\chapter{Transport computation: Doped LaTiO$_3$}
\label{chap:latio3_revisited}
In this chapter we will revisit the doped LaTiO$_3$ system that we considered
in the first chapters of this thesis.  We will not be able to treat the full
realistic bandstructure model but will rather consider a slightly simpler
model where only the correlated d-bands of the titanium are retained.  This is
a better realization of the real bandstructure than the model we considered
earlier insofar as it treats properly the symmetry and the degeneracy of the
band and we are also able to treat degenerate models that have a finite value
for the interaction.  The interaction term that we treat in this model is
$SU(N)$ symmetric Hubbard type interaction, i.e. there is penalty of $U$
associated with each pair of electrons present at each site.  This is probably
a different interaction than exist for these orbitals in the real material
since in general we do not expect the same interaction to be between electrons
of opposite spins in the same orbital as there is between two electrons in
different orbitals.  So far, methods to deal with this, more general, case
have not been developed so we will here consider only the $SU(N)$ case.

The solution method is based on recent progress made in solving DMFT
equations and and impurity models for multiband systems and these
calculations are essentially the first fruits of that work.

\section{The LaTiO$_3$ system}
As we have mentioned before (see \article\ref{sec:titanate}) LaTiO$_3$
is a Mott insulator.  What that means of course is that the insulating
nature is due to correlations and not due to the band structure.
Indeed with the valence of titanium being equal to 1.0 the system
should be a metal according to band theory and that is what we see in
the LDA computation of the bandstructure and density of states for
this system in figure \ref{fig:LDAdos}.
\begin{figure}[t]
\centering
\begin{tabular}{c@{\;}c}
\mbox{\epsfig{file=Figures-8/latio3-dos.ps,angle=-90,width=\halfpage}} &
\mbox{\epsfig{file=Figures-8/partial-dos.ps,angle=-90,width=\halfpage}} \\
\mbox{\epsfig{file=Figures-8/latio3Bands1.ps,angle=-90,width=\halfpage}} &
\mbox{\epsfig{file=Figures-8/latio3Bands2.ps,angle=-90,width=\halfpage}} \\
\end{tabular}
\caption{The LDA bandstructure and density of states for the LaTiO$_3$
system computed in the atomic spheres approximation.  The left figures
shows the bandstructure and total DOS over a wide energy range.  The
right figures shows the bandstructure and total DOS around the
Fermi-level and the different components that make up the complete
DOS.}
\label{fig:LDAdos}
\end{figure}   
In the figures on the left we see the bandstructure and the total
density of states per spin over a large energy range, where the filled
oxygen p-bands are clearly visible below the Fermi-surface.  In the
figure on the right we have shown the same data in a smaller region
around the Fermi-level and also displayed the various components that
make up the total density of states.  As we can see the main
contribution to the total density of states is from the titanium
d-bands, and the rest is due to contributions from lanthanum and
oxygen.  We see that some of the spectral density is due to the oxygen
p-bands and thus the deep levels beneath the Fermi-level can't be
wholly due to oxygen and indeed there is some contribution from
titanium d-levels down there.

In the discussion that follows we will make the assumption that the
three bands that cross the fermi-surface and extend from around 11.5eV
to 14.2eV can be characterized as degenerate titanium t$_{2g}$ bands.
With this we aim to model the correlations and the low temperature
behavior of the doped LaTiO$_{3}$ system with a simple tight binding
$t_{2g}$ model with a bandwidth of 2.7eV.

\section{Doped LaTiO$_3$: The Sr$_{x}$La$_{1-x}$TiO$_{3}$ series}
Upon doping with Sr the Mott insulating state of LaTiO$_{3}$ is
immediately destroyed and the system becomes metallic as we can see in
the graph on the right in figure \ref{fig:TokuraData}.  The
temperature dependence of the resistivity in this figure shows an
activated behavior at LaTiO$_{3}$ but as the material is doped it
reverts to a correlated metal showing a Fermi-liquid, $T^{2}$,
dependence at low temperatures.
\begin{figure}[htb]
\centering
\begin{tabular}{c@{\;}c}
\mbox{\epsfig{file=Figures-8/Tokura2.eps,width=0.5\linewidth}} &
\mbox{\epsfig{file=Figures-8/Tokura1.eps,width=0.4\linewidth}} \\
\end{tabular}
\caption{Experimental data showing the temperature and doping
dependence of the resistivity and the doping dependence of the
magnetic susceptibility and the linear coefficient of specific heat.
Data taken from reference \cite{Tokura:1993a}.}
\label{fig:TokuraData}
\end{figure}   
The graph on the left in figure \ref{fig:TokuraData} shows the
magnetic susceptibility and the linear coefficient of specific heat
and as we can see the data shows a dramatic increase in both these
parameters as we get closer to the LaTiO$_{3}$ end indicating that the
Mott insulating behavior can seen as being due to a divergence in the
effective mass, i.e. the vanishing of the quasiparticle residue.

\section{The model}
In LaTiO$_3$ the crystal structure around the titanium ions has cubic
symmetry so if we only want to retain the d-band of the titanium in the
Hamiltonian we must first change the d-band basis of spherical harmonics,
\begin{eqnarray}
  \Psi^s_{2,0}(\theta,\phi) 
  &=& \sqrt{\frac{5}{16\pi}}(3\cos^2\theta-1)
  \nonumber \\
  \Psi^s_{2,\pm 1}(\theta,\phi) 
  &=& \mp\sqrt{\frac{15}{8\pi}}\sin\theta\cos\theta e^{\pm i\phi} 
  \nonumber \\
  \Psi^s_{2,\pm 2}(\theta,\phi) 
  &=& \sqrt{\frac{15}{32\pi}}\sin^2\theta e^{\pm 2i\phi} 
\end{eqnarray}
to a basis that respects this symmetry, the so called cubic harmonic
basis:
\begin{gather}
  \label{eq:cubicHarmonics}
  \Psi^c_{2,1}(x,y,z) = \frac{1}{4}\sqrt{\frac{15}{\pi}}\frac{2xy}{r^2}
  \qquad
  \Psi^c_{2,2}(x,y,z) = \frac{1}{4}\sqrt{\frac{15}{\pi}}\frac{2xz}{r^2}
  \qquad
  \Psi^c_{2,3}(x,y,z) = \frac{1}{4}\sqrt{\frac{15}{\pi}}\frac{2yz}{r^2}
  \nonumber\\
  \Psi^c_{2,4}(x,y,z) = \frac{1}{4}\sqrt{\frac{15}{\pi}}\frac{x^2-y^2}{r^2}
  \qquad
  \Psi^c_{2,5}(x,y,z) = \frac{1}{4}\sqrt{\frac{15}{\pi}}
        \frac{3z^2-r^2}{\sqrt{3}r^2}.
\end{gather}
The Coulomb interaction from the eight oxygen ions surrounding the
titanium ion splits the 5-fold degenerate d-band into a lower lying
3-fold band comprised of $\Psi^c_{2,1}$, $\Psi^c_{2,2}$ and
$\Psi^c_{2,3}$, and a upper 2-fold degenerate band of $\Psi^c_{2,4}$
and $\Psi^c_{2,5}$.  The lower band is referred to as the $t_{2g}$
band and the upper one as the $e_g$ band.  The splitting is referred
to as crystal field splitting.  Since the band has only one electron
in it, or less upon doping with strontium, the fermi-level will be far
below the $e_g$-band edge.  Hence in our model we will simply assign
some large energy to the $e_g$-band and no dynamics, i.e.~no hopping
of $e_g$ electrons between sites.  The hopping between the $t_{2g}$
orbitals can be found by considering the overlap between the $t_{2g}$
orbitals on neighboring sites and using symmetry consideration we find
the following Hamiltonian:
\begin{eqnarray}
  H &=& \sum_{i}E_{t_{2g}}
  \left(
    d^{+}_{xy,i}d_{xy,i}+d^{+}_{xz,i}d_{xz,i}+d^{+}_{yz,i}d_{yz,i}
  \right) 
  \nonumber\\ &+&
  \sum_{i}E_{e_{g}}
  \left(
    d^{+}_{x^2+y^2,i}d_{x^2+y^2,i}+d^{+}_{3z^2-1,i}d_{3z^2-1,i}
  \right) 
  \nonumber\\ &-&
  \sum_{i,j=\pm x}
  \left( 
    t_1 d^{+}_{xy,i}d_{xy,i+j}+
    t_1 d^{+}_{xz,i}d_{xz,i+j}+
    t_2 d^{+}_{yz,i}d_{yz,i+j}
  \right)
  \nonumber\\ &-&
  \sum_{i,j=\pm y}
  \left( 
    t_1 d^{+}_{xy,i}d_{xy,i+j}+
    t_2 d^{+}_{xz,i}d_{xz,i+j}+
    t_1 d^{+}_{yz,i}d_{yz,i+j}
  \right)
  \nonumber\\ &-&
  \sum_{i,j=\pm z}
  \left( 
    t_2 d^{+}_{xy,i}d_{xy,i+j}+
    t_1 d^{+}_{xz,i}d_{xz,i+j}+
    t_1 d^{+}_{yz,i}d_{yz,i+j}
  \right).
\end{eqnarray}
Fourier transforming the variables in the Hamiltonian we get:
\begin{eqnarray}
  \label{eq:t2gHamiltonian}
  H &=& \sum_{k}E_{e_g}
  \left(
    d^{+}_{x^2+y^2,k}d_{x^2+y^2,k}+d^{+}_{3z^2-1,k}d_{3z^2-1,k}
  \right) 
  \nonumber\\ &+&
  \sum_{k}d^{+}_{xy,k}d_{xy,k}
  \left(E_{t_{2g}}-2t_1\cos(k_x)-2t_1\cos(k_y)-2t_2\cos(k_z)\right)
  \nonumber\\ &+&
  \sum_{k}d^{+}_{xz,k}d_{xz,k}
  \left(E_{t_{2g}}-2t_1\cos(k_x)-2t_2\cos(k_y)-2t_1\cos(k_z)\right)
  \nonumber\\ &+&
  \sum_{k}d^{+}_{yz,k}d_{yz,k}
  \left(E_{t_{2g}}-2t_2\cos(k_x)-2t_1\cos(k_y)-2t_1\cos(k_z)\right).
\end{eqnarray}
The reason why we get two hopping integrals, $t_1$ and $t_2$ is because of
orientation of the different orbitals -- there is more overlap between $xy$
orbitals and $xz$ orbitals at $i$ and $i+x$ than there is between $yz$ at the
same sites.  Thus we have $t_2 < t_1$.   The half width of the $t_{2g}$ band
is easily found from the Hamiltonian above and is given by:
\begin{equation}
  \label{eq:t2gHalfWidth}
  D_{t_{2g}} = 2(2|t_1|+|t_2|).
\end{equation}
\begin{figure}[t]
\centering
\begin{tabular}{c@{\;}c}
\mbox{\epsfig{file=Figures-8/t2gDOS.ps,angle=-90,width=\halfpage}} &
\mbox{\epsfig{file=Figures-8/t2gPHI.ps,angle=-90,width=\halfpage}} \\
\mbox{\epsfig{file=Figures-8/t2gBands.ps,angle=-90,width=\halfpage}} &
\mbox{\epsfig{file=Figures-8/cubicBands.ps,angle=-90,width=\halfpage}} 
\end{tabular}
\caption{The upper two figures show the bare density of states and the
bare transport density for the $t_{2g}$ band of bandwidth 2.7eV using
hopping parameters with $t_{2} = t_{1}/10$.  For comparison we show a
generic cubic s-band of the same bandwidth.  The lower two figures
show the corresponding bandstructure, t$_{2g}$ band on the left and
cubic band on the right.}
\label{fig:t2gDensities}
\end{figure}         
It is easy to compute the density of states for this Hamiltonian since
we have already diagonalized it -- the total density of states is
simply the sum of the density of states for the individual degrees of
freedom.  For the $e_g$ band the density of states is simply a delta
function centered at $E_{e_g}$ and we will ignore that.  For the
$t_{2g}$ band the density of states is best computed using Fourier
transforms, namely:
\begin{equation}
  \label{eq:t2gDOS}
  D(\epsilon) = 3\sum_{k}\delta\left(
  \epsilon-[-2t_1\cos(k_x)-2t_1\cos(k_y)-2t_2\cos(k_z)]\right)
  = \int_{-\infty}^{\infty}ds e^{is\epsilon}\tilde{D}(s)
\end{equation}
where (see equation (\ref{eq:BesselFunction}))
\begin{eqnarray}
  \tilde{D}(s) &=& \int_{-\infty}^{\infty}\frac{d\epsilon}{2\pi}
  e^{-is\epsilon}D(\epsilon)
  = 3\frac{1}{2\pi}\sum_{k}e^{2ist_1\cos(k_x)+2ist_1\cos(k_y)+2ist_2\cos(k_z)}
  \nonumber\\ &=&
  3\frac{1}{2\pi}
  \left(\int_{-\pi}^{\pi}\frac{dk}{2\pi}e^{2ist_1\cos(k)}\right)^2
  \left(\int_{-\pi}^{\pi}\frac{dk}{2\pi}e^{2ist_2\cos(k)}\right)
  \nonumber\\ &=&
  3\frac{1}{2\pi}\left(J_0(2t_{1}s)\right)^{2}J_0(2t_{2}s).
\end{eqnarray}
Here we have taken $E_{t_{2g}} = 0$.  In figure \ref{fig:t2gDensities}
we have displayed the density of states for the $t_{2g}$-band of
bandwidth 2.7eV where the ratio of the larger and the smaller hopping
parameter is taken to be 10.  For comparison we plot the generic cubic
s-band with the same bandwidth.  It is seen that by tuning the $t_2$
parameters we can go continuously from a realization of the generic
s-band, tight binding, density of states in 2 dimensions ($t_2 = 0$)
to the 3 dimensional one ($t_2 = t_1$).  In addition we have also
displayed in figure \ref{fig:t2gDensities} the transport density:
\begin{equation}
  \label{eq:t2gPHI}
  \Phi(\epsilon) = 3\sum_{k}(\vec{v}_k)^2\delta\left(
  \epsilon-[-2t_1\cos(k_x)-2t_1\cos(k_y)-2t_2\cos(k_z)]\right)
\end{equation}
where $\vec{v} =
\grad{k}[-2t_1\cos(k_x)-2t_1\cos(k_y)-2t_2\cos(k_z)]$. This quantity
can also be easily computed through Fourier transforms and allows us
to compute the transport function without having to do the full
Brillouin zone integral.  The lower two graphs in figure
\ref{fig:t2gDensities} show the bandstructure for the the t$_{2g}$
band and the cubic band.

\section{DMFT solution}
The method \cite{Savrasov:2000} that is used here to solve the impurity model
associated with the DMFT equations is based on interpolated perturbation
theory.  By construction the method is exact in the following limits:
\begin{itemize}
\item $i\omega \rightarrow 0$ for all values of $U$
\item $i\omega \rightarrow \infty$ for all values of $U$
\item $U \rightarrow 0$ for all values of $i\omega$ 
\item $U \rightarrow \infty$ for all values of $i\omega$ 
\end{itemize}
and interpolates smoothly between these limits for intermediate values of $U$
and $i\omega$.

By properly treating these limits the method ensures that both high energy
properties, such as total energy as well as low energy properties such as
quasiparticle renormalization and the Mott transition are captured correctly.

\section{Results: Spectral densities and basic parameters}
\label{sec:basicResults}
In this section we will look at solutions for the $t_{2g}$ model and
consider the variation in the physical quantities as a function of
interaction strength.  In our computation here we use the following
choices for the parameters of the band:
\begin{equation}
t_1 = 0.05 
\qquad\mbox{and}\qquad
t_2 = 0.005.
\end{equation}
These parameters can be regarded as the internal parameters within the
numerical code and the results will be scaled after the computation to
produce the correct bandwidth.  In what follows we will always assume
that the bandwidth is 2.7eV but we should note that the units could
also be determined a-posteriori by comparing the results of the
computation to available experimental data.  Hopefully this way of
determining the units on the hopping parameters would also lead to a
bandwidth of 2.7eV.  The reason why we chose to fix the bandwidth
a-priori is because that is more in line with the DMFT methodology,
i.e. we compute the effects of the correlations on top of previously
determined LDA or single particle results.  

In the following discussion all energy values will be quoted in eV but
since the units in the code are somewhat different the values of some
of the parameters might seem a little arbitrary.

In the data-series that we consider in this section we have taken the
density, $n$, to be 0.95, 0.9 and 0.80 which corresponds to 5, 10 and
20\% doping from the Mott insulating state at $n = 1.0$.  The
temperature is taken as $T = 74K$.  The calculation is started at a
small value of interaction $U = 1.93eV$ and then it is slowly
increased to $U = 11.57eV$.  The computation is done is two steps:
\begin{enumerate}
\item The self energy is computed on the Matsubara axis, where computations 
are usually more stable. 
\item The solution from the Matsubara axis is used to compute the retarded 
self energy on the real axis.  
\end{enumerate}
The second step of this procedure is usually more likely to incur
numerical instabilities.  For this reason we have not been able to
obtain retarded self energies for values of the interaction larger
than about $U = 5.2eV$ and hence we cannot compute the full density of
states or the transport parameters for $U$ larger than that.  On the
Matsubara axis the model can be solved for larger values of $U$ and
here we have gone all the way up to $U = 11.57eV$.

\begin{figure}[t]
\centering
\begin{tabular}{c@{\;}c}
\mbox{\epsfig{file=Figures-8/dos0.3.ps,angle=-90,width=\halfpage}} &
\mbox{\epsfig{file=Figures-8/dos0.5.ps,angle=-90,width=\halfpage}} \\
\mbox{\epsfig{file=Figures-8/dos0.7.ps,angle=-90,width=\halfpage}} &
\mbox{\epsfig{file=Figures-8/dos0.8.ps,angle=-90,width=\halfpage}} \\
\end{tabular}
\caption{The renormalized density of states for the t$_{2g}$ band computed 
for various values of interaction. $U$. }
\label{fig:t2gDOS}
\end{figure}         

If we start by looking at the real axis data we see from figure
\ref{fig:t2gDOS} that the density of states around the Fermi-surface for the
smallest value of the interaction is quite similar to the bare density of
states in figure \ref{fig:t2gDensities}, but in addition we have an upper
Hubbard peak centered around 2.5eV.  We also see a small satellite peak below
the Fermi-surface and since this peak is close to 5eV below the upper Hubbard
peak it does not seem to correspond to lower Hubbard band and looks like it
might be spurious.  However, as the interaction is increased this satellite
peak moves closer to the Fermi-level and increases in weight and for $U =
5.14$eV this peak should clearly be identified as the lower Hubbard band.  At
the same time we see that the peak that is originally at the Fermi-level
narrows, most of the spectral weight is transferred up to the upper Hubbard
band and for $U = 5.14$eV this peak should be identified as the
quasiparticle peak.  The width of the quasiparticle peak as compared to the
width of the original band is given by the quasiparticle residue, defined by:
\begin{equation}
Z = \left(
1-\left.
\frac{\partial \real \Sigma(\omega)}{\partial \omega}
\right|_{\omega = 0}
\right)^{-1} = 
\left(
1-\left.
\frac{\partial \imag \Sigma(i\omega_{n})}{\partial (i\omega_{n})}
\right|_{i\omega_{n} \rightarrow 0}
\right)^{-1}.
\end{equation}
We see from this equation that the quasiparticle residue can be computed from
the Matsubara axis data and in figure \ref{fig:t2gMatsubara} we have displayed
$Z$ along with the chemical potential, the real part of the self energy at the
Fermi-surface and the density of states at the Fermi-surface which we compute
simply by finding the zero frequency limit of the imaginary part of the
Matsubara Green's function.
\begin{figure}[t]
\centering
\begin{tabular}{c@{\;}c}
\mbox{\epsfig{file=Figures-8/t2gChem.ps,angle=-90,width=\halfpage}} &
\mbox{\epsfig{file=Figures-8/t2gQuasi.ps,angle=-90,width=\halfpage}} \\
\mbox{\epsfig{file=Figures-8/t2grhoFermi.ps,angle=-90,width=\halfpage}} &
\mbox{\epsfig{file=Figures-8/t2gReSigZero.ps,angle=-90,width=\halfpage}} \\
\end{tabular}
\caption{The chemical potential, quasiparticle residue, density of states at
  the Fermi-level and the real part of the self energy at the Fermi-level for
  the t$_{2g}$ model as a function of interaction strength, U.}
\label{fig:t2gMatsubara}
\end{figure}     
We see clearly from the figure that there is a qualitative change in
the solution occurring around $U = 6eV$.  The quasiparticle residue
becomes less a function of the interaction strength, and depends
essentially only on doping.  The large $U$ values of the quasiparticle
residue are in close agreement with previous estimates for the Mott
insulator \cite{Kajueter:1996a}.  It seems likely that at half filling
the quasiparticle residue would go to zero for interaction strengths
above 5.5-6eV and thus the model would be insulating in that regime
and metallic for lower values.  Upon doping away from half filling the
metallic state does not change very drastically as we can see from the
doping independence of the quasiparticle residue whereas the
insulating state is always rendered metallic for any value of $U$.

The behavior of the chemical potential is indicative of the transfer of
spectral weight as we increase the interaction strength.  For small values of
the interaction the chemical potential increases which tells us that the
spectral weight from the noninteracting band is being transferred to the upper
Hubbard band.  For higher values of the interaction strength this trend is
reversed and now the spectral weight is being transferred to energies below
the Fermi-level.  Hence the scenario as the interaction strength is increased
unfolds as follows: At zero $U$ we start out with a noninteracting band and
when $U$ is increased the spectral weight is transferred to an upper Hubbard
band and the original noninteracting peak can be regarded as the lower Hubbard
band.  As we increase $U$ even further the peak that we identified as the
lower Hubbard band loses weight to a structure that forms below the
Fermi-level -- the lower Hubbard band, and the original noninteracting band
can now be identified with the quasiparticle peak.

We should note that the value of interaction at which the metal insulator
crossover takes place here is about two times the bandwidth or four times the
half bandwidth.  This is a somewhat larger number than is obtained in single
band models where the critical $U$ is found to be around 2.5-3.0 times the
half bandwidth \cite{Georges:1996}.  Using estimates for multiband models from
\cite{Fresard:1997} we find that the critical $U$ for the Mott transition
takes place at $U_c \simeq 6.1eV$ which agrees very well with our data.


\subsection{Comparison with experiment: photoemission}
\label{subsec:photoemission}
It is instructive to compare the spectral densities that we have obtained for
our model to experimental measurements of photoemission in this system.  The
photoemission data can be thought of as essentially the local spectral density
below the Fermi-level, convolved with some response function that
characterizes the apparatus being used, say a Gaussian\cite{Georges:1996}.
The resulting spectra is simply a smoothed out version of the spectral density
below the Fermi-level and here we can simply compare the photoemission data
directly to the spectral densities in figure \ref{fig:LDAdos} and
\ref{fig:t2gDOS}.

\begin{figure}[htb]
\centering
\epsfig{file=Figures-8/Spectra.eps,width=0.95\linewidth}
\caption{(a) Valence band photoemission spectra of
La$_{1-x}$Sr$_{x}$TiO$_{3+y/2}$ taken from \cite{Yoshida:1999}.
Figure (b) shows the photoemission spectrum in the titanium 3d band
region.}
\label{fig:photoemission}
\end{figure} 

From photoemission measurements \cite{Yoshida:1999} conducted on the system
La$_{1-x}$Sr$_{x}$TiO$_{3+y/2}$ (see figure \ref{fig:photoemission}) we see
clearly the presence of an oxygen 2p-band deep beneath the Fermi-surface just
as in the LDA spectral density from figure \ref{fig:LDAdos}.  Both the
position of the band compared to the Fermi-level and its width are quite
consistent with the LDA data.  Closer to the Fermi-level we see a small hump,
about half an electron volt beneath the Fermi-surface with a width of about
2eV.  This hump clearly corresponds well to the lower Hubbard band in density
of states that we obtain with our model at, say 4.5eV (see figure
\ref{fig:t2gDOS}).  We note here that both the position of the peak as well as
the width is quite consistent with our data for that interaction value.  Other
experiments \cite{Fujimori:1992a,Fujimori:1992b,Hase:1993} on the
La$_{1-x}$Sr$_{x}$TiO$_{3}$ show the same photoemission features.

\section{Results: transport Functions}
We now turn to the computation of the transport functions.  In figure
\ref{fig:t2gPHI} we have displayed the transport function for the t$_{2g}$
model for various values of the interaction strength, $U$, using the same
doping and temperature values as in the previous section.  For clarity we have
also displayed the transport function for the largest values of the
interaction in a smaller window around the Fermi-level.

It should be noted here that the method that is used to compute the self
energy at low Matsubara energies assumes that the self energy is zero at the
Fermi-level.  Then in the subsequent computation of the self energy on the
real axis a small broadening is added by hand to the self energy in order to
stabilize the computation.  This is of course bad news when it comes to
computing the transport properties since it is clear that the transport
function will depend very strongly on the broadening.  We will nevertheless
push forward with this calculation and we can probably expect to obtain fairly
good results for the thermopower, but the absolute values of the
conductivities should be taken with a grain of salt.

In addition to presenting the values computed by the method developed in the
previous chapters (denoted BZI in the figures) we have also displayed the
exact value computed from the dispersion relation using Fourier transforms
(denoted FFT in the figures).  We see that there is excellent agreement
between the two methods although the direct integration method is quite noisy
which is either due to the smallness of the imaginary part of the self energy
or because of the degeneracy of the eigenvalues as discussed in
\article\ref{sec:afterthoughts}.
\begin{figure}[t]
\centering
\begin{tabular}{c@{\;}c}
\mbox{\epsfig{file=Figures-8/phi0.3.ps,angle=-90,width=\halfpage}} &
\mbox{\epsfig{file=Figures-8/phi0.5.ps,angle=-90,width=\halfpage}} \\
\mbox{\epsfig{file=Figures-8/phi0.7.ps,angle=-90,width=\halfpage}} &
\mbox{\epsfig{file=Figures-8/phi0.8.ps,angle=-90,width=\halfpage}} \\
\mbox{\epsfig{file=Figures-8/phi0.7s.ps,angle=-90,width=\halfpage}} &
\mbox{\epsfig{file=Figures-8/phi0.8s.ps,angle=-90,width=\halfpage}} \\
\end{tabular}
\caption{The transport function for the t$_{2g}$ model computed for various
  values of the interaction, $U$.  BZI denotes Brillouin zone integration, FFT
  denotes exact calculation.}
\label{fig:t2gPHI}
\end{figure} 

We see clearly how the increase in interaction strength decreases the value of
the transport function at the Fermi-surface; compared to the change in the
quasiparticle peak of the density of states the change in the transport
function is much more dramatic.  

\subsection{Transport parameters}
In figures \ref{fig:t2gTrans} we have displayed the transport
coefficients for the t$_{2g}$ model taking the lattice constant as 7.4
Bohr radii which corresponds to the LaTiO$_{3}$ system.

As we can see the conductance of the model decreases as we increase
the interaction; the electrical resistivity increases tenfold from the
smallest to the largest $U$ and a concomitant decrease is seen in the
thermal conductivity.  We note that the Lorentz ratio is very close to
the classical value of 24.4nW$\Omega$/K$^2$ for impurity scattering
and the values of the resistivity and thermal conductivity for the
lower values of $U$ are in the range of typical metals \cite[chapter
1]{Ashcroft}.  We note also that as the interaction increases the
Lorentz ratio decreases indicating a crossover to Fermi-liquid type
behavior in the self energy around the Fermi-level, in accordance with
our discussion in Appendix \ref{app:fermi-liq}.

\begin{figure}[htb]
\centering
\begin{tabular}{c@{\;}c}
\mbox{\epsfig{file=Figures-8/t2gElres.ps,angle=-90,width=\halfpage}} &
\mbox{\epsfig{file=Figures-8/t2gThcon.ps,angle=-90,width=\halfpage}} \\
\mbox{\epsfig{file=Figures-8/t2gLorRat.ps,angle=-90,width=\halfpage}} &
\mbox{\epsfig{file=Figures-8/t2gThpow.ps,angle=-90,width=\halfpage}}
\end{tabular}
\caption{The electrical resistivity, thermal conductivity, thermopower and 
Lorentz ratio for the t$_{2g}$ model as a function of interaction
strength, U.}
\label{fig:t2gTrans}
\end{figure}

The thermopower is negative, indicating that the carriers are electron-like
and both the magnitude and the sign of the thermopower are consistent with the
La$_{1-x}$Sr$_{x}$TiO$_{3}$ data from ref.~\cite{Hays:1999} shown in figure
\ref{fig:goodenough}.  The increase in the magnitude of the thermopower with
increasing value of $U$ agrees with the intuitive notion of localization of
carriers with increasing interaction leading to larger entropy per carrier and
therefore larger values of thermopower. 

\section{Results: specific heat and thermopower}
\label{subsec:specificHeat}
The linear coefficient of specific heat is another quantity that we can easily
compare to experiment and has a simple connection to the quasiparticle residue
and the transition from the metallic to the insulating state.  The linear
coefficient of specific heat, $\gamma$ is given by the following equation:
\begin{eqnarray}
        \gamma &=& \frac{\pi^2}{3Z}\kb^2 \rho_{tot}(E_f) 
        = 2.357\left[\frac{mJ}{mol K^2}\right]
        \frac{\rho_{tot}(E_f)[1/eV]}{Z}
        \nonumber \\
        &=& 5.631\left[\frac{10^{-4}cal}{mol K^2}\right]
        \frac{\rho_{tot}(E_f)[1/eV]}{Z}
\end{eqnarray}
where $Z$ is the quasiparticle residue and $\rho_{tot}$ is the total density
of states per unitcell at the Fermi-surface.  Table \ref{table:ZandGamma}
contains the computed values of $Z$ and $\gamma$ for the t$_{2g}$ model over
the whole interaction range.
\begin{table}[htb]
\centering
\begin{tabular}{|c||r|r|r||r|r|r||r|r|r|}\hline
Quantity & 
\multicolumn{3}{c||}{$Z$} & 
\multicolumn{3}{c||}{$\gamma$  $[mJ / mol K^2]$}  &
\multicolumn{3}{c|}{$S$  $[\mu V / K]$} 
\\\hline
$U$ $[eV]$ &  5\%  & 10\%  & 20\%  & 5\%  & 10\% & 20\% & 5\%  & 10\% & 20\% 
\\\hline\hline
1.93 & 0.821 & 0.824 & 0.834 &  5.8 &  5.7 &  5.5 &  -2.9 &  -3.0 & -3.3 \\ \hline
2.57 & 0.728 & 0.731 & 0.746 &  6.5 &  6.4 &  6.1 &  -3.2 &  -3.4 & -3.7 \\ \hline
3.21 & 0.627 & 0.627 & 0.642 &  7.6 &  7.5 &  7.1 &  -3.8 &  -4.0 & -4.2 \\ \hline
3.86 & 0.514 & 0.511 & 0.512 &  9.2 &  9.2 &  9.0 &  -4.7 &  -4.9 & -5.1 \\ \hline
4.50 & 0.390 & 0.376 & 0.365 & 12.1 & 12.5 & 12.7 &  -6.2 &  -6.5 & -7.0 \\ \hline
5.14 & 0.265 & 0.206 & 0.219 & 17.8 & 22.8 & 21.3 &  -9.2 & -12.1 & -11.7 \\ \hline
5.79 & 0.089 & 0.112 & 0.175 & 53.4 & 41.7 & 27.1 & -26.3 & -22.4 & -13.6 \\ \hline
6.43 & 0.072 & 0.099 & 0.157 & 63.0 & 47.3 & 29.8 & -39.0 & -24.5 & -16.0 \\ \hline
7.07 & 0.066 & 0.092 & 0.146 & 71.8 & 50.9 & 31.9 & -35.7 & -26.3 & -17.4 \\ \hline
7.71 & 0.063 & 0.090 & 0.140 & 74.9 & 52.8 & 33.6 & -37.0 & -26.5 & -17.5 \\ \hline
8.36 & 0.062 & 0.087 & 0.136 & 76.4 & 54.6 & 34.7 & -37.0 & -27.0 & -17.7 \\ \hline
9.00 & 0.061 & 0.085 & 0.132 & 77.9 & 56.1 & 35.9 & -37.0 & -27.2 & -18.1 \\ \hline
9.64 & 0.061 & 0.084 & 0.130 & 79.2 & 56.9 & 36.6 & -37.1 & -27.4 & -18.2 \\ \hline
10.29 & 0.060 & 0.082 & 0.126 & 79.6 & 58.3 & 37.7 & -38.1 & -27.7 & -18.6 \\ \hline
10.93 & 0.058 & 0.082 & 0.127 & 84.1 & 59.1 & 37.7 & -35.6 & -26.9 & -18.2 \\ \hline
11.57 & 0.059 & 0.081 & 0.127 & 81.9 & 59.3 & 37.7 & -36.8 & -27.2 & -18.3 \\ \hline
\end{tabular}
\caption{The quasiparticle residue and the linear coefficient of
specific heat as a function of the interaction strength.}
\label{table:ZandGamma}
\end{table}
\begin{figure}[htb]
\centering
\epsfig{file=Figures-8/t2gGamma.ps,angle=-90,width=\fullpage}
\caption{The linear coefficient of specific heat for the t$_{2g}$ 
  model as a function of the interaction strength, $U$. }
\label{fig:t2gGamma}
\end{figure}
In figure \ref{fig:t2gGamma} we have also displayed a graph of $\gamma$ as a
function of $U$ for the three values of doping.  Since the density of states
at the Fermi-surface is almost independent of interaction strength the $U$
dependence of $\gamma$ mirrors that of the quasiparticle residue and the
dramatic increase in $\gamma$ around $U = 5.4eV$ signifies the metal insulator
transition which is driven by the divergence of the effective mass
\cite{Tokura:1993a}.

\subsection{Evaluation of thermopower from Matsubara data}
Since the transport parameters depend essentially only on the value of
the transport function at and around the Fermi-surface (see
appendix \ref{app:fermi-liq}) it is tempting to try to evaluate these
parameters from the Matsubara data.  Here we will try to do this for
the thermopower since for small scattering we expect the thermopower to
be essentially independent of the scattering.  Thus we approximate the 
self energy around the Fermi-level in the following way:
\begin{equation}
\Sigma(\omega) = \Sigma'+\omega\left(1-\frac{1}{Z}\right)-i\Sigma''
\end{equation}
where $\Sigma'$ is the zero frequency limit of the real part of the
Matsubara self energy, and $\Sigma''$ is a similar limit for the
imaginary part.  The second parameter, $\Sigma''$, we actually put in
by hand and simply pick a small value (of the order of $T$) for it.

Once we have a functional form for the self energy we can compute the
transport function around the Fermi-surface and proceed to evaluate
the thermopower.  
\begin{figure}[htb]
\centering
\epsfig{file=Figures-8/t2gAnalThPow.ps,angle=-90,width=\fullpage}
\caption{The results of the analytic evaluation of the thermopower
from the Matsubara data.}
\label{fig:analyticS}
\end{figure} 
The results from this are shown in figure \ref{fig:analyticS} and the data
itself can be seen in table \ref{table:ZandGamma}.  In the small $U$ regime
this evaluation agrees well with our previous calculation but as we can see
there is a significant change in the thermopower around the metal-insulator
crossover.  This change reflects essentially the change in the quasiparticle
residue, in a similar fashion as the linear coefficient of specific heat.  The
qualitative behavior of the thermopower above the crossover does not however
agree with the experimental data from figure \ref{fig:goodenough}.

The reason for this discrepancy is quite simple and relates to the
distribution of spectral weight around the Fermi-surface.  To recall how the
doping affects the thermopower we need to look back at the Hubbard model
discussed earlier in this thesis. In the figure we obtained for the
thermopower in the Hubbard model using the IPT (see figure \ref{fig:latio3-1})
the behavior of the thermopower is qualitatively very similar to the
experimental data.  In this case the reason why there is a positive change in
the thermopower as we decrease the doping is due to the disappearance of the
quasiparticle peak.  The quasiparticle peak is an electron-like phenomenon and
contributes to negative thermopower but as the temperature is increased this
peak vanishes and the intrinsic contribution from the Hubbard bands takes over
-- which in the case of small doping leads to positive thermopowers.  The
temperature at which the quasiparticle peak melts away increases as we
increase the doping and thus the smaller doping values have larger, i.e. less
negative or more positive, thermopower.  In our case here - where we are
trying to estimate the thermopower from Matsubara data - we do not have any
information about the lower Hubbard band in our simple model for the self
energy.  The only contribution is therefore an electron-like contribution from
the quasiparticle peak which increases in magnitude as the quasiparticle
residue decreases.  Hence the doping dependence of our estimate of the
thermopower is exactly the opposite to what is seen in experiments.  

We should note that this is also the case in our previous calculation for the
thermopower seen in figure \ref{fig:t2gTrans}.  There we calculated the
transport parameters using the actual real axis self energy and nevertheless
we seem to obtain the wrong doping dependence for the thermopower.  The
conclusion that we can draw from this calculation is therefore that the
distribution of spectral weight is not properly captured in the IPT solution
to the Hubbard model.  Somehow the lower Hubbard band seems to be either too
far below the Fermi-surface or it has too little spectral weight. In this
sense this model, or at least the solution of it, is somewhat flawed.   

\section{Doping dependence}
In addition to the data discussed in the previous section we have also
computed the DMFT self energies for several other doping values.  The
computation is done using the same set of parameters as in
\article\ref{sec:basicResults} except that we take $T = 298K$.
\begin{figure}[htb]
\centering
\epsfig{file=Figures-8/gammaMulti.ps,angle=-90,width=\fullpage}
\caption{Comparison of the various methods for computing $\gamma$. DMFT
  denotes the results from the DMFT computation with $U = 5.4eV$, EXP denotes
  the experimental values obtained from \cite{Tokura:1993a}, TB is the
  free-electron value obtained from our tight binding model, TB+DMFT is the
  same free-electron value divided by the DMFT quasiparticle residue, LDA
  denotes the free-electron value obtained from the ASA LDA computation for
  the LaTiO$_{3}$ system.}
\label{fig:gammaMulti}
\end{figure} 
In figure \ref{fig:gammaMulti} we have displayed the linear coefficient of
specific heat as a function of doping.  In addition to our results, which
correspond $U = 5.4eV$, and the experimental data we have also displayed
the free-electron values obtained from the tight-binding model and also the
LDA results.  We chose $U = 5.4eV$ since that value corresponds to twice the
value of the bandwidth.  As we can see the DMFT result at this particular
value of $U$ overestimates $\gamma$ quite significantly.  It is therefore
instructive to extract from the data the value of $U$ at which the DMFT value
coincides with the experimental value.  This data is displayed in figure
\ref{fig:effectiveU} and we can see that the effective value of $U$ decreases
roughly linearly from $U = 5.4eV$ at the LaTiO$_{3}$ to $U = 0$ at the
SrTiO$_{3}$ end.  
\begin{figure}[htb]
\centering
\epsfig{file=Figures-8/effectiveU.ps,angle=-90,width=\fullpage}
\caption{The effective value of the interaction $U$ at which the DMFT value
  for $\gamma$ coincides with the experimental value from \cite{Tokura:1993a}.
  The solid straight line crosses the $y$ axis at $U = 5.4eV$.}
\label{fig:effectiveU}
\end{figure} 
A possible reason for this behavior is screening.  It is clear that in the
Mott insulating state the carriers are very localized and the effective
interaction between them should be close to the unrenormalized value.  However
as we increase the doping we expect some screening to take place which will
lead to a decrease in the effective interaction between the carriers and in
the limit of 100\% doping one would expect the effective interaction to scale
to zero.  Since this screening is not built into the DMFT formalism that we
use here it is reasonable to expect that we have to modify our $U$ as the
doping changes.

It is interesting to note that in spite of the lack of success that the LDA
has with predicting the linear coefficient of specific heat the same is not
true for the thermopower.
\begin{table}[htb]
\label{table:expVSlda_S}
\centering
\begin{tabular}{|c|c|c|c|}
\hline
 Doping   &  $S_{exp}$    &  $S_{LDA}$ \\\hline\hline
  5\%     &     -5.2      &    -5.6    \\\hline
 25\%     &     -9.3      &    -7.8    \\\hline
 50\%     &    -18.3      &    -9.3    \\\hline
 75\%     &    -29.4      &   -18.2    \\\hline
 80\%     &    -41.2      &   -22.8    \\\hline
\end{tabular}
\caption{Comparison of the experimental thermopower and the thermopower
 calculated using the LDA bandstructure data at T = 300K.  The units on 
 the thermopower are $[\mu V/K]$}
\end{table}
This we can see from table \ref{table:expVSlda_S} where we have tabulated the
experimental and the LDA data for the thermopower at 300K and as we see the
experiment and the LDA calculation seem to agree in the range of 5\% to 25\%
but at higher doping values the agreement is not as good.  Of course this
agreement with experiment should probably be regarded as accidental and it is
clear that the LDA will not be able to predict the transition from
electron-like to hole-like behavior that is seen at smaller dopings.


\section{Results from Quantum Monte Carlo}
In this section we will briefly look at results for the t$_{2g}$ model
obtained from Quantum Monte Carlo (QMC) calculations.  It has to be
admitted that we are cheating slightly here since the QMC calculation,
although being a real multiorbital calculation, uses the semicircle
bare density of states from the Bethe lattice.  We will nevertheless
use the self energy with our t$_{2g}$ Hamiltonian.  

The QMC results that we consider here are obtained from a semicircular
density of states with a bandwidth of 2.0, which we will scale to
2.7eV.  The value of the interaction is $U = 4.0 = 5.4eV$ and the
inverse temperature is $\beta = 16$ which corresponds to a rather
large temperature of $T = 980K$.

\begin{figure}[htb]
\centering
\begin{tabular}{c@{\;}c}
\mbox{\epsfig{file=Figures-8/qmcDOS1.ps,angle=-90,width=\halfpage}} &
\mbox{\epsfig{file=Figures-8/qmcDOS2.ps,angle=-90,width=\halfpage}} \\
\mbox{\epsfig{file=Figures-8/qmcDOS3.ps,angle=-90,width=\halfpage}} &
\mbox{\epsfig{file=Figures-8/qmcDOS4.ps,angle=-90,width=\halfpage}} \\
\end{tabular}
\caption{Spectral densities computed with QMC.  The temperature is $T
= 980K$ and $U = 5.4eV$.}
\label{fig:qmcDOS}
\end{figure}   

The QMC method is by its very nature a complex time method and thus
does not provide us with results on the real frequency axis.  Hence,
in order to obtain retarded self energies the results from the
Matsubara axis have to be continued to the real axis.  Here this is
done using the maximum entropy method (MAXENT) which here seems to be
much more reliable than doing analytic continuation using Pad\'e
approximants.

The dataset that we present here corresponds to four different values
of density and the resulting spectral densities are displayed in
figure \ref{fig:qmcDOS}.  The corresponding transport functions are
displayed in figure \ref{fig:qmcPHI}.
\begin{figure}[htb]
\centering
\begin{tabular}{c@{\;}c}
\mbox{\epsfig{file=Figures-8/qmcPHI1.ps,angle=-90,width=\halfpage}} &
\mbox{\epsfig{file=Figures-8/qmcPHI2.ps,angle=-90,width=\halfpage}} \\
\mbox{\epsfig{file=Figures-8/qmcPHI3.ps,angle=-90,width=\halfpage}} &
\mbox{\epsfig{file=Figures-8/qmcPHI4.ps,angle=-90,width=\halfpage}} \\
\end{tabular}
\caption{Transport functions computed with QMC.  Parameters are the
same as in figure \ref{fig:qmcDOS}.}
\label{fig:qmcPHI}
\end{figure}  

The transport parameters computed from the QMC data are listed in table
\ref{table:qmcTrans}.  We note that the values for the resistivity are in
reasonable agreement with figure \ref{fig:TokuraData} although the value of
the temperature here is quite a bit larger than the values shown in that
figure.  It is also encouraging that we obtain the correct qualitative doping
dependence.

\begin{table}[htb]
\label{table:qmcTrans}
\centering
\begin{tabular}{|c|c|c|c|}
\hline
Density  & $\rho$ [$\mu \Omega m$] & $\kappa$ [$W/mK$] & $S$ [$\mu V/K$] \\ 
\hline\hline
0.99280  & 10.739086  &  0.564135  &  -137.984832 \\ \hline
0.92820  & 4.917915   &  2.048371  &  -107.592107 \\ \hline
0.88440  & 6.450519   &  4.124970  &  -112.103186 \\ \hline
0.73904  & 2.450386   &  2.991632  &  -35.632503 \\  \hline
\end{tabular}
\caption{Transport parameters computed from the QMC data.}
\end{table}

For the thermal conductivity we do not have any data for this system
but the values that we obtain here is close to experimental values
obtained for manganite oxides \cite{Visser:1997}.

In contrast with the reasonable experimental agreement for the two other
transport parameters the thermopower is in striking disagreement with
experimental data where there is positive change in the thermopower as the
doping is decreased.  One could argue that this trend is encouraging since it
agrees with the IPT data that we obtained earlier in the chapter.  We are
certainly more inclined to believe the QMC results than the IPT and if further
investigations support these results we will likely have to look for a more
realistic Hamiltonian in order to model this material.

\section{Conclusion}
In this chapter we have looked at simple model for doped LaTiO$_{3}$ using the
dynamical mean field theory and a newly developed method for solving the
associated DMFT impurity model.  This calculation is one of the first steps in
the extension of the DMFT to realistic models.  We investigated the solution
of the model as a function of interaction strength and found strong
indications of a metal insulator crossover at interaction strength
corresponding to about twice the bandwidth, which agrees with previous
estimates in the literature.  Breakdown of the solution on the real axis
prevented us from computing the density of states and the transport function
in the insulating regime but in the metallic regime we obtained parameters in
the range of typical metals with electron majority carriers.  However, since
undoped LaTiO$_{3}$ is an insulator it is of prime importance to be able to
compute the parameters of the model in the insulating regime and further
investigations are needed.


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