse arc.
Under this scenario the regions have a linear
size given by their apparent angular size times
the scattering distance --  $D(1-s) \sim 140$~pc.
With angular extents of 0.1-1 mas their maximum implied linear sizes
are 0.014-0.14 AU.

Considering that the discrete reverse arcs were only prominent
in some of the observations (ie days 0-31 and 288), the
regions of localized enhanced scattering are irregularly distributed
in the ionized ISM.  In 31 days the pulsar moves about 10 AU
transverse to the line of sight.  Thus the persistence of
the reverse arcs through 31 days suggests that the
localized regions of enhanced scattering are clumped over
about 10 AU.

If localized scattering regions are indeed responsible for the
discrete reverse arcs, the next question is what
are these regions physically.  However, this is beyond the
reach of our present investigation, and we refer interested readers
to Rickett, Lyne and Gupta (1997)
and the proceedings of the workshop on
``Small Ionized and Neutral Structure in the ISM'',
held in Socorro May 2006, particularly the papers by Stinebring and Rickett.

\section{Conclusion}

The ISS of pulsar B1133+16 showed pronounced changes in
the character of its secondary spectra in 12 days of observation
spread over 15 months.  While these data which were centered at 408 MHz
were recorded over 20 years ago, they often show the parabolic arc
phenomenon that was discovered quire recently by Stinebring et al. (2001).  However,
the basic parabolic arc varied substantially in its extent and shape.
Discrete reverse arcs were visible for about two months early in the sequence
and then again after about nine months.  There was a two month interval when
the arc was barely detectable, corresponding to when the
primary dynamic spectra had a very wide decorrelation bandwidth.

We concentrated our analysis on two days with clearly visible
discrete reverse arcs, which were fortuitously sampled rapidly enough
to resolve them in Doppler frequency.   Using the theory of CRSC,
which ignores dispersion,
we modelled these features and hence were able to estimate
the scattered brightness distribution.  We tested the applicability
of the theory, by simulating and electromagnetic wave propagation
code that includes dispersion, and found that
the reverse arcs are not altered substantially by including dispersion.

Our estimate of the scattered brightness function is highly eleongated
with an axial ratio of more than 10:1.  It is asymmetrical about the origin
and has a small number of discrete maxima.    These properties are not consistent
with expectations from a scattering region that behaves as
a statistically homogeneous random medium with a Kolmogorov spectrum.
When combined witth the conclusions from Hill et al. (2005), our results suggest
that there are highly localized discrete scattering regions
which subtend 0.1-1 mas, but can scatter (or refract) the radiation by
angles that are five or more times larger.  The asymmetry in
the scattered image may well be caused by the chance configuration of these regions.

\acknowledgements  We thank Andrew Lyne for completing the series of
observations at Jodrell Bank in 1984-85 and we thank Yashwant Gupta for work on
the original data reduction.  We thank the National Science Foundation for support under grants AST 9988398 and AST 0507713.


\section{Introduction}

The heavy fermion materials have been one of the most challenging
problems for the condensed matter community in the last three
decades. The physics of these systems is dominated by the
competition of many interactions, manifesting in a variety of
phases\cite{Stewart01}. Close to the zero-temperature
transition between two such phases the materials typically show
non-Fermi liquid behavior on a large temperature range. This
behavior has been often attributed to an underlying quantum
critical point (QCP).

In a subclass of these materials (including CeCu$_{6-x}$Au$_x$,
YbRh$_2$Si$_2$ and Ce$T$In$_5$, with $T$=Ir, Co, Rh) it is
possible to drive the system from an antiferromagnetic (AF) phase
to a paramagnetic (PM) phase by suppressing continuously the
N\'{e}el temperature via application of pressure, magnetic field
or doping. The conventional spin-density wave (SDW) theory of a
QCP\cite{Hertz75, Millis93, Moriya95} disagrees with the
experimental evidence in many of these
materials\cite{Stewart01,Schroeder00,Gegenwart02,Custers03,Coleman01}.
This has triggered a renewed interest in microscopic descriptions
of the heavy fermion state and its possible instabilities to
ordered states.
More generally, not only the critical point but also the formation
of the heavy fermion state in the regime where the RKKY interactions
are sizeable requires nontrivial techniques for its microscopic description
\cite{Nakatsuji04,Sun05b}.
Heavy fermion materials can be
modeled by the periodic Anderson model which describes a wide
conduction band of $spd$ electrons and a narrow band of $f$
electrons which is strongly correlated and hybridized with the
$spd$ electrons.

Doniach\cite{Doniach77} established that for large hybridization
the $f$ electrons combine with the conduction electrons to form a
heavy Fermi liquid, while for small hybridization the $f$ moments
order magnetically and hence do not undergo the Kondo effect.
Thus, changes in the hybridization not only lead to the appearance
of magnetic long range order but introduce also a change in the
electronic structure referred to as $f-c$ decoupling or Kondo
breakdown (KB). Namely, for large values of the
hybridization the quasiparticles are composites of $f$ electrons
and conduction $c$ electrons with a Fermi surface which obeys
Luttinger theorem, counting both the $f$ and $c$ electrons. For
small hybridization the $f$ electrons decouple from the conduction
electron quasiparticles, and the Fermi surface contains only the
latter. The decoupling between the $f$ and $c$ electrons can be
viewed as an orbitally selective Mott transition, and has been
studied
with single site DMFT\cite{demedici05a}
and slave boson techniques\cite{pepin-2006}.
As stressed in \cite{Senthil04}, a precise formulation of KB
requires the elimination of antiferromagnetic long range
order which breaks the translational symmetry of the original
lattice. So far this has been achieved by studying models with
disorder within DMFT\cite{Burdin02}, or by modifying the symmetry
group of the spin from SU(2) to SU(N)\cite{Senthil04}.
For earlier studies of the KB using slave bosons see
\cite{Andrei89,Castellani91}.

Single site Extended DMFT of various models have been carried out.
For the Anderson lattice model\cite{Sun03,Sun05a}, the KB
and the onset of AF should be viewed as distinct
instabilities which occur at two different values of the RKKY
coupling (relative to the Kondo
coupling). The same situation occurs in the field theory approach
of Ref. \cite{Senthil04}.

On the other hand, extended DMFT studies of a Kondo lattice model,
in which the conduction electron band is constrained so as to
prevent its magnetic polarization, reveal that the KB
and the onset of magnetism take place at the same point in the
phase diagram\cite{Si01,Grempel03,Zhu03,zhu-2006}.
See however Ref.\cite{glossop-2006}. Justification for the study
of this model is given in Ref.\cite{Si05}.

In DMFT based approaches the breakdown of Fermi liquid theory at
the transition is understood in terms of local physics.
In other works the same breakdown is explained in terms of the
coupling of the electrons to the long wave length fluctuations of
an emergent gauge field\cite{Senthil04} and to the Kondo order
parameter fluctuations \cite{Paul07,pepin-2006}.

In this paper we handle the competition between magnetism and formation of a
heavy PM state in an unbiased way using Cellular Dynamical
Mean Field Theory (CDMFT)\cite{Kotliar2001spb}. This approach
improves the treatment of non-local correlations over single-site
DMFT considering a cluster of sites and providing a
self-consistency condition on the cluster action. All the
correlations whose range is contained within the cluster are then
treated beyond a simple mean-field level. In this paper we
consider a cluster of two sites embedded in a three dimensional
lattice, which is the minimal unit able to capture the competition
between PM and AF correlations. A similar approach was introduced
in an earlier publication\cite{Sun05b} based on a quantum Monte
Carlo impurity solver, but that study was limited to large
temperatures and small values of $U$. We overcome this limitation
by using Exact Diagonalization (ED) as an impurity
solver\cite{Caffarel94}. This solver gives access to the ground
state properties of the system with a finite energy resolution
(typically 0.005 of the bandwidth in our case).

In the theory of the Mott transition, the investigation of the PM
insulating phase has been very fruitful and allowed an
understanding of the finite temperature state, above the magnetic
order in many compounds. We pursue a similar approach to the
periodic Anderson model by investigating both the magnetic and a
``parent compound'' PM state, to understand better the character
of the electronic structure. This allows us to describe the
magnetic state as a result of an additional instability of the PM
state.

We show that a phase transition separates the heavy fermion
paramagnetic phase from an antiferromagnetic phase. Approaching
the transition from the paramagnetic side the Fermi liquid is
strongly renormalized. An analysis of the mean field PM solution
into the AF phase reveals an orbitally selective Mott decoupling
and localization of the $f$-electrons. Within the resolution of
our method, the antiferromagnetic transition and the decoupling of
the $f$ and $c$ electrons take place for the same value of the
hybridization. Through a comparison with a Wilson's Numerical
Renormalization Group (NRG) calculation, we further relate the
nature of the transition to the proximity of the system in the
parameter space to the quantum critical point of the two-impurity
Kondo model (2IKM)\cite{Jones88,Affleck95}.

The transition has dramatic consequences on the quasiparticle
spectra and on the topology of the Fermi surface which can be
detected by photoemission experiments and by Hall effect and de
Haas-van Alphen frequencies measures.

The paper is organized as follows. In Sec. \ref{methods} we
introduce the model Hamiltonian and the CDMFT method employed. In
Sec. \ref{results} we analyze the results of the CDMFT
calculations and compare them with single site DMFT results and
Numerical Renormalization Group results on the underlying impurity
problem.
Conclusion are presented in Sec. \ref{conclusions}.

\section{Methods \label{methods}}

The three-dimensional periodic Anderson model Hamiltonian is
\begin{eqnarray}\label{Hamiltonian}
    H \!\!&=& \!- t \sum_{\langle ij \rangle}
    c^\dagger_{i\sigma} c^\dagga_{j\sigma}
    - \mu \sum_i c^\dagger_{i\sigma} c^\dagga_{i\sigma}
    + V \sum_i \left( f^\dagger_{i\sigma} c^\dagga_{i\sigma}
    + h.c. \right) \nonumber\\
    &&+ (E_f-\mu) \sum_i f^\dagger_{i\sigma} f^\dagga_{i\sigma}
    + U \sum_i n^f_{i\uparrow} n^f_{i\downarrow}
\end{eqnarray}
where $c^\dagger_{i\sigma}$ creates a conduction band electron at
site $i$ with spin $\sigma$ and $f^\dagger_{i\sigma}$ does the
same for the localized orbitals. The conduction electron bandwidth
$W = 12 |t| = 2$ sets the energy unit.

In the large $U/V$ limit the model maps onto the Kondo lattice
model, where each $f$ orbital is replaced by a spin $\frac{1}{2}$
which is Kondo-coupled to the local conduction electrons with a
coupling $J_K = ( \frac{1}{|E_f-\mu|} + \frac{1}{|U+E_f-\mu|}
)V^2$, where $\rho$ is the conduction electron density of states
at the chemical potential.
It is also possible to define two energy scales in the
problem. The first is the single-impurity Kondo temperature $T_0 =
\exp(-1/2 \rho J_K)$. It is not known a priori how the ``lattice''
Kondo energy is related to $T_0$. The second scale is set by the
magnetic interactions. These are not contained explicitly in this
Hamiltonian, but an effective RKKY coupling is generated in fourth
order perturbation theory in $V$ (second order in the Kondo
coupling, $J_{RKKY} \sim J_K^2/W$). The RKKY coupling oscillates
in real space with a period that is related to the bare conduction
band Fermi momentum, ranging from antiferromagnetic at half
filling to ferromagnetic in the limit of an empty band. A
comparison of these two energy scales allows to understand the
qualitative features of the phase diagram of the
model\cite{Doniach77}. The ratio between Kondo temperature and
RKKY interaction is an increasing function of $V$, hence for small
$V$ we will find a phase in which the magnetic fluctuations
dominate giving rise to an ordered state, while for large $V$ the
Kondo effect dominates and the $f$ moments are screened by the
conduction electrons.

At half filling the Kondo screening of the moments by the
conduction electrons results in an insulator with a gap induced
by the hybridization. In this paper we are interested in metallic
heavy fermions, and we restrict ourselves to study the system away
from half filling.

In the path integral formalism the conduction band can be
integrated out in the action and replaced by an effective
long-range retarded hopping for $f$ electrons. After these
manipulations the Lagrangian of the system is
\begin{eqnarray}\label{Lagrangian}
    {\cal L} &=& \sum_i f^\dagger_{i\sigma}(\tau) \left( \partial_\tau -\mu
    + E_f \right) f^\dagga_{i\sigma}(\tau) \\
    &&+ \sum_{ij} f^\dagger_{i\sigma}(\tau) \Delta^c_{ij}(\tau-\tau')
    f^\dagga_{j\sigma}(\tau')
    + U \sum_i n^f_{i\uparrow} n^f_{i\downarrow} \nonumber
\end{eqnarray}
where the effect of the conduction band is absorbed in the Weiss
field
\begin{equation}\label{Delta-cond}
    \Delta^c(i\omega,\vect{q}) = \frac{V^2}{i\omega+\mu-\varepsilon_{\vect{q}}}
\end{equation}
and $\varepsilon_{\vect{q}} = -2t \sum_{\alpha=x,y,z}
\cos(q_\alpha)$.

In CDMFT the original three-dimensional lattice is now tiled with
a superlattice of clusters and an effective Anderson impurity
action is derived for a single cluster. Finally a self-consistency
condition relates the cluster Green's function to the local
Green's function of the superlattice. The self-consistency is more
easily written as a matrix form
\begin{equation}
   G_{loc}(i\omega) = \sum_{\vec{k}} \left[
   \left(
   \begin{array}{cc}
      i\omega +\mu -E_f & 0 \\
      0 & i\omega +\mu -E_f
   \end{array} \right) \right.
\end{equation}
\begin{displaymath}
   \left. - \frac{V^2}{(i\omega+\mu)^2-\varepsilon_{\vec{k}}^2}
   \left(
   \begin{array}{cc}
      i\omega +\mu & e^{ik_x} \varepsilon_{\vec{k}} \\
      e^{-ik_x} \varepsilon_{\vec{k}} & i\omega +\mu
   \end{array} \right) -
   \left(
   \begin{array}{cc}
      \Sigma_{11}(i\omega) & \Sigma_{12}(i\omega) \\
      \Sigma_{21}(i\omega) & \Sigma_{22}(i\omega)
   \end{array} \right)
   \right]^{-1}
\end{displaymath}
where spin index has been suppressed to simplify the notation. It
is very important to notice that in the large $U/V$ limit the
impurity action becomes equivalent to the action of a two impurity
Kondo model away from particle-hole (p-h) symmetry. At p-h
symmetry this model is known to have a QCP separating a Kondo screened
phase from a phase in which the impurities are decoupled from the
conduction band\cite{Jones88,Affleck95}. Away from p-h symmetry
the QCP is not accessible but still influences the physics in an
intermediate energy range\cite{DeLeo04f,varma-2006}.

In order to solve the cluster-impurity problem we employ ED. In
this method the cluster action is recast in the form of an
Anderson impurity Hamiltonian where the continuous degrees of
freedom of the free electron bath are parametrized by a finite
number $N_b$ of sites:
\begin{eqnarray}\label{H_cluster_ED}
    H_{ED} &=& \sum_{i=1,2} f^\dagger_{i\sigma} \left( E_f - \mu
    \right) f^\dagga_{i\sigma} + U \sum_{i=1,2} n^f_{i\uparrow}
    n^f_{i\downarrow} \\
    && + \sum_{l=1,N_b} \varepsilon_{l\sigma} a^\dagger_{l\sigma}
    a^\dagga_{l\sigma}
    + \sum_{i,l} \left( V_{il\sigma} a^\dagger_{l\sigma}
    f^\dagga_{i\sigma} + h.c. \right). \nonumber
\end{eqnarray}

In this representation the hybridization-function
$\Delta_{\mu\nu}= \mathcal{G}^0_{\mu\nu}-\left(i\omega + \mu
- E_f \right)\delta_{\mu\nu}$, related to
the non interacting impurity Green's function $\mathcal{G}^0_{\mu\nu}=
- \langle f^\dagga_{\mu}f^{\dagger}_{\nu} \rangle_{U=0}$ of the cluster
impurity problem, becomes
\begin{equation}\label{hybrid_cluster}
    \Delta_{ij\sigma}(i\omega) = \sum_{l=1,N_b} \frac{ V^*_{jl\sigma}
    V_{il\sigma} }{i\omega - \varepsilon_{l\sigma}}
\end{equation}
where the indices $i,j=1,2$ represent the cluster sites. In the
figures presented in this work we used 8 sites in the bath, but
the results have been tested in few points using 10 sites.

The ground state and Green's functions of this Hamiltonian are
determined via the Lanczos procedure and the self-consistency
equation in turns allows to derive a new set of bath parameters.
The self-consistency condition is used to determine the $\Delta$
function on a finite mesh of points $\omega_n=(2n+1)\pi/\beta$.
This implies a finite resolution in frequency. We used
$\beta=100$. The process is iterated until convergence is reached.
To study the interplay of antiferromagnetism and Kondo screening
we restrict the study to the metallic regime close to half-filling
where the RKKY interaction is predominantly antiferromagnetic.
This is achieved by fixing the values $U=10$, $E_f=-5.5$ and
$\mu=0.2$ throughout the paper.
Being a mean field theory, DMFT gives the possibility to
selectively allow some instabilities and forbid others. Thus we
will allow the development of AF order and study the realistic
material phase diagram. However we will also force the system to
remain in a PM state to study the ``normal state'' underlying the
AF phase. This will prove important to study the KB
point and to understand the origin of the
transitions in the system.

\section{Results \label{results}}

We compute the
staggered magnetization $M=\langle n^f_{1\uparrow} \rangle -
\langle n^f_{1\downarrow} \rangle = - ( \langle n^f_{2\uparrow}
\rangle - \langle n^f_{2\downarrow} \rangle)$ as a function of $V$
(Fig.\ref{magn}).
\begin{figure}
  \begin{center}
    \includegraphics[width=8cm]{fig1.eps}
    \caption{(Color online) Magnetization $M$ and average of the
    quasiparticle residue $Z$ eigenvalues for the AF
    (both for majority and minority spin) and PM solutions
    as a function of $V$.
    Inset: Ratio between lattice and
    single-impurity Kondo energies in DMFT and CDMFT.
    \label{magn}}
  \end{center}
\end{figure}
For small $V$ the RKKY interaction is dominating and the solution
describes an AF state with a magnetization close to one. In the
opposite limit of large $V$, where the Kondo screening is more
effective, the system is a PM metal. Increasing $V$ in the AF
phase, the magnetization decreases smoothly and approaches zero at
a value $V^* \sim 0.585$ where a phase transition occurs. Above
this critical value the system becomes paramagnetic. The value of
$M$ at the transition jumps from $\sim 0.22$ for $V=0.58$ to $\sim
0.006$ for $V=0.59$ and in between the convergence becomes
extremely slow. The numerical accuracy
does not allow us to exclude a weakly first order transition
scenario, although in this case the
coexistence region would be very small.

To clarify the nature of the transition we investigate the
cluster quasiparticle residue
$\hat{Z}=(\hat{1}-\partial\hat{\Sigma}/\partial\omega)^{-1}|_{\omega=0}$ (in this
case, a matrix in spin and cluster indices). The AF
three-dimensional SDW theory predicts that antiferromagnetism
develops from a well defined Fermi liquid state and hence a finite
quasiparticle residue
$Z$ at the transition. On the contrary KB scenarios
predict
that $Z$ goes to zero at the transition.
In Fig. \ref{magn} we plot the $\hat{Z}$ eigenvalues.
In the AF phase the two curves correspond to minority and majority
species.
In the PM phase we plot only the average of the eigenvalues
since the difference is not appreciable on the scale of the plot.
In both phases the eigenvalues of the $\hat{Z}$ matrix
decrease as we approach the transition.
Precisely at the transition one eigenvalue goes to zero,
suggesting that the KB is indeed the correct
scenario in this model.

Approaching the transition from the PM phase by reducing $V$,
we observe a strong reduction of the $\hat{Z}$ eigenvalues leading to a large effective
mass enhancement.
This result suggests that increasing the correlations
by decreasing $V$ from the PM side, the Fermi liquid becomes
heavier and heavier, and eventually it is more favorable to get
rid of the entropy by developing AF long range order. The
fundamental difference with the SDW picture is that here the
mechanism responsible for the instability is short-ranged in
space.
The cluster self-energies are in fact approximately local
on the PM side of the transition. This may lead to think that the
results are essentially similar to a single site DMFT solution of the
periodic Anderson model. However there are essential differences.
Indeed, even setting the off-diagonal self-energy
$\Sigma_{12}=0$ the CDMFT self-consistency equations do not reduce
to the corresponding single-site DMFT equations.
To demonstrate this
we compare in the inset of Fig.\ref{magn} the ratio between the
lattice Kondo temperature obtained through the relation $T_K =
\Delta(0) Z$ and the corresponding single-impurity energy $T_0$ in
the two cases. While the DMFT result gives a smooth and finite
$T_K$ across $V^*$, in CDMFT $T_K$ appears to approach a zero
value at the transition.

We take advantage now of the possibility of suppressing the AF
long range order to study the normal PM solution in the AF phase.
For $V<V^*$ this solution
is obviously unstable to antiferromagnetism and it corresponds
physically to a highly frustrated system.
\begin{figure}
  \begin{center}
    \includegraphics[width=8cm]{fig2.eps}
    \caption{(Color online) Spectral function of the $f$ electrons
    on the two sides of the transition with
    AF order allowed (upper panel) and forbidden (lower
    panel). The two Hubbard bands are outside the energy range.
    \label{spectral_function}}
  \end{center}
\end{figure}
The analysis of the spectral function along the two solutions
(allowing and forbidding magnetic ordering,
Fig.\ref{spectral_function}) reveals that the transition takes
place also in absence of antiferromagnetism.
Namely, at small V, a phase with no spectral weight at the Fermi
level is realized in both cases.
The Kondo peak present in the PM phase disappears in going across the
transition, leaving behind a pseudogap, and the $f$ electrons
effectively decouple from the conduction band, at least within the
energy resolution of our calculations.
The transition is an orbitally selective Mott transition
for the $f$ band.
This implies that the $f$
electrons do not contribute to the Fermi surface for $V<V^*$.
This is in marked contrast with the paramagnetic single site DMFT
solution where there is only a Fermi liquid Kondo screened phase
and no sign of such a transition or of a pseudogapped phase.
Notice also that in the CDMFT treatment the $f$ transfer of spectral
weight takes place from the Fermi level to its immediate vicinity
as the hybridization is reduced. This should be contrasted with
the single site DMFT description of the Mott transition in the Hubbard model where the
spectral weight is transferred from the Fermi level directly to the
Hubbard bands at much higher energies.

\begin{figure}
  \begin{center}
    \includegraphics[width=8cm]{fig3.eps}
    \caption{(Color online) Imaginary part of the even self-energy
    eigenvalue on the imaginary axis for different values of the
    hybridization. Inset: odd eigenvalue (in a
    different range of hybridization values).
    \label{sigma_even_odd}}
  \end{center}
\end{figure}
To clarify the Mott character of the transition we plot in
Fig.\ref{sigma_even_odd} the imaginary frequency behavior of the
two $\Sigma$-eigenvalues within the PM solution, named
respectively $\Sigma_{even}=\Sigma_{11}+\Sigma_{12}$ and
$\Sigma_{odd}=\Sigma_{11}-\Sigma_{12}$. Approaching the transition
from the PM phase the slope at zero frequency of both eigenvalues
increases, signaling the growth of the effective mass in the Fermi
liquid. At the transition the even eigenvalue develops a
divergence signaling the end of the Fermi liquid description and
the beginning of the Mott insulating phase. The odd eigenvalue has
a similar divergence at a lower value of the hybridization as
shown in the inset. A qualitative understanding of this behavior
can be proposed considering that $\Sigma_{even}$ and
$\Sigma_{odd}$ represent in a loose sense the lattice self-energy
calculated around momenta $(0,0,0)$ and $(\pi,\pi,\pi)$
respectively. Hence the Mott transition corresponds to the first
appearance of a surface poles in the self-energy around the origin
of the Brillouin zone. Decreasing further the hybridization the
surface of poles moves towards the corner of the zone and finally
disappears for $V\sim0.43$.

While for $V > V^*$ the self-energy was approximately local,
in the PM solution the development of magnetism for $V < V^*$ is
replaced by a dramatic increase of non-local correlations as shown
in Fig.\ref{sigma12}, where we plot the real part of $\Sigma_{12}$
at the lowest available frequency in the imaginary axis.
\begin{figure}
  \begin{center}
    \includegraphics[width=8cm]{fig4.eps}
    \caption{(Color online) Non local correlations (${\rm Re}
    \Sigma_{12}$, calculated at the lowest available imaginary frequency)
    along the AF and PM solutions. The vertical
    dashed line marks the transition. For comparison
    with the PM case the NRG results (without self-consistency).
    \label{sigma12}}
  \end{center}
\end{figure}

The behavior of $T_K$, of the spectral function and of the
off-diagonal self-energy near the transition is very similar to
the corresponding behavior found in the 2IKM close to its
QCP\cite{DeLeo04f,varma-2006,in_prep}. To establish this
connection we used Wilson's Numerical Renormalization Group
to solve the impurity model underlying the self-consistency in the
PM case for the same set of parameters.
The difference with the ED calculation is that we used a constant
density of states for the NRG bath, so the results are non
self-consistent.
The results are nevertheless
qualitatively similar. The main difference between the results of
the two calculations is that the transition found in CDMFT is
replaced in NRG by a crossover. We ascribe this effect to the lack
of a self-consistent bath in the NRG calculation. In Fig.\ref{sigma12}
we compare the non local correlations, finding a very good
agreement. Other features like the pseudogap formation around
$V^*$ are also
reproduced. The similarity of these two results suggests that most
of the physics relevant to the lattice is already captured at the
level of the two impurity problem. In this case the onset of
magnetism in the lattice model can be understood as arising from
the vicinity to the impurity QCP.
Indeed at the 2IKM critical point there are three relevant operators
that correspond to possible symmetry breaking instabilities.
Susceptibilities in the antiferromagnetic, superconducting and
p-h symmetry breaking channels are diverging at the impurity QCP.
Since p-h symmetry is already broken in the lattice system,
only the other two instabilities are left to provide a way to
quench the residual entropy associated to the QCP. In this work
we did not consider the superconducting instability but we
expect it to play a relevant role (and possibly compete with
antiferromagnetism) depending on the specific details of the
band structure of the realistic material.

Notice however several important differences with the picture
of local quantum criticality of Ref.\cite{Si01}. The reference
system describing the physics of the lattice model requires a
cluster of two sites, rather than one, embedded in a medium.
The critical point of this two impurity model, has two relevant
directions corresponding to varying the hybridization and breaking
p-h symmetry. So the impurity model without the self-consistency
condition has a multicritical point rather than a critical point.
Generic problems, as the one considered in this paper, are p-h
asymmetric, and therefore in these cases the two impurity QCP is
not reachable.
However the analysis of the NRG low-energy spectrum confirms that in going
from one side to the other of the crossover (correspondingly, from
one side to the other of the transition in the CDMFT case) the
system evolves from a state in which the impurities are screened
by the conduction electrons (for large $V$) to a state in which
the impurities take advantage of the RKKY interaction to form a
singlet and decouple from the conduction band (for small $V$).
Hence the low-energy picture for $V<V^*$ is that of a conduction
band asymptotically decoupled from a collection of spins arranged
in singlets.
In the lattice model the self-consistency condition transforms
the crossover of the impurity model into a true phase transition.

Following the paramagnetic solution we can also observe a
violation of the Luttinger theorem for $V<V^*$ in the form of a
mismatch between the number of particles and the volume of the
Fermi surface. This is clear from the plot of the number of $c$
and $f$ electrons per site as a function of $V$ (Fig.
\ref{densities}). In the PM phase both the conduction and the $f$
bands are doped ($n_f, n_c \neq 1$). In this case the Fermi
surface has a volume equal to the doping $\delta = n_c + n_f - 2$.
The total occupation is not constant because we fixed the chemical
potential rather than the density. For $V<V^*$ the occupation in
both bands becomes essentially constant, implying that $V$ is no
more effective. In this regime the only contribution to the Fermi
surface comes from the $c$ electrons which, being non-interacting
and decoupled from the $f$ band, have a Fermi surface with volume
$n_c = 1 + \delta$.
These results once again support the orbitally
selective Mott transition picture of the KB
\cite{demedici05a,pepin-2006}.
Namely, an $f$-like fluid that becomes incompressible beyond the
KB point.

\begin{figure}
  \begin{center}
    \includegraphics[width=8cm]{fig5.eps}
    \caption{(Color online) $f$ and $c$ electron occupation
    per site. The transition is marked by the vertical dashed
    line.
    \label{densities}}
  \end{center}
\end{figure}

In Fig.\ref{bands} we plot the $k$-resolved total spectral density
$-\frac{1}{\pi} \rm{Im} (G_c(\vect{k},\omega) +
G_f(\vect{k},\omega))$ at low energies along the paramagnetic
solution in the two phases. We intentionally consider points far
from the transition to show that the results are robust and do not
depend on the vicinity to special points in the phase diagram. To
obtain this quantity we reconstruct the rotational and
translational lattice symmetry, broken by the cluster-tiling procedure, by
periodization of the cumulants \cite{Stanescu06b}. This scheme is
more appropriate when correlations are stronger. Indeed the simple
periodization of the self-energy does not reproduce the local
quantities correctly upon integration over momenta, introducing
spurious states in gapped regions of the spectrum. In panel (a)
the system is in the PM phase ($V=0.65$). It is possible to track
the origin of the small value of $Z$ in the narrow band crossing
the chemical potential. The Fermi surface encloses a ``small''
region of the Brillouin zone centered around the origin.
In panel (b) we show the PM solution in the AF phase ($V=0.30$).
It is clear that the volume of the Fermi surface in this phase has
increased. The difference between the Fermi surface volume and the
density is equal to one within the numerical precision, confirming
the violation of the Luttinger theorem as expected in the
orbital selective Mott
transition \cite{pepin-2006}. It is also possible to track the
non-dispersive $f$-band at around $\omega \sim -0.2$.
\begin{figure}
  \begin{center}
    \includegraphics[width=8.5cm]{fig6.eps}
    \caption{(Color online) Total spectral function as a function
    of momentum and energy on the path connecting the origin
    $\Gamma=(0,0,0)$ and the corner $\Pi=(\pi,\pi,\pi)$ of
    the Brillouin zone. (a) $V=0.65$ PM phase. (b) $V=0.30$
    PM solution in the AF phase.
    \label{bands}}
  \end{center}
\end{figure}

Since the PM solution for $V<V^*$ is unstable to
antiferromagnetism, to make contacts with experiments we plot in
Fig.\ref{bands_AF} the spectral density for the AF solution
($V=0.50$). The bands are doubled because of the AF ordering.
These results are representative for the whole AF phase down to
small value of $V$. The striking difference with respect to the PM
phase is the big rearrangement of the spectral weight: now the
occupied region is centered around the $(\frac{\pi}{2},
\frac{\pi}{2}, \frac{\pi}{2})$ point. This result implies a sudden
jump across the antiferromagnetic transition in all the properties
connected with the topology of the Fermi surface and it is
consistent with the experimental observation of a discontinuous
behavior of the Hall coefficient\cite{Paschen04} and of the de
Haas-van Alphen frequencies\cite{Shishido05}. The contribution of
the $f$ electrons in this phase is manifest in the two
non-dispersing bands at $\omega \sim -0.1$ and $\omega \sim 0.45$,
corresponding to the two peaks in Fig.\ref{spectral_function},
while the bands crossing the chemical potential have mostly $c$
character.
\begin{figure}
  \begin{center}
    \includegraphics[width=8.0cm]{fig7.eps}
    \caption{(Color online) Total spectral function as a function
    of momentum and energy along the same path of Fig. \ref{bands}
    for the AF solution. ($V=0.50$)
    \label{bands_AF}}
  \end{center}
\end{figure}

\section{Conclusions \label{conclusions}}

To summarize, we have constructed and investigated a cluster
dynamical mean field theory for the periodic Anderson model using
a link as a reference frame to capture the competition of the
Kondo effect and the RKKY interaction. The mean field phase
diagram away from half-filling features a transition separating a
paramagnetic heavy electron phase from an antiferromagnetic
phase. In the PM phase, the phase transition is accompanied by a
strong renormalization of quasiparticles, marked by a large
enhancement of the effective mass on all the Fermi surface.

By constraining a mean-field metastable paramagnetic solution
into the AF region of the phase diagram, we have been able to show
that the the transition can be
described as an orbitally selective Mott transition.
The decoupling of the $f$-band from the
conduction band marks a localization of the $f$-electrons, as
shown by the opening of a pseudogap in the single particle spectrum
and the appearance of poles in the self-energy at the Fermi
level, a clear fingerprint of ``Mottness''. This is further
supported by the incompressibility of the system, which presents
a constant density by varying the hybridization, and by the violation
of the Luttinger theorem for $V<V^*$ along the PM solution.
It is important to compare this transition, which presents a
orbital-selective Mott character, with the traditional DMFT Mott
transition of the single band Hubbard model. In the present case
the weight is shifted from the chemical potential to the immediate
vicinity forming a pseudogap rather than being transferred to the
Hubbard bands at much higher energy. This feature is the product
of our link-DMFT analysis in three dimensions, strikingly
different from the infinite dimensional DMFT result where the
physics is purely local. It would be important in future DMFT
investigations to better understand the nature of this Mott-like
transition by studying its cluster-size dependence.

In the SDW scenario, the fermions which are not coupled by the
ordering vector are unaffected by the transition. On the contrary
in our CDMFT study, the transition is accompanied by a
dramatic rearrangement of the Fermi surface, in agreement
with recent experiments\cite{Paschen04,Shishido05}.

Our findings can be understood in terms of proximity to a
decoupling QCP with the same properties of the QCP found in the
2IKM, namely the impurity model underlying the self-consistency
equations. Although the lattice system cannot reach that critical point due
to the particle-hole symmetry breaking, the region in which the
effect of the critical point is relevant is large enough to trigger
the lattice antiferromagnetic instability\cite{DeLeo04f}. If we follow the
paramagnetic solution the vicinity to the critical point results
instead in an increase of the non-local correlations $\Sigma_{12}$ in
agreement with the behavior of the 2IKM. In real systems this effect is
preempted by the lattice antiferromagnetic instability.
It would be interesting to realize heavy fermion systems in highly
frustrated lattices to suppress the antiferromagnetic instability.
This would provide a physical realization of the paramagnetic
solution of the CDMFT equations.

In this framework the antiferromagnetic transition is the result
of the response of the system to the instability of the nearby
decoupling QCP, rather than being an intrinsic feature. Notice
that other correlations are diverging at the 2IKM critical point.
We expect that depending on the details of the model other
instabilities (mainly superconductivity) compete with
antiferromagnetism to order the system and quench the residual
entropy of the unstable QCP.

We believe that our description captures the correct physics of
the periodic Anderson model in an intermediate asymptotic
regime of energies and proximity
to the transition.
At higher energies, the single site DMFT description is regained.
At very low energies and very close to the transition, further
non local effects could turn out to be very important.
We do not know if our treatment fits within the local quantum
criticality paradigm where no spatial anomalous dimensions are
generated. To elucidate these questions would require the
investigation of the dynamical susceptibility beyond the cluster
DMFT approximation, which is beyond the scope of this work.

We would like to thank M. Fabrizio, A. Georges, F. Steglich, P. Coleman, C.
Pepin, I. Paul, O. Parcollet, P. Sun, K. Haule, E. Lebanon, M.
Capone, T. Stanescu, J.C. Domenge and L. de' Medici for helpful
discussions and comments. This work was supported by the NSF under
Grant No. DMR 0528969.