and
Mott_Anderson_SCES2004_100205.tex
%----------------------------------------------------------------------
% Template article for SCES '04 conference:
% document class 'phb-proc4-auth'
%
% This template should be used for all manuscripts submitted to the
% SCES '04 - The International Conference on Strongly Correlated Electron Systems
% Universitat Karlsruhe (TH) July 26-30, 2004
%
% Send any comments to the SCES '04 Administration <sces04@physik.uni-karlsruhe.de>
%----------------------------------------------------------------------
% The template contains a number of LaTeX commands of the form :
%
% \command{}
%
% In order to complete this abstract, fill in the blank fields between
% the curly braces with the requested information.
%
% e.g. in order to add an abstract title, replace
% \title{}    with   \title{Evidence of Solitons in Tedium Diboride}
%
% The \author and \address commands can take an optional label
% in square brackets of the form :
%
% \command[label]{}
%
% The text of the abstract should be inserted between the two commands
% \begin{abstract} and \end{abstract}.
%
% Please leave all commands in place even if you don't fill them in.
% Do not alter commands which have non-blank arguments unless you are
% familiar with them.
%
% N.B. In LaTeX, all text following the % sign is treated as a comment.
%      Some commands, e.g. \thanks{}, have been "commented
%      out".  Remove the % sign if you wish to use it.
%      Blank lines are ignored (except in the manuscript text where
%      they will be treated as a paragraph break).
%
%----------------------------------------------------------------------
% Invoke the LaTeX document class for these conference proceedings

\documentclass{phb-proc4-auth}

%----------------------------------------------------------------------
% Install LaTeX packages to allow enhanced math formatting and
% graphics installation.  These are the recommended packages.  If you
% are experienced in using LaTeX, you may choose a different list.

\usepackage{graphicx}
\usepackage{amssymb}

%----------------------------------------------------------------------
% Start processing the document and front page

\begin{document}
\begin{frontmatter}

%----------------------------------------------------------------------
% Specify destination and version number of the manuscript

\journal{SCES '04}

%----------------------------------------------------------------------
% Title of manuscript

\title{Mott--Hubbard and Anderson Transitions in Dynamical Mean--Field Theory}

%----------------------------------------------------------------------
% List of authors
%
% List each author using a separate \author{} command
%
% If there is more than one author address, add a label to each author
% of the form \author[label]{name}.  This label should be identical to
% the corresponding label provided with the \address command.
%
% e.g. if there are three authors from two institutions in USA and
% France, you can link them to their respective addresses, using
%
% \author[US]{John Doe}
% \author[US,FR]{Jane Doe}
% \author[FR]{Jean Dupont}
% \address[US]{University of Life, Somewhere, USA}
% \address[FR]{Universite de la Vie, Quelque Part, France}
%
% N.B. Unlike the document class used for abstract submissions, it is
% possible to have the author associated with more than one address,
% as shown in the example above.
%

\author[kb]{Krzysztof Byczuk\corauthref{11}}
\author[wh]{Walter Hofstetter}
\author[dv]{Dieter Vollhardt}

%----------------------------------------------------------------------
% List of addresses
%
% If there is more than one address, list each using a separate
% \address command using a label to link it to the respective author
% as described above

\address[kb]{Institute of Theoretical Physics,
Warsaw University, ul. Ho\.za 69, PL-00-681 Warszawa, Poland}
\address[wh]{Condensed Matter Theory Group, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA}
\address[dv]{Theoretical Physics III, Center for Electronic Correlations and Magnetism,
Institute for Physics,University of Augsburg, D-86135 Augsburg, Germany}

%----------------------------------------------------------------------
% Title page footnotes
%
% If you need to add qualifying information to any of the authors,
% use the \thanksref{} command within the \author command.  The
% argument is the label of a corresponding \thanks[label]{text}
% command which contains the footnote text
%
% e.g. you can acknowledge a funding authority for John Doe, using
%
% \author{John Doe\thanksref{ABC}}
% \thanks[ABC]{This work was supported by Institute of Unphysical
%    Phenomena under contract no. ABC-123}
%

%\thanks[]{}

%----------------------------------------------------------------------
% Contact Information
%
% Add the complete postal address, telephone number, fax number, and
% email address of the corresponding author as a special footnote using
% the \corauth[]{} command.  This works in a similar way to the \thanks
% command.  Add the \corauthref{} command within the \author command.
% The argument is the label of a corresponding \corauth[label]{text}
% command which contains the contact information.  Prefix the text with
% Corresponding Author:
%
% e.g. if the contact author is John Doe,
%
% \author{John Doe\corauthref{1}}
% \corauth[1]{Corresponding Author: University of Life, 123 Some St.,
%    Somewhere, MI 12345, USA.  Phone: (555) 555-5555
%    Fax: (555) 555-7777, Email: JDoe@uol.edu}
%

\corauth[11]{Corresponding Author: Institute of Theoretical Physics,
Warsaw University, ul. Ho\.za 69, PL-00-681 Warszawa, Poland. Phone: (48) 22
5532256, Fax: (48) 22  6219475, Email: byczuk@fuw.edu.pl}

%----------------------------------------------------------------------
% Text of abstract

\begin{abstract}
The Anderson--Hubbard Hamiltonian at half--filling is
investigated within dynamical mean--field theory at zero
temperature. The local density of states is calculated by taking
the geometric and arithmetic mean, respectively.
The non--magnetic ground state phase diagrams obtained within the different
averaging schemes are compared.

\end{abstract}

%----------------------------------------------------------------------
% Manuscript keywords
%
% Please give two or three keywords in the form: keyword \sep keyword
% e.g. NMR \sep superconductivity
%
% NB The syntax is different from the abstract document class

\begin{keyword}
Mott--Hubbard and Anderson Metal--Insulator Transitions,
  Dynamical Mean--Field Theory
\end{keyword}

%----------------------------------------------------------------------
% End of front page

\end{frontmatter}

%----------------------------------------------------------------------
% Manuscript text
%
% Fill in the following space with the manuscript text.
%
% A number of LaTeX commands may be invoked in this space, e.g.
%
% \section{} : to insert a new section title
% \label{}   : to label the numbered section for use in \ref{}
% \cite{}    : to add a reference using the label in \bibitem{}
%
% A number of LaTeX environments may be used, e.g.
% \begin{equation}
%     An equation inserted here will be automatically numbered
% \end{equation}
%
% Please refer to other LaTeX documentation for help on using these
% environments.

%Insert the main text of the manuscript here.

The metal insulator transitions (MIT) due to electronic interactions
(Mott-Hubbard MIT) \cite{Mott90} and due to impurity scattering (Anderson
localization) \cite{Anderson58} are
subtle quantum mechanical phenomena which require non-perturbative
investigation schemes.
In this respect the dynamical mean-field theory (DMFT) \cite{georges96} is
very useful.
However, it cannot describe the physics of
Anderson localization if the local disorder is included by taking the
arithmetic average over the disorder \cite{ulmke95}.
Recently it was proposed to use the geometric average over the disorder
to include also the Anderson MIT \cite{Dobrosavljevic97,Dobrosavljevic03,Schubert03}.
In this contribution we compare
 the entire non-magnetic ground state phase diagrams
of correlated, disordered electrons at half filled lattice as obtained by
taking
the geometric  and the arithmetic averages over the disorder
within the DMFT \cite{byczuk04}.



We study the system described by a single--orbital Anderson--Hubbard model
\begin{equation}
H_{AH}=-t\sum_{\langle ij\rangle \sigma }a_{i\sigma }^{\dagger }a_{j\sigma
}+\sum_{i\sigma }\epsilon _{i}n_{i\sigma }+U\sum_{i}n_{i\uparrow
}n_{i\downarrow },  \label{1}
\end{equation}
where $t>0$ is the amplitude for hopping between nearest neighbors, $U$ is
the on--site repulsion, $n_{i\sigma }=a_{i\sigma }^{\dagger }a_{i\sigma }^{{%
\phantom{\dagger}}}$ is the local electron number operator,
$a_{i\sigma }$ ($a_{i\sigma }^{\dagger}$) is the annihilation
(creation) operator of an electron with spin $\sigma$, and the
local ionic energies $\epsilon _{i}$ are independent random
variables. We assume a continuous probability distribution for
$\epsilon _{i}$, i.e., $\mathcal{P}(\epsilon _{i})=\Theta (\Delta
/2-|\epsilon _{i}|)/\Delta ,$ with $\Theta $ as the step function.
Here,
 $\Delta $ is a measure of the disorder strength.

This model is solved within DMFT by mapping it \cite{georges96} onto an ensemble of
effective single--impurity Anderson Hamiltonians with different
$\epsilon _{i}$:
\begin{eqnarray}
H_{\mathrm{SIAM}} &=&\sum_{\sigma }(\epsilon _{i}-\mu )a_{i\sigma }^{\dagger
}a_{i\sigma }+Un_{i\uparrow }n_{i\downarrow }  \label{2} \\
&&+\sum_{\mathbf{k}\sigma }V_{\mathbf{k}}a_{i\sigma }^{\dagger }c_{\mathbf{k}%
\sigma }+V_{\mathbf{k}}^{\ast }c_{\mathbf{k}\sigma }^{\dagger }a_{i\sigma
}+\sum_{\mathbf{k}\sigma }\epsilon _{\mathbf{k}}c_{\mathbf{k}\sigma
}^{\dagger }c_{\mathbf{k}\sigma }.  \nonumber
\end{eqnarray}
Here $\mu = U/2$ is the chemical potential corresponding to a half-filled
band, and $V_{\mathbf{k}}$ and $\epsilon _{\mathbf{k}}$ are the
hybridization matrix element and the dispersion relation of the auxiliary
bath fermions $c_{\mathbf{k}\sigma }$, respectively.

For each ionic energy $\epsilon _{i}$ we calculate the local Green
function $G(\omega ,\epsilon _{i})$, from which we can obtain
either the geometrically averaged local density of states (LDOS)
$\rho _{\mathrm{geom}}(\omega )=\exp \left[ \langle \ln \rho
_{i}(\omega )\rangle \right]$ or the arithmetically averaged LDOS
$\rho _{\mathrm{arith}}(\omega )= \langle \rho _{i}(\omega
)\rangle  $, where $\rho _{i}(\omega )=-\mathrm{{Im}}G(\omega
,\epsilon _{i})/\pi $, and $\langle O_{i}\rangle =\int d\epsilon
_{i}\mathcal{P}(\epsilon _{i})O(\epsilon _{i})$ is the arithmetic
mean of $O_{i}$. The lattice Green function is given by the
corresponding Hilbert transform as $G(\omega )=\int d\omega
^{\prime }\rho _{\mathrm{\alpha}}(\omega ' )/(\omega -\omega
^{\prime })$, where the subscript $\mathrm{\alpha}$ stands for
either "$\mathrm{geom}$" or "$\mathrm{arith}$". The local
self--energy $\Sigma (\omega )$ is determined from the
$\mathbf{k}$-integrated Dyson equation $\Sigma(\omega )=\omega
-\eta (\omega )-1/G(\omega )$ where the hybridization function
$\eta (\omega )$ is defined as $\eta (\omega
)=\sum_{\mathbf{k}}|V_{\mathbf{k}}|^{2}/\left( \omega -\epsilon
_{\mathbf{k}}\right) $. The self--consistent DMFT equations are
closed through the Hilbert transform $ G(\omega )=\int d\epsilon
N_{0}(\epsilon )/\left[\omega -\epsilon -\Sigma (\omega )\right]
$, which relates the local Green function for a given lattice to
the self--energy; here $N_{0}(\epsilon )$ is the non--interacting
DOS.

The Anderson--Hubbard model (\ref{1}) is solved for
a semi-elliptic DOS, $N_{0}(\epsilon )=4\sqrt{1-4\epsilon ^{2}}/\pi$.
Then $\eta (\omega )=G(\omega )/16$.
 The DMFT equations are solved
at zero temperature by the numerical renormalization group technique
\cite{NRG}.
For numerical integrations  we use discrete values of $\epsilon_i$
selected according to the Gauss-Legendre algorithm.
The number of $\epsilon_i$ levels depends on $\Delta$ 
and is adjusted to obtain  smooth spectral functions \cite{ulmke95}.


 \begin{figure}
     \centering
     \includegraphics[width=0.4\textwidth]{phase_diagram.eps}
     \caption{Non--magnetic ground state phase diagram of the Anderson--Hubbard model at
half-filling as calculated by DMFT with the geometrically averaged  local density of states.
I -- correlated, disordered metal, II -- coexistence regime, III -- crossover
regime, IV -- Mott insulator, V -- Anderson insulator. At the dashed line all
states within the Hubbard subbands become localized. $W$ is a bare energy
band-width.}
 \end{figure}


 \begin{figure}
     \centering
     \includegraphics[width=0.4\textwidth]{phase_diagram_cpa.eps}
     \caption{Non--magnetic ground state phase diagram of the Anderson--Hubbard model at
half-filling as calculated by DMFT with the arithmetically averaged local
density of states.
Labels I -- IV are the same as in Fig. 1.}
 \end{figure}


The main results are presented in Fig.~1 and 2, where the
non--magnetic ground state phase diagrams of the Anderson--Hubbard
model (1) are plotted. At weak disorder both averaging schemes are
seen to lead to very similar results for the Mott-Hubbard
transition from the paramagnetic correlated, disordered  metal (I)
to the paramagnetic, disordered Mott insulator (IV). For
$\Delta\lesssim 2$ we find hysteresis and a coexistence regime
(II). Above $\Delta\approx 2$ there is a crossover regime (III).
At strong disorder the DMFT with geometrically averaged LDOS
describes an Anderson transition between a correlated, disordered
metallic and insulating phase, respectively (V) \cite{comment}. 
Since neither the
paramagnetic Mott nor the Anderson insulator is characterized by a
broken symmetry they are continously connected, i.e., the phases
IV and V in Fig.~1 are connected by a continuous line which does
not cross a metallic phase. Obviously, the Anderson transition is
missed within the DMFT supplied by the arithmetic averaging.

In summary, although the geometric averaging procedure does not
capture all aspects of Anderson localization
\cite{Dobrosavljevic03,alvermann04} it provides valuable new
insights into MITs in correlated and disordered electron systems,
which are not obtained by taking the arithmetic disorder average
\cite{ulmke95}.

This work was supported in part by the
Sonderforschungsbereich 484 of the Deutsche Forschungsgemeinschaft (DFG).
Financial support of KB through KBN-2 P03B 08 224, and of WH through the DFG
and a Pappalardo Fellowship is gratefully acknowledged.



%----------------------------------------------------------------------
% Reference section
%
% List each reference with a separate \bibitem{} command.  The
% argument contains the label that is used in the \cite{} command
% in the main text
%
% e.g.
%
%    This follows our pioneering work on TdB2\cite{TdB2}.
%
% \bibitem{TdB2}
% J. Doe, J. Doe, and J. Dupont, J. Irrep. Res. 10 (2000) 1000.

\begin{thebibliography}{00}

%\bibitem{}

\bibitem{Mott90} N.~F.~Mott, Proc. Phys. Soc. A {\bf 62}, 416 (1949);
{\em Metal--Insulator Transitions}, 2nd edn. (Taylor and Francis,
London 1990).

\bibitem{Anderson58} P.~W.~Anderson, Phys. Rev. {\bf 109}, 1492 (1958).

\bibitem{georges96} A.~Georges {\it et al.},
Rev.\ Mod.\ Phys.\ {\bf 68}, 13 (1996).


\bibitem{ulmke95} M.~Ulmke, V.~Jani\v{s}, and D.~Vollhardt, Phys. Rev. B {\bf 51}, 10411 (1995).



\bibitem{Dobrosavljevic97} V.~Dobrosavljevi\'c and G.~Kotliar, Phys. Rev. Lett. {\bf 78}, 3943 (1997).

\bibitem{Dobrosavljevic03} V.~Dobrosavljevi\'c, A.~A.~Pastor, and B.~K.~Nikoli\'c,
Europhys. Lett. {\bf 62}, 76 (2003).

\bibitem{Schubert03} G.~Schubert, A.~Wei{\ss}e, and H.~Fehske, cond-mat/0309015.

\bibitem{byczuk04} K.~Byczuk, W.~Hofstetter, and D.~Vollhardt,
  cond-mat/0403765, Phys. Rev. Lett. {\bf 94}, (2005) - in press.

\bibitem{NRG}
 R.~Bulla, Phys. Rev. Lett. {\bf 83}, 136 (1999).

\bibitem{comment} Currently we try to improve the results when $U\to 0$ using
  different route to obtain the self-energy. This should reduce
  effects due to a logarithmic broadening within NRG scheme \cite{NRG}. 

\bibitem{alvermann04} A.~Alvermann, {\it at al.}, cond-mat/0406051.



\end{thebibliography}

%----------------------------------------------------------------------
% Figures and Tables
%
% Insert figures and tables at the end of the document unless you
% are familiar with the LaTeX positional options.
%
% \begin{figure}
%     \centering
%     \includegraphics{filename.eps}
%     \caption{Insert figure caption here}
% \end{figure}
%
% \begin{table}
%     \centering
%     \begin{tabular}
%     Insert table here
%     \end{tabular}
%     \caption{Insert table caption here}
% \end{table}
%
% Please refer to other LaTeX documentation for help on using these
% environments.

%----------------------------------------------------------------------
% Terminate document

\end{document}

%----------------------------------------------------------------------

cond-mat0502257.extracted_bib
\begin{thebibliography}{00}

%\bibitem{}

\bibitem{Mott90} N.~F.~Mott, Proc. Phys. Soc. A {\bf 62}, 416 (1949);
{\em Metal--Insulator Transitions}, 2nd edn. (Taylor and Francis,
London 1990).

\bibitem{Anderson58} P.~W.~Anderson, Phys. Rev. {\bf 109}, 1492 (1958).

\bibitem{georges96} A.~Georges {\it et al.},
Rev.\ Mod.\ Phys.\ {\bf 68}, 13 (1996).


\bibitem{ulmke95} M.~Ulmke, V.~Jani\v{s}, and D.~Vollhardt, Phys. Rev. B {\bf 51}, 10411 (1995).



\bibitem{Dobrosavljevic97} V.~Dobrosavljevi\'c and G.~Kotliar, Phys. Rev. Lett. {\bf 78}, 3943 (1997).

\bibitem{Dobrosavljevic03} V.~Dobrosavljevi\'c, A.~A.~Pastor, and B.~K.~Nikoli\'c,
Europhys. Lett. {\bf 62}, 76 (2003).

\bibitem{Schubert03} G.~Schubert, A.~Wei{\ss}e, and H.~Fehske, cond-mat/0309015.

\bibitem{byczuk04} K.~Byczuk, W.~Hofstetter, and D.~Vollhardt,
  cond-mat/0403765, Phys. Rev. Lett. {\bf 94}, (2005) - in press.

\bibitem{NRG}
 R.~Bulla, Phys. Rev. Lett. {\bf 83}, 136 (1999).

\bibitem{comment} Currently we try to improve the results when $U\to 0$ using
  different route to obtain the self-energy. This should reduce
  effects due to a logarithmic broadening within NRG scheme \cite{NRG}. 

\bibitem{alvermann04} A.~Alvermann, {\it at al.}, cond-mat/0406051.



\end{thebibliography}