Matrix product variational formulation for lattice gauge theory
Takanori Sugihara
For hamiltonian lattice gauge theory, we introduce the matrix product anzats inspired from density matrix renormalization group. In this method, wavefunction of the target state is assumed to be a product of finite matrices. As a result, the energy becomes a simple function of the matrices, which can be evaluated using a computer. The minimum of the energy function corresponds to the vacuum state. We show that the $S=1/2$ Heisenberg chain model are well described with the ansatz. The method is also applied to the two-dimensional $S=1/2$ Heisenberg and U(1) plaquette chain models.
aipcheck.tex
%
% $Id: aipcheck.tex,v 1.7 2002/01/11 19:50:46 frank Exp $
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Testing for potential problems with this class
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newif\ifproblem
\newif\iftimesok

\makeatletter
\def\IfStandaloneCheck{\def\next{aipcheck}
  \edef\currjob{\jobname}
  \edef\next{\meaning\next}
  \edef\currjob{\meaning\currjob}
  \ifx\currjob\next
    \expandafter\@firstoftwo
  \else
    \expandafter\@secondoftwo
  \fi
}
\makeatother

\typeout{***********************************************}
\typeout{*}
\typeout{* Testing if all files required for the aipproc}
\typeout{* class are available ...}
\typeout{*}
\typeout{***********************************************}

\typeout{*}
\typeout{* Looking for LaTeX2e ... }
\ifx\documentclass\undefined
 \typeout{*}
 \typeout{* Sorry this is a fatal error:}
 \typeout{*}
 \typeout{* The aipproc class can only be used with LaTeX2e which is}
 \typeout{* the standard LaTeX since 1994!}
 \typeout{*}
 \typeout{* Please make sure that your version of LaTeX is up-to-date}
 \typeout{* before attempting to use this class.}
 \typeout{*}
 \expandafter\stop
\else
 \typeout{* ... ok }
\fi


\def\next#1/#2/#3\next{#1#2}
\typeout{*}
\typeout{* Testing that LaTeX2e is not too old ... }
\ifnum\expandafter\next\fmtversion\next<199612 \relax
 \typeout{* ... what a vintage! }
 \typeout{*}
 \typeout{* Sorry this is a fatal error:}
 \typeout{*}
 \typeout{* The aipproc class can only be used with a recent version}
 \typeout{* of LaTeX2e. Your version is dated \fmtversion\space --- but}
 \typeout{* at least the 1996/12/01 version is required!}
 \typeout{*}
 \typeout{* Please make sure that your version of LaTeX is up-to-date}
 \typeout{* before attempting to use this class.}
 \typeout{*}
 \expandafter\stop
\else
 \ifnum\expandafter\next\fmtversion\next<199806 \relax
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   \typeout{*}
   \typeout{* Your version of LaTeX2e is quite old --- the aipproc class}
   \typeout{* hasn't been tested with your release.}
   \typeout{*}
   \typeout{* We believe that it will probably work, but if you encounter}
   \typeout{* problems you will need upgrade your installation.}
   \typeout{*}
   \typein{* Type <return> to continue ...}
   \problemtrue
 \else
   \typeout{* ... ok }
 \fi
\fi


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\IfFileExists{aipproc.cls}
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     \typeout{* ... ok }
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     \typeout{*}
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     \typeout{* Before you can use the aipproc class you have to unpack}
     \typeout{* it from the documented source.}
     \typeout{*}
     \typeout{* Run LaTeX on the file 'aipproc.ins', e.g.,}
     \typeout{*}
     \typeout{* \space\space latex aipproc.ins}
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     \typeout{* or whatever is necessary on your installation to process}
     \typeout{* a file with LaTeX. This should unpack a number of files for you:}
     \typeout{*}
     \typeout{* aipproc.cls \space and \space aip-*.clo}
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     \typeout{* After that retry processing this guide.}
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     \typeout{*}
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     \typeout{* The aipxfm.sty file which is part of the aipproc distribution}
     \typeout{* must be installed in a directory which is searched by LaTeX.}
     \typeout{*}
     \typeout{* Please install this file and retry.}
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     \stop
}

\typeout{*}
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\IfFileExists{aip-8s.clo}
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     \typeout{* ... ok }
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     \typeout{*}
     \typeout{* Sorry this is a fatal error:}
     \typeout{*}
     \typeout{* The aip-8s.clo file which is part of the aipproc distribution}
     \typeout{* must be installed in a directory which is searched by LaTeX.}
     \typeout{*}
     \typeout{* Please install this file and retry.}
     \typeout{*}
     \stop
}

\typeout{*}
\typeout{* Looking for aip-8d.clo ... }
\IfFileExists{aip-8d.clo}
    {
     \typeout{* ... ok }
    }
    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* Sorry this is a fatal error:}
     \typeout{*}
     \typeout{* The aip-8d.clo file which is part of the aipproc distribution}
     \typeout{* must be installed in a directory which is searched by LaTeX.}
     \typeout{*}
     \typeout{* Please install this file and retry.}
     \typeout{*}
     \stop
}


\typeout{*}
\typeout{* Looking for aip-6s.clo ... }
\IfFileExists{aip-6s.clo}
    {
     \typeout{* ... ok }
    }
    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* Sorry this is a fatal error:}
     \typeout{*}
     \typeout{* The aip-6s.clo file which is part of the aipproc distribution}
     \typeout{* must be installed in a directory which is searched by LaTeX.}
     \typeout{*}
     \typeout{* Please install this file and retry.}
     \typeout{*}
     \stop
}


\typeout{*}
\typeout{* Looking for aip-arlo.clo ... }
\IfFileExists{aip-arlo.clo}
    {
     \typeout{* ... ok }
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    {
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     \typeout{*}
     \typeout{* Sorry this is a fatal error:}
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     \typeout{* The aip-arlo.clo file which is part of the aipproc distribution}
     \typeout{* must be installed in a directory which is searched by LaTeX.}
     \typeout{*}
     \typeout{* Please install this file and retry.}
     \typeout{*}
     \stop
}

\typeout{*}
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	 {
	  \typeout{* ... ok }
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	  \typeout{*}
	  \typeout{* Sorry this is a fatal error:}
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	  \typeout{* Your LaTeX distribution contains neither fixltx2e.sty}
	  \typeout{* nor fix2col.sty.}
	  \typeout{*}
	  \typeout{* This means that it is either too old or incompletely}
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	  \typeout{*}
	  \typeout{* fixltx2e.sty is part of the standard LaTeX distribution}
	  \typeout{* since 1999; fix2col.sty is an earlier version of this}
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	  \typeout{*}
	  \typeout{* Best solution is to get the latest LaTeX distribution.}
	  \typeout{* If this is impossible for you, download fix2col.sty.}
	  \typeout{* You can get this software from a CTAN host.}
          \typeout{* Refer to http://www.ctan.org and search for "fix2col".}
	  \typeout{*}
	  \typeout{* After you have updated your LaTeX distribution}
	  \typeout{* retry processing this guide.}
	  \stop
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}

\typeout{*}
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\IfFileExists{fontenc.sty}
    {
     \typeout{* ... ok }
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    {
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     \typeout{*}
     \typeout{* Sorry this is a fatal error:}
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     \typeout{* The fontenc package, which is part of standard LaTeX}
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     \typeout{* run the aipproc class.}
     \typeout{*}
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     \typeout{* this LaTeX release is too old or that it was installed}
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     \typeout{*}
     \typeout{* Please make sure that your version of LaTeX is okay}
     \typeout{* before attempting to use this class. The LaTeX distribution}
     \typeout{* contains the file "ltxcheck.tex" which can be used to}
     \typeout{* test the basic functionality and integrity of your installation.}
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    {
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     \typeout{*}
     \typeout{* Sorry this is a fatal error:}
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     \typeout{* The calc package, which is part of standard LaTeX}
     \typeout{* (tool distribution) has to be installed at the site}
     \typeout{* to run the aipproc class.}
     \typeout{*}
     \typeout{* The fact that it cannot be found either means that}
     \typeout{* this LaTeX release is too old or that it was installed}
     \typeout{* only in parts.}
     \typeout{*}
     \typeout{* Please make sure that the tools distribution of LaTeX}
     \typeout{* is installed before attempting to use this class.}
     \typeout{*}
     \typeout{* (You might be able to get calc.sty separately for your}
     \typeout{* installation if you are unable to upgrade to a recent}
     \typeout{* distribution for some reason.)}
     \typeout{*}
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    {
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     \typeout{*}
     \typeout{* Problem detected:}
     \typeout{*}
     \typeout{* The varioref package, which is part of standard LaTeX}
     \typeout{* (tool distribution) is not installed at this site.}
     \typeout{*}
     \typeout{* The fact that it cannot be found either means that}
     \typeout{* this LaTeX release is too old or that it was installed}
     \typeout{* only in parts.}
     \typeout{*}
     \typeout{* You can use the aipproc class without this package but }
     \typeout{* you cannot make use of the options "varioref" or "nonvarioref".}
     \typeout{*}
     \typeout{* Please also note that the aipguide.tex documentation}
     \typeout{* normally uses the "varioref" option to show its}
     \typeout{* effects (which  will now fail).}
     \typeout{*}
     \typein{* Type <return> to continue ...}
     \problemtrue
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\typeout{*}
\typeout{* Looking for times.sty ... }
\IfFileExists{times.sty}
    {
     \begingroup
% load times and forget it immediately again
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     \typeout{* this LaTeX release is too old or that it was installed}
     \typeout{* only in parts with old files remaining!}
     \typeout{*}
     \typeout{* You can use the aipproc class without this package but}
     \typeout{* you have to specify the option "cmfonts" which result in}
     \typeout{* documents which are not conforming to the AIP layout specification!}
     \typeout{*}
     \typeout{* You can also try using the class in the following way:}
     \typeout{*}
     \typeout{* \space\space \string\documentclass[cmfonts]{aipproc}}
     \typeout{* \space\space \string\usepackage{times}}
     \typeout{* \space\space ...}
     \typeout{*}
     \typeout{* With luck this will result in Times Roman output but chances}
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     \typeout{*}
     \typein{* Type <return> to continue ...}
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    {
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     \typeout{*}
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     \typeout{*}
     \typeout{* The fact that this package cannot be found either means that}
     \typeout{* this LaTeX release is too old or that it was installed}
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     \typeout{*}
     \typeout{* You can use the aipproc class without this package but }
     \typeout{* you have to specify the option "cmfonts" which result in}
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     \typeout{*}
     \typein{* Type <return> to continue ...}
     \problemtrue
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\iftimesok % don't bother testing other font options if times already
           % bad

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           \typeout{* .fd file (defining how to load Times Roman) is missing.}
           \typeout{*}
           \typeout{* The fact that this package is only partially installed}
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           \typeout{*}
           \typeout{* You can use the aipproc class without this package but }
           \typeout{* you have to specify the option "cmfonts" which result in}
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           \typeout{*}
           \typein{* Type <return> to continue ...}
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\newcommand\CheckFDFile[3]{%
  \typeout{*}
  \typeout{* Looking for #1#3.fd or #2#3.fd ... }
  \IfFileExists{#1#3.fd}
    {
     \typeout{* ... ok }
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    {
     \IfFileExists{#2#3.fd}
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%\CheckFDFile{ot1}{OT1}{ot1ztmcm}
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%\CheckFDFile{oms}{OMS}{omsztmcm}
%\CheckFDFile{omx}{OMX}{omxztmcm}

\typeout{*}
\typeout{* Looking for mathptm.sty ... }
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     \CheckFDFile{ot1}{OT1}{ptmcm}
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      \typeout{*}
      \typeout{* Problem detected:}
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      \typeout{* The mathptm package, which is part of standard LaTeX}
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      \typeout{* support files describing which fonts to load are missing!}
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      \typeout{*}
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	  \typeout{* The mathptmx package, which is part of standard LaTeX}
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	  \typeout{* support files describing which fonts to load are missing!}
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	\typeout{* This is unfortunate but not a disaster as the older}
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    {
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     \typeout{*}
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     \typeout{*}
     \typeout{* The fact that this package cannot be found either means that}
     \typeout{* this LaTeX release is too old or that it was installed}
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     \typeout{* You can use the aipproc class without this package but }
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     \typein{* Type <return> to continue ...}
     \problemtrue
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\typeout{* Looking for mathtime.sty ... }
\IfFileExists{mathtime.sty}
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     \typeout{* ... ok }
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    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* The mathime package can not be found.}
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\typeout{*}
\typeout{* Looking for graphicx.sty ... }
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     \typeout{* ... ok }
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    {
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     \typeout{*}
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     \typeout{* The graphics package, which is part of standard LaTeX}
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     \typeout{* The fact that this package cannot be found either means that}
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     \typeout{* You can use the aipproc class without this package but }
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     \typeout{* Please note that you will get a further error message below}
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     \typeout{* As a result the illustrations in aipguide will look strange.}
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\typeout{*}
\typeout{* Looking for textcomp.sty ... }
\IfFileExists{textcomp.sty}
    {
     \typeout{* ... ok }
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    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* Problem detected:}
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     \typeout{* The textcomp package, which is part of standard LaTeX}
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     \typeout{*}
     \typeout{* The fact that this package cannot be found either means that}
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     \typeout{* Type return in response to that error.}
     \typeout{*}
     \typein{* Type <return> to continue ...}

     \def\textdegree{$^\circ$}         % used below but now
                                       % not set up
     \problemtrue
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\typeout{*}
\typeout{* Looking for url.sty ... }
\IfFileExists{url.sty}
    {
     \typeout{* ... ok }
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    {
     \typeout{* ... not found! }
     \typeout{*}
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     \typeout{*}
     \typeout{* Without this package you will not be able to use the \string\url}
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     \typeout{* Refer to http://www.ctan.org and search for "url".}
     \typeout{*}
     \typein{* Type <return> to continue ...}

     \problemtrue
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\makeatletter

\typeout{*}
\typeout{* Looking for natbib.sty ... }
\IfFileExists{natbib.sty}
    {
     \IfStandaloneCheck
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sugihara.tex
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%  This manuscript has been written by T. Sugihara
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\begin{document}

\title{Matrix product variational formulation for lattice gauge theory}


\author{Takanori Sugihara}{
  address={RIKEN BNL Research Center, 
Brookhaven National Laboratory, Upton, New York 11973, USA 
}
}


\begin{abstract}
For hamiltonian lattice gauge theory, 
we introduce the matrix product anzats 
inspired from density matrix renormalization group. 
In this method, wavefunction of the target state is assumed to be 
a product of finite matrices. As a result, the energy becomes a simple 
function of the matrices, which can be evaluated using a computer. 
The minimum of the energy function corresponds to the vacuum state. 
We show that the $S=1/2$ Heisenberg chain model are well described 
with the ansatz. The method is also applied to the 
two-dimensional $S=1/2$ Heisenberg and U(1) plaquette chain models. 
\end{abstract}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% MAINMATTER
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

RHIC experiments have started to test the fundamental properties 
of quantum chromodynamics (QCD). The importance of first-principle 
analysis of QCD has increased largely in the context of 
color confinement. There is the expectation that quarks and gluons 
deconfine in extreme conditions such as heavy ion collision. 
However, lattice gauge theory at finite density has been stuck 
for a long time. 
The most ideal treatment of non-equilibrium quantum physics 
is to trace time-evolution of quantum states 
based on the Schr\"odinger equation. 

Density Matrix Renormalization Group (DMRG) is the variational 
method that gives the most accurate results in one-dimensional 
quantum systems \cite{white}. In these days, DMRG has been used 
as a standard method to complement quantum Monte Carlo and 
successful in solving one-dimensional (zero and finite temperature) 
and two-dimensional (zero temperature) quantum systems. 
Application of DMRG to elemanrary particle and molecular physics has 
started some years ago. Recent interesting progress of DMRG is its 
application to non-equilibrium quantum physics and quantum 
information theory \cite{dmrg}. 

The first application of DMRG to particle physics was the massive 
Schwinger model with the $\theta$ term \cite{Byrnes:2002nv}. 
There is an old prediction by S. Coleman that quarks 
deconfine at $\theta=\pi$ \cite{Coleman:1976uz}. 
The model has not been analyzed accurately with Monte Carlo 
because of the sign problem. On the other hand, 
DMRG has been successful in describing the details of the phase 
transition with large lattices because DMRG is free from 
the sign problem. 

The second application has been 
given by the author \cite{Sugihara:2004qr}. It is a preliminary 
work for study of gauge theory. In bosonic lattice systems, 
each site has infinite degrees of freedom and therefore hamiltonian 
is infinite dimensional differently from spin and fermion systems. 
It is not evident whether DMRG truncation 
works for bosonic degrees of freedom. DMRG needs to be tested 
in a simpler bosonic model before going to gauge theory. 
In Ref. \cite{Sugihara:2004qr}, DMRG has been applied to 
a (1+1)-dimensional $\lambda\phi^4$ model. The DMRG result for 
the critical exponent $\beta= 0.1264\pm 0.0073$ 
is consistent with the exact one $\beta=1/8=0.125$. 

Matrix product variational method is a result of large 
simplification of DMRG \cite{or}. In this method, 
the energy function has a simple form and easy to evaluate. 
However, the advantage is lost if a constraint is imposed 
to variational space directly. The Gauss law in gauge theory is 
one of the most important examples of constraints. 
To avoid this difficulty, the author has developed a method to introduce 
constraints with undetermined multipliers in Hamiltonian \cite{sugi1}. 
Also, the author has generalized the matrix product variational method 
so that it can be applied to higher-dimensional general systems \cite{sugi2}. 

We introduce matrix product states according to Ref. \cite{or}. 
With the knowledge from DMRG, 
wavefunction is represented as a product of matrices. 
\begin{equation}
  |\Psi\rangle = \sum_{s_1,\dots,s_L=1}^K
   {\rm tr} [A[s_1]\dots A[s_L]] |s_1\rangle\dots |s_L\rangle, 
\end{equation}
where periodicity is assumed. 
$A[s]$ has the following normalization condition. 
\begin{equation}
  \sum_{\beta=1}^M \sum_{s=1}^K
    A^*[s]_{\alpha\beta} A[s]_{\alpha'\beta}=\delta_{\alpha\alpha'}. 
\end{equation}
Each of the matrices $A[s]$ can be parameterized with an appropriate 
number of independent variables \cite{sugi1}. It is expected that 
better results is obtained for larger $M$. 
Let us apply the ansatz to 
the $S=1/2$ Heisenberg chain model with periodic boundary conditions
\begin{equation}
  H = \sum_{i=1}^L
  (S^x_i S^x_{i+1} + S^y_i S^y_{i+1} + S^z_i S^z_{i+1}). 
\end{equation}
Periodicity simplifies calculation of the energy function 
\begin{equation}
  E[A]=\frac{\langle\Psi|H|\Psi\rangle}{\langle\Psi|\Psi\rangle}
  = \sum_a \frac{{\rm tr}(\hat{S}^a \hat{S}^a \hat{1}^{L-2})}
    {{\rm tr} (\hat{1}^L )},
\end{equation}
where
$\hat{S}^a = \sum_{s,s'}\langle s|S_1^a|s'\rangle A^*[s] \otimes A[s']$
and $\hat{1} = \sum_s A^*[s] \otimes A[s]$. 
The minimum of the energy function corresponds to the ground state. 
In actual numerical calculation, the matrix $\hat{1}$ is 
diagonalized to simplify calculation of the powers of $\hat{1}$. 

\begin{table}
\begin{tabular}{rllll}
\hline
     $M\backslash L$ 
      & $10$          & $100$         & $1000$        & $10000$ \\
\hline
$6$   & $-0.4092$ & $-0.4372$ & $-0.4371$ & $-0.4368$ \\
$12$  & $-0.4092$ & $-0.4427$ & $-0.4425$ & $-0.4425$ \\
Exact & $-0.4515$ & $-0.4438$ & $-0.4431$ & $-0.4431$ \\
\hline
\end{tabular}
\caption{Numerical results for ground-state energy per site 
in the $S=1/2$ Heisenberg chain model. 
The exact values have been obtained with the Bethe ansatz. 
}
\label{table1}
\end{table}

Table \ref{table1} shows numerical results for ground-state energy 
per site, which are compared with the exact ones. 
$M$ is the size of the matrices $A[s]$. 
When the lattice size $L$ is small, convergence is poor even for 
large $M$. On the other hand, when $L$ is large, 
the numerical results approaches the exact one 
as $M$ becomes larger. In the best case, error is less than $1\%$. 
The lattice size dependence of energy is small 
when $L\ge 100$. 

\begin{figure}
  \includegraphics[height=.13\textheight]{tube}
  \put(-248,37){\small$L_1$}
  \put(-192,-15){\small$L_2$}
  \put(-47,17){\small$L_2$}
  \caption{Reduction of two dimensional lattice to one dimensional}
  \label{fig1}
\end{figure}

\begin{figure}
  \includegraphics[height=.27\textheight]{heisenberg}
  \caption{Energy per bond in the two-dimensional $S=1/2$ Heisenberg model.}
  \label{fig2}
\end{figure}
In the same way, the two-dimensional $S=1/2$ Heisenberg model is analyzed. 
To use the matrix product ansatz, one-dimensional structure needs to be 
found on the two-dimensional lattice (see Fig. \ref{fig1}). 
The tube and ring sizes are denoted as $L_1$ and $L_2$, respectively. 
Figure \ref{fig2} plots energy per bond as a function of $1/L_1$ 
for a very large ring with $L_2=10000$. 
The circle for $L_1=1$ is the exact result from the Bethe ansatz. 
The other three circles have been calculated using the matrix product ansatz. 
We are going to make a comparison in the thermodynamic limit $L_1\to\infty$ 
by extrapolating the obtained points to the limit. 
In Table \ref{table2}, energy per bond is compared among 
various results in the thermodynamic result. 
``Lattice size'' means the largest lattice size used for calculation. 
\begin{table}
\begin{tabular}{lllll}
\hline
Method & $E$ & Lattice size & Year & Reference \\
\hline
Monte Carlo    & $-0.3347$    & $16^2$   & 1999 & \cite{sandvik} \\
DMRG           & $-0.3347$    & $12^2$   & 2001 & \cite{xiang} \\
DMRG           & $-0.3321$    & $20^2$   & 2003 & \cite{farnell} \\
TPVA           & $-0.3272$    & Very large & 2004 & \cite{tpva} \\
Matrix product & $-0.3292\pm 0.0005$     & $10000\times 4$ & 2004 & This work\\
\hline
\end{tabular}
\caption{
Energy per bond of the two-dimensional $S=1/2$ Heisenberg model 
in the thermodynamic limit. 
}
\label{table2}
\end{table}

The method is applied to a U(1) plaquette chain model. 
The hamiltonian is 
\[
 H = \sum_l E^2_l
 - x\sum_{\rm p} (U_{\rm p}+U_{\rm p}^\dagger), 
\]
where $x\equiv 1/g^4$ \cite{ks}. 
Figure \ref{fig3} plots energy as a function of the parameter $x$ 
and compares it with the results of Ref. \cite{Hamer:2000iz}. 
In this calculation, the Gauss law $\nabla \cdot {\mathbf E} = 0$ 
has not been imposed on the variational space. 
Since vacuum wavefunction is available, we can check gauge invariance 
of the obtained vacuum state 
by calculating the vacuum expectation value of the electric field. 
The obtained result $\sim 10^{-3}$ is small compared to energy, 
which shows approximate gauge invariance of the vacuum state. 
Currently, calculation with the Gauss law constraint is being conducted. 
Further precise analysis with larger lattices will be given elsewhere 
to refine the results shown in this presentation. 
\begin{figure}
  \includegraphics[height=.27\textheight]{gauge_energy}
  \caption{Energy per site vs $x$ 
in the U(1) hamiltonian lattice gauge model.}
  \label{fig3}
\end{figure}


%\begin{theacknowledgments}
The numerical calculations were carried on the RIKEN RSCC system. 
This work has been partially supported by RIKEN BNL. 
%\end{theacknowledgments}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% You may have to change the BibTeX style below, depending on your
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%% following lines and see the aipguide.pdf for further information.
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\bibliographystyle{aipproc}   % if natbib is available
%\bibliographystyle{aipprocl} % if natbib is missing

\begin{thebibliography}{99}

\bibitem{white} S.~R.~White, 
Phys.\ Rev.\ Lett. {\bf 69}, 2863 (1992);
Phys.\ Rev.\ B {\bf 48}, 10345 (1993). 

\bibitem{dmrg}
U.~Sch\"ollwoeck, 
cond-mat/0409292. 

\bibitem{Byrnes:2002nv}
T.~Byrnes, P.~Sriganesh, R.~J.~Bursill, and C.~J.~Hamer,
Phys.\ Rev.\ D {\bf 66}, 013002 (2002). 

\bibitem{Coleman:1976uz}
S.~R.~Coleman,
%``More About The Massive Schwinger Model,''
Annals Phys.\  {\bf 101}, 239 (1976).
%%CITATION = APNYA,101,239;%%

\bibitem{Sugihara:2004qr}
T.~Sugihara,
%``Density matrix renormalization group in a two-dimensional lambda phi**4
%Hamiltonian lattice model,''
JHEP {\bf 0405}, 007 (2004). 

\bibitem{or}
S.~\"Ostlund and S.~Rommer, 
Phys.\ Rev.\ Lett. {\bf 75}, 3537 (1995). 

\bibitem{sugi1}
T.~Sugihara, in preparation. 

\bibitem{sugi2}
T.~Sugihara, in preparation. 

\bibitem{sandvik}
A.~W.~Sandvik, 
Phys.\ Rev.\ B {\bf 56}, 678 (1997). 

\bibitem{xiang}
T.~Xiang, J.~Lou, and Z.~Su, 
Phys.\ Rev.\ B {\bf 64}, 104414 (2001). 

\bibitem{farnell}
D.~J.~J.~Farnell, 
Phys.\ Rev.\ B {\bf 68}, 134419 (2003). 

\bibitem{tpva}
Y.~Nishio, N.~Maeshima, A.~Gendiar, and T.~Nishino, 
cond-mat/0401115. 

\bibitem{ks}
J.~B.~Kogut and L.~Susskind,
Phys.\ Rev.\ D {\bf 11}, 395 (1975). 

\bibitem{Hamer:2000iz}
C.~J.~Hamer, R.~J.~Bursill and M.~Samaras,
Phys.\ Rev.\ D {\bf 62}, 054511 (2000). 

\end{thebibliography}


\end{document}

\endinput


hep-lat0411017.extracted_bib
\begin{thebibliography}{99}

\bibitem{white} S.~R.~White, 
Phys.\ Rev.\ Lett. {\bf 69}, 2863 (1992);
Phys.\ Rev.\ B {\bf 48}, 10345 (1993). 

\bibitem{dmrg}
U.~Sch\"ollwoeck, 
cond-mat/0409292. 

\bibitem{Byrnes:2002nv}
T.~Byrnes, P.~Sriganesh, R.~J.~Bursill, and C.~J.~Hamer,
Phys.\ Rev.\ D {\bf 66}, 013002 (2002). 

\bibitem{Coleman:1976uz}
S.~R.~Coleman,
%``More About The Massive Schwinger Model,''
Annals Phys.\  {\bf 101}, 239 (1976).
%%CITATION = APNYA,101,239;%%

\bibitem{Sugihara:2004qr}
T.~Sugihara,
%``Density matrix renormalization group in a two-dimensional lambda phi**4
%Hamiltonian lattice model,''
JHEP {\bf 0405}, 007 (2004). 

\bibitem{or}
S.~\"Ostlund and S.~Rommer, 
Phys.\ Rev.\ Lett. {\bf 75}, 3537 (1995). 

\bibitem{sugi1}
T.~Sugihara, in preparation. 

\bibitem{sugi2}
T.~Sugihara, in preparation. 

\bibitem{sandvik}
A.~W.~Sandvik, 
Phys.\ Rev.\ B {\bf 56}, 678 (1997). 

\bibitem{xiang}
T.~Xiang, J.~Lou, and Z.~Su, 
Phys.\ Rev.\ B {\bf 64}, 104414 (2001). 

\bibitem{farnell}
D.~J.~J.~Farnell, 
Phys.\ Rev.\ B {\bf 68}, 134419 (2003). 

\bibitem{tpva}
Y.~Nishio, N.~Maeshima, A.~Gendiar, and T.~Nishino, 
cond-mat/0401115. 

\bibitem{ks}
J.~B.~Kogut and L.~Susskind,
Phys.\ Rev.\ D {\bf 11}, 395 (1975). 

\bibitem{Hamer:2000iz}
C.~J.~Hamer, R.~J.~Bursill and M.~Samaras,
Phys.\ Rev.\ D {\bf 62}, 054511 (2000). 

\end{thebibliography}