Formation of clusters in the $2D$ $t-J$ model: The mechanism for phase separation.
A. Fledderjohann\dag\
\leftskip 2cm \rightskip 2cm %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% The emergence of phase separation is investigated in the framework of a $2D$ $t-J$ model by means of a variational product ansatz, which covers the infinite lattice by two types of $L\times L$ clusters. Clusters of the first type are completely occupied with electrons, i.e. they carry maximal charge $Q_e=L^2$ and total spin 0, and thereby form the antiferromagnetic background. Holes occur in the second type of clusters -- called ``hole clusters''. They carry a charge $Q_h<L^2$. The charge $Q_h$ and the number $N(Q_h)$ of hole clusters is fixed by minimizing the total energy at given hole density and spin exchange coupling $\alpha=J/t$. For $\alpha$ not too small ($\alpha>0.5$) it turns out that hole clusters are occupied with an even number $Q_h<L^2$ of electrons and carry a total spin 0. For increasing $\alpha$ the charge $Q_h(\alpha)$ of the hole clusters decreases. %For $\alpha>3.5$ it turns out to be zero. %Moreover, the interaction energy between neighbouring clusters %plaquettes ($L=2$) favours %favours the condensation of clusters with equal charge, %which implies phase separation. %abdollah-a1 %It's Some points on the boundary curve can be extracted from $Q_h(\alpha)$. %product ansatz for the ground state with $L\times L$ %clusters ($L=2$, $L=4$). The minimum of the ground state energy is %characterized by two cluster charges $Q_1,Q_2$, which depend on the charge %density $\rho=Q/N$ and the spin exchange coupling $\alpha=J/t$. Signatures %for phase separation are found in those domains of the $\rho-\alpha$ plane, %where the two cluster charges $Q_1,Q_2$ are well ``separated'' [($Q_1,Q_2$)= %(0,4), (2,4) for $L=2$; ($Q_1,Q_2$)= (0,16), (2,16), (6,16), (10,16), (14,16) %for $L=4$]. Moreover, the interaction energy between neighbouring plaquettes %clusters ($L=2$) favours the condensation of clusters with equal charge. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
draft.tex
%  Ref: CM/218152/PAP/43701
%
%  Resubmission to Journal of Physics: Condens. Matter
%
%
%  TITLE: Formation of clusters in the $2D$ $t-J$ model:
%         The mechanism for phase separation.
%                              
%
%
%  AUTHORS: A. Fledderjohann, A. Langari, K.H. M"utter
%
%
%  ADDRESS: Dr. Abdollah Langari
%           Physics Department
%           Sharif University of Technology
%           P.O.Box: 11365-9161
%           Tehran-14588-89694,
%           IRAN
%
%  FAX:     +98-21- 6602 2711
%
% E-mail: langari@mpipks-dresden.mpg.de
%
%
%
%  MANUSCRIPT:
%
%
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%\draft
%\preprint{}
\title{Formation of clusters in the $2D$ $t-J$ model:
The mechanism for phase separation.}

\author{A. Fledderjohann\dag\, A. Langari\ddag
\footnote[3]{To whom correspondence should be addressed
(langari@mpipks-dresden.mpg.de)}
and K.-H. M\"utter\dag\
}

\address{\dag\ Physics Department, University of Wuppertal, 42097
Wuppertal, Germany}

\address{\ddag\ Department of Physics, Sharif University of Technology, 
Tehran 11365-9161, Iran
%\ddag\ Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran 19395-5531, Iran
}


\date{\today}

%\maketitle

\begin{abstract}
\leftskip 2cm
\rightskip 2cm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The emergence of phase separation is investigated in the framework of a $2D$
$t-J$ model by means of a variational product ansatz, which covers the infinite
lattice by two types of $L\times L$ clusters. Clusters of the first type are
completely occupied with electrons, i.e. they carry maximal charge $Q_e=L^2$ and
total spin 0, and thereby form the antiferromagnetic background. Holes occur
in the second type of clusters -- called ``hole clusters''. They carry a 
charge $Q_h<L^2$. The charge $Q_h$ and the number $N(Q_h)$ of hole clusters is fixed
by minimizing the total energy at given hole density and spin exchange
coupling $\alpha=J/t$. For $\alpha$ not too small ($\alpha>0.5$) it turns out
that hole clusters are occupied with an even number $Q_h<L^2$ of electrons and
carry a total spin 0. For increasing $\alpha$ the charge $Q_h(\alpha)$ of the hole
clusters decreases. %For $\alpha>3.5$ it turns out to be zero.
%Moreover, the interaction energy between neighbouring clusters %plaquettes ($L=2$) favours 
%favours the condensation of clusters with equal charge,
%which implies phase separation. 
%abdollah-a1
%It's
Some points on the
 boundary curve can be extracted from $Q_h(\alpha)$.
%product ansatz for the ground state with $L\times L$
%clusters ($L=2$, $L=4$). The minimum of the ground state energy is 
%characterized by two cluster charges $Q_1,Q_2$, which depend on the charge
%density $\rho=Q/N$ and the spin exchange coupling $\alpha=J/t$. Signatures
%for phase separation are found in those domains of the $\rho-\alpha$ plane,
%where the two cluster charges $Q_1,Q_2$ are well ``separated'' [($Q_1,Q_2$)=
%(0,4), (2,4) for $L=2$; ($Q_1,Q_2$)= (0,16), (2,16), (6,16), (10,16), (14,16)
%for $L=4$]. Moreover, the interaction energy between neighbouring plaquettes
%clusters ($L=2$) favours the condensation of clusters with equal charge.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{abstract}

\pacs{71.10.Fd,71.27.+a,75.10.-b, 75.10.Jm}
% Uncomment for Submitted to journal title message
\submitto{Journal of Physics: Condensed Matter}

\maketitle


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction\label{sec1}}
%\section{Introduction: Motivation and questions of interest}
%\setcounter{equation}{0}

Holes play a fundamental role in our understanding of high $T_c$
superconductivity.\cite{bednorz86} The parent materials like
$La_2CuO_4$ are insulators with antiferromagnetic order and
doping with holes (missing electrons) opens the superconducting
phase. \cite{anderson87}

Experimental evidence has been found in $La_2CuO_{4+\delta}$
for phase separation.\cite{jorgensen88,hammel90,hammel91,chou96}
This means that the holes in the $CuO_2$ planes are not
distributed uniformly but concentrate in ``hole rich''
domaines. 
%abdollah-new
The compound phase separates for $0.01\leq \delta \leq 0.06$ below $T_{ps} \sim 300 K$
into the nearly  stoichiometric antiferromagnetic  $La_2CuO_{4+\delta_1}$ with
$\delta_1 < 0.02$ and N\'eel temperature ($T_N \simeq 250 K$), and a metallic superconducting
oxygen-rich phase $La_2CuO_{4+\delta_2}$ with $\delta_2 \approx 0.02$ and $T_c \simeq 34 K$.
The other evidence is related to the $Sr$ doped compound $La_{2-x}Sr_xCuO_{4+\gamma}$.
This compound phase separates for $x \leq 0.03$ into the superconducting $La_{2-x}Sr_xCuO_{4+\gamma_1}$
($\gamma_1\approx 0.08$) and the nonsuperconducting $La_{2-x}Sr_xCuO_{4+\gamma_2}$
($\gamma_2\approx 0.00$) phases. \cite{cho93}
%end-abdollah-new


Intensive studies have been performed to understand this phenomenon
in models for strongly correlated electrons like the Hubbard --
and $t-J$ model.
In the $2D$ $t-J$ model various attempts have been made
to exploit the phase diagram in the plane spanned by the
charge density $\rho=Q_{\mbox{tot}}/N$ and the spin coupling
$\alpha=J/t$ ($N$, $Q_{\mbox{tot}}$ denote the total numbers
of sites and electrons, respectively).

Phase separation occurs, if both the charge density $\rho$ and
the spin coupling $\alpha$, are large enough
\begin{eqnarray}
1\geq\rho\geq\rho_1 & , & \alpha>\alpha_p(\rho_1)\,.\label{large_rho}
\end{eqnarray}
In this regime, the ground state can be represented by
a product ansatz
\begin{equation}
\psi(\rho,N) =\psi_e\Big(N_e(\rho)\Big)\psi_h\Big(
N_{h}(\rho,\rho_1)\Big)
\label{psiprod}
\end{equation}
of two clusters with site numbers $N_e(\rho)$ and $N_{h}(\rho,\rho_1)$,
which cover the whole lattice:
\begin{eqnarray}
N_e(\rho)+N_{h}(\rho,\rho_1) & = & N\,.\label{clusters}
\end{eqnarray}
The $N_e(\rho)$ sites in the ``electron'' cluster are all occupied with
electrons. The corresponding ground state $\psi_e(N_e(\rho))$
is just the ground state of the $2D$ Heisenberg model with $N_e(\rho)$ sites.
Holes occur in the ``hole'' cluster with $N_h(\rho,\rho_1)$ sites.
The corresponding ground state $\psi_h(N_h(\rho,\rho_1))$ is given by the
$t-J$ model with $N_h(\rho,\rho_1)$ sites and $Q_h=N_h(\rho,\rho_1)\rho_1$
electrons. The numbers $N_e(\rho)$ and $N_h(\rho,\rho_1)$ for the cluster
sites are fixed by the total number of sites (\ref{clusters}) and the
total number of electrons
%
%
%In the second cluster, the charge density is fixed to be $\rho_1$.
%The corresponding wave function $\psi_{\rho_1}(
%N_{\rho_1}(\rho))$ is given by the $t-J$ model on a cluster with
%$N_{\rho_1}(\rho)$ sites and $Q_1=N_{\rho_1}(\rho)\rho_1$ electrons.
%The numbers $N_1(\rho)$ and $N_{\rho_1}(\rho)$ for the cluster sites
%are fixed by the total number of sites (\ref{clusters}) and the total
%number of electrons: % (\ref{clusters})
%
\begin{eqnarray}
N_e(\rho)+\rho_1N_{h}(\rho,\rho_1) & = & Q_{\mbox{tot}}\,.\label{clusters2} 
\end{eqnarray} 
From (\ref{psiprod})-(\ref{clusters2}) one derives that the
ground state energy per site $\varepsilon(\rho,\alpha)$:
\begin{eqnarray}
\varepsilon(\rho,\alpha) & = & \frac{1}{1-\rho_1}\left[
\varepsilon(\rho_1,\alpha)(1-\rho)+\varepsilon(1,\alpha)(\rho-\rho_1)\right]
\label{e_lin_rho}
\end{eqnarray}
is linear in $\rho$. This holds for (\ref{large_rho}), i.e.
in the region with phase separation.

The focus %of 
%abdollah-a1
%interest
%some studies
 is the boundary curve $\alpha_p(\rho_1)$
for phase separation.
Two controversary points of view can be found in the literature:
Emery et al.\cite{emery90} suggested that in the $2D$ $t-J$
model the phase separation curve $\alpha_p(\rho_1)$ starts at 
$\alpha_p(\rho_1=1)=0$, which means that phase separation exists
already for low doping $(1-\rho_1)\simeq 0$ and small values of $\alpha$,
as observed experimentally.\cite{jorgensen88}
Their point of view is supported by Hellberg and Manousakis
\cite{hellberg95,hellberg97,hellberg99,hellberg00}.

On the other hand Putikka et al.\cite{putikka92,putikka94} concluded
from a high temperature expansion, that phase separation
at low doping only emerges for larger $\alpha$ values
$\alpha>\alpha_p(\rho_1=1)\simeq 1.2$, which would exclude
the $\alpha$ regime realized in the experiments. 
This point of view is supported by DMRG calculationss on
ladders (Rommer, White, Scalapino\cite{rommer00}), 
variational wave functions of
the Luttinger-Jastrow-Gutzwiller type (Valenti, Gros\cite{valenti92}),
Lanczos calculations (Dagotto\cite{dagotto92,dagotto98}). 

%abdollah-a1
Finally, Green's function Monte-Carlo simulations performed by Calandra et al. (CBS)\cite{calandra98}
led to a value $\alpha_p(\rho_1=1)\simeq 0.5$.
%and their modifications (Fehske\cite{fehske91}).

%In this paper we would like to propose a new method
%which extracts $\alpha_p(\rho_1)$ from the ground state energies
%$E^{(L)}(Q,\alpha=\alpha(\rho_1=Q_1/N))$
%on $L\times L=N$ clusters with charges $Q=L^2, Q=Q_1, Q=Q_1-L$.
%These energies can be computed analytically for plaquette
%clusters ($L=2$) as will be discussed in Section \ref{sec2}.
%Phase separation can be observed in these small clusters on a
%``microscopic scale''.
%The interaction energies $W(Q,Q')$ between neighbouring
%plaquette clusters are discussed in Section \ref{sec3}.
%They favour clustering of plaquettes with equal charge,
%which can be interpreted as phase separation on a
%``macroscopic scale''.

%abdollah-a1
%However
In this paper, we would like to study the mechanism of phase separation by
means of a product ansatz with $L\times L$ clusters of charge $Q$
which cover the infinite lattice.
Our method is in the spirit of the coupled cluster method (CCM)
designed almost 50 years ago \cite{coester58} as an 
approximation scheme for quantum many body problems. In order to
handle the interaction between clusters, perturbative \cite{gros93}
and variational \cite{potthoff03} methods have been
implemented. 
In our study of the phase diagram in the ($\rho=Q/N,\alpha=J/t$)
plane, we proceed as follows: We start from the ground states
$\psi^{(L)}(\rho_1,\alpha)$ with energies $E^{(L)}(Q_1,\alpha)$
on $L\times L$ clusters with charge $Q_1=\rho_1L^2$. These ground states
can be computed analytically for plaquette clusters ($L=2$).
In Section \ref{sec2} we will discuss the product ansatz with
plaquette clusters, which minimizes the total energy at fixed
charge density $\rho=Q/N$. Phase separation can be observed
on the small plaquette cluster on a ``microscopic scale''.
%The interaction energies $W(Q,Q')$ between neighbouring plaquette
%clusters with total charges $Q,Q'$ are discussed in Section \ref{sec3}. 
%For $\alpha$ values not too small, they favour clustering
%of plaquettes with equal charge $Q=Q'$, which can be interpreted as phase
%separation on a ``macroscopic scale''.

In Section \ref{sec4} we extend our considerations to 
clusters of size $L\times L$. Results for the phase diagram
obtained from a numerical calculation
of ground state energies on a $4\times 4$ cluster are shown. 
%abdollah-a1
We finally summerize and present the discussion in Section \ref{sec6}.

%presented.
%Section \ref{sec5} presents results on hole--hole correlators
%obtained from our cluster product ansatz.





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Plaquette cluster in the $t-J$ model: Phase
separation ``in statu nascendi''\label{sec2}}

We start from the $t-J$ model in two dimensions which is built up
as a lattice of $L\times L$ clusters (cf. Fig. \ref{fig1} for the
example of $L=2$):
\begin{eqnarray}
H & = & t\sum_i h_{p_i}(\alpha=J/t)+t''\sum_{\langle i,j\rangle}
h_{p_i,p_j}(\alpha)\label{H_t_tpp}
\end{eqnarray}
with $h_{p_i}$ and $h_{p_i,p_j}$ containing all nearest neighbour
bonds $\langle k,l\rangle$ in a cluster $p_i$ or between neighbouring
clusters ($p_i$, $p_j$), respectively.
In general, each nearest neighbour bond $\langle k,l\rangle$ of the lattice either belongs
to a cluster (parameters $t,J=\alpha t$) or connects neighbouring
clusters ($t'',J''=\alpha t''$) and contributes with $th^{(k,l)}(\alpha)$
or $t''h^{(k,l)}(\alpha)$
to the respective intra- or inter-cluster part of the Hamiltonian (\ref{H_t_tpp}):
%(i.e. $th^{(k,l)}(\alpha)$ or $t''h^{(k,l)}(\alpha)$):
%-- e.g. for $\langle k,l\rangle$ as 
%nearest neighbour bond
%in a cluster $p_i$ the corresponding contribution $h_{p_i}^{(j,k)}(\alpha)$ 
%to $h_{p_i}(\alpha)$ reads
\begin{equation}
h^{(k,l)}(\alpha)= 
{\cal P}\Big[-\sum_{\sigma}
\left(
c_{k,\sigma}^+
c_{l,\sigma} + \mbox{h.c.} \right)
+\alpha \Big({\bf S}_{k}\cdot {\bf S}_{l}
-\frac{1}{4}n_{k}n_{l}\Big)\Big]{\cal P} 
\label{element_h_plaq}
\end{equation}
%\begin{eqnarray}
%H & = & t\sum_{i, j\in p_i} h_{i, j}(\alpha=J/t)+t''\sum_{i\in p_i ,j\in p_j}
%h_{i, j}(\alpha).%\nonumber\\
%\end{eqnarray}
%Each nearest neighbour bond in Fig. \ref{fig1}
%either belongs to a cluster ($p_i$; parameters $t,J=\alpha t$) or connects neighbouring
%clusters ($p_i, p_j$; $t'',J''=\alpha t''$).
%The Hamiltonian of each bond ($h_{j, k}(\alpha)$) reads
%\begin{eqnarray}
%h_{j, k}(\alpha) & = &
%{\cal P}\Big[-\sum_{\sigma}
%\left(
%c_{j,\sigma}^+
%c_{k,\sigma} + \mbox{h.c.} \right)\nonumber\\
% &  &
%\hspace{0.5cm}+\alpha \Big({\bf S}_{j}{\bf S}_{k}
%-\frac{1}{4}n_{j}n_{k}\Big)\Big]{\cal P},\nonumber\\ \label{element_h_plaq}
%\end{eqnarray}
where
\begin{eqnarray}
n_{l} & = & \sum_{\sigma}n_{l,\sigma}=\sum_{\sigma}c_{l,\sigma}^+
c_{l,\sigma}
\end{eqnarray}
and ${\cal P}$ projects onto the subspace where 
the occupation numbers $n_{l}$ are restricted to zero or one.
%abdollah-2 The following line has been added:
The electron spin operator is represented by ${\bf S}_{k}$ at the k-th site.


%The intercluster terms are built correspondingly 
%with hopping parameter $t''$. 
%except the double-primed
%The latter are meant to emphasize the difference
%to the zeroth order situation
%of uncoupled clustersr. 
%Finally, we are aiming to obtain lowest order
%perturbation results for the %homogeneous 
%isotropic $2D$ $t-J$ lattice (i.e.
%$t''=t\equiv 1$). Before, we will discuss the
%physical content obtainable from single clusters ($t''=0$).
%The question of choosing appropriate boundary conditions for the
%$L\times L$ clusters is of particular interest (see e.g.
%[\onlinecite{note1}] ***(??)***). \ldots

%\vspace{2.0cm}
%(which are --due to the projection operator $\cal{P}$-- restricted 
%to values zero or one, i.e. no double occupancy of lattice
%sites by electrons of opposite spin)
%as shown in Fig. \ref{fig1}. 

%In particular for the isotropic case ($t''=t$, $\alpha''=\alpha$)
%we write
%\begin{eqnarray}
%H & = & t\left(H_t+\alpha H_J\right);\hspace{1.0cm} \alpha=\frac{J}{t}\,.
%\end{eqnarray}


The interaction between $L=2$ clusters is mediated by
the two dashed links for each spatial dimension. It is treated in the following by a
perturbation expansion in the hopping parameter $t''$, which
allows the hopping of electrons between neighbouring plaquettes.
%(****** remark on $\alpha''$?? ******)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[ht!]
\centerline{\hspace{0.0cm}\includegraphics[width=6.0cm,angle=0]{Fig1.eps}
\hspace{-0.0cm}}
\caption{$2D$ lattice structure with underlying $2\times 2$ plaquette
structure.}
\label{fig1}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

To zeroth order in $t''$, the eigenstates of the $2D$ $t-J$ 
model decay into a product
\begin{eqnarray}
%|Q_1\ldots Q_{N^{(p)}}\rangle & = & \prod_p |Q_p\rangle
 & & \prod_p |Q_p\rangle\label{prod_ansatz}
\end{eqnarray}
of plaquette eigenstates $|Q_p\rangle$ with charge $Q_p$.
The ground state is characterized by a minimum of the energy:
\begin{eqnarray}
E_0=\sum_p E^{(p)}(Q_p) & = & \sum_{Q=0}^4 E^{(p)}(Q)
N(Q)\label{e0_I}
\end{eqnarray}
where $N(Q)$ is the number of plaquettes with charge $Q=0,1,2,3,4$.
$E^{(p)}(Q)$ is the ground state energy of a plaquette with charge
$Q$.

The numbers $N(Q)$ are constrained by the total number of plaquettes ($N/4$)
and the total charge ($Q_{\mbox{tot}}$):
%\begin{itemize}
%\item
%the total number of plaquettes $N^{(p)}$
\begin{eqnarray}
\sum_{Q=0}^4 N(Q)=N^{(p)}=\frac{N}{4}%\label{e0_II}
 & \quad,\quad & \sum_{Q=0}^4 QN(Q)=Q_{\mbox{tot}}\label{e0_II}
\end{eqnarray}

%\item
%the total charge
%\begin{eqnarray}
%\sum_{Q=0}^4 QN(Q) & = & Q_{\mbox{tot}}\label{e0_III}
%\end{eqnarray}
%\end{itemize}
The plaquette ground state energies $E^{(p)}(Q)$ have been computed in
Appendix A of Ref. \cite{fl04}. % for the general case with
%different couplings $t,\alpha$ %and $t',\alpha'$ 
%in the vertical
%and horizontal direction. 
The there obtained formulas simplify for the
symmetric case %$t=t',\,\alpha=\alpha'$ 
we are considering here.

%hier_8
In the charge sectors $Q=0,1,2,4$
there is no level crossing and the plaquette
ground state energy for all $\alpha>0$ reads
\begin{eqnarray}
E^{(p)}(0) & = & 0\hspace{4.0cm}S=0\label{eQ0_1}\\
E^{(p)}(1) & = & -2t\hspace{3.4cm}S=1/2\label{eQ1_1}\\
E^{(p)}(2) & = & -\frac{t}{2}\Big(\alpha+\sqrt{\alpha^2+32}\Big)\hspace{0.55cm}S=0\label{eQ2_1}\\
E^{(p)}(4) & = & -3t\alpha\hspace{3.15cm}S=0\label{eQ4_1}
\end{eqnarray}
Note that the total spin ${\bf S}^{(p)}$ of the plaquette
electrons commutes with the plaquette Hamiltonian and the
eigenvalues of ${{\bf S}^{(p)}}^2=S(S+1)$ can be used to
characterize the plaquette ground state.
This plays an important role in the sectors with
$Q=3$ where the ground state changes with $\alpha$:
\begin{eqnarray}
E_{I}^{(p)}(3,\alpha) & = & -2t,\hspace{31mm}S=3/2 \hspace{3mm}\mbox{for}
 \hspace{3mm}0\leq\alpha\leq\alpha_m\nonumber\\
[4pt]
E_{II}^{(p)}(3,\alpha) & = &  -t\Big(\alpha+\sqrt{\frac{\alpha^2}{4}
+3}\Big),\hspace{5mm}S=1/2\hspace{5mm}\mbox{for}
\hspace{5mm}\alpha_m\leq\alpha<2\nonumber\\[4pt]
E_{III}^{(p)}(3,\alpha) & = & -t\Big(\frac{3\alpha}{2}+1\Big),
\hspace{17mm}S=1/2\hspace{5mm}\mbox{for}
\hspace{5mm}\alpha>2. \label{eQ3}
\end{eqnarray}
with $\alpha_m=\frac{2}{3}(4-\sqrt{13})=0.262..$.

%It is remarkable to note,
Note, that the Nagaoka ferromagnetic state\cite{nagaoka66} --
with maximal spin $S=Q/2$ -- is already visible on a 4-site plaquette 
with one hole, i.e. $Q=3$, if $\alpha$ is small enough $\alpha<
\alpha_m=0.262$. Eigenstates with maximal plaquette spin $S=Q/2$
exist in all charge channels; the corresponding energy eigenvalues
$E(Q,S=Q/2)$ do not depend on the spin exchange coupling $\alpha$. However,
these eigenstates are not ground states except for the one hole case.

Let us next look for the minimum of the total energy (\ref{e0_I}),
which depends on the relative magnitude of the plaquette energies.
\begin{itemize}
\item[a)]
In the regime, where the inequality
\begin{eqnarray}%\label{ineq_4233}
E^{(p)}(4)+E^{(p)}(2) & < & 2E^{(p)}(3)
%\nonumber\\ 
\label{ineq_4233}
\end{eqnarray}
holds, two plaquettes with charge $Q=3$ are %eliminated and
substituted by two plaquettes with charges $Q=2$
and $Q=4$, respectively.
Insertion of the ground state energies [(\ref{eQ2_1}), (\ref{eQ4_1}),
(\ref{eQ3})]
%for $Q=2$, (\ref{eQ4_1}) for $Q=4$ and
%(\ref{eQ3})(I,II,III) for $Q=3$
yields the validity of (\ref{ineq_4233}) for
\begin{eqnarray}\label{al_ineq_4233}
\alpha & > & \alpha(3)=\frac{2}{\sqrt{21}}\simeq 0.436 \,.
\label{regime_a}
\end{eqnarray}



\item[b)]
The inequalitiy
\begin{eqnarray}
E^{(p)}(0)+E^{(p)}(4) & < & 2E^{(p)}(2)
\label{ineq_0422}
\end{eqnarray} 
holds for
\begin{eqnarray}
\alpha & \geq & \alpha(2)=4\sqrt{2/3}\simeq 3.266
\label{regime_b}
\end{eqnarray}
and allows the substitution of the charge $Q=2$ plaquettes.



\item[c)]
The inequality
\begin{eqnarray}
E^{(p)}(0)+E^{(p)}(2) & < & 2E^{(p)}(1)%\nonumber
\label{ineq_0211}
\end{eqnarray}    
holds for 
\begin{eqnarray}
\alpha & > & 2
\label{regime_c}
\end{eqnarray}
and allows to substitute $Q=1$ plaquettes.

\end{itemize}


Fig. \ref{fig2}(A) shows a schematic view of the phase diagram in
the $\rho -\alpha$ plane with the pairs of plaquette charges,
which fix the ground state energy per site according to eqs.
(\ref{e0_I})-(\ref{e0_II}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[ht!]
\centerline{\hspace{0.0cm}\includegraphics[width=15.0cm,angle=0]{Fig_23b.ps}
\hspace{0.0cm}}
\caption{Schematic phase diagram obtained from $2\times 2$ plaquettes (A)
%The charges $Q_1,Q_2$ that constitute the respective ground states
%in the $\rho-\alpha$-plane are given and the boundaries %$\alpha_{1,2}$
%of the $(2,4)$ regime are: $\alpha(2)=2/\sqrt{21}$, $\alpha(0)=4\sqrt{2/3}$;
and $4\times 4$ plaquettes (B).
The charges $Q_1,Q_2$ that constitute the respective ground states
in the $\rho-\alpha$-planes and the boundaries $\alpha(2j)$ are given in the text.}
\label{fig2}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In each of the domains with plaquette pairs $Q_1,Q_2$ 
\begin{eqnarray}
Q_1<Q_2 & \quad\quad & \frac{Q_1}{N_c}<\rho<\frac{Q_2}{N_c}
\end{eqnarray}
%abdollah-a1
where the  plaquette sites $N_c=4$,  the ground state energy per site:
\begin{equation}
\varepsilon(\rho,\alpha)=\frac{1}{Q_2-Q_1}\Big\{
E^{(p)}(Q_1)\Big(\frac{Q_2}{N_c}-\rho\Big)+
E^{(p)}(Q_2)\Big(\rho-\frac{Q_1}{N_c}\Big)\Big\}
\label{ge_Q1Q2}
\end{equation}
is linear in $\rho$ and the chemical potential
$\mu(\rho,\alpha)=d\varepsilon/d\rho$
%\begin{eqnarray}
%\mu(\rho,\alpha) & = & \frac{d\varepsilon}{d\rho}
%\label{chemical_pot}
%\end{eqnarray}
is constant with respect to $\rho$.
%\begin{eqnarray}
%\mu(\rho,\alpha) & = & \frac{1}{Q_2-Q_1}\Big\{
%E^{(p)}(Q_2,\alpha)-E^{(p)}(Q_1,\alpha)\Big\}\,.
%\end{eqnarray}
Note that two domains ($Q_1,Q_2$) and ($Q_2,Q_3$) with a common
plaquette charge $Q_2$ have also a common horizontal boundary at charge
density $\rho_2=Q_2/4$.
%
At this boundary the chemical potential $\mu(\rho,\alpha)$
as function of $\rho$ is discontinuous with a jump
\begin{eqnarray}
\Delta(Q_2,\alpha) & = & \mu(\rho+0,\alpha)-\mu(\rho-0,\alpha)\nonumber\\
 & = & \frac{E^{(p)}(Q_3)}{Q_3-Q_2}+\frac{E^{(p)}(Q_1)}{Q_2-Q_1}-
\frac{Q_3-Q_1}{(Q_3-Q_2)(Q_2-Q_1)}E^{(p)}(Q_2)
\end{eqnarray}
which vanishes for a specific value of $\alpha=\alpha(Q_2)$.
%\begin{eqnarray}
%\Delta(Q_2,\alpha=\alpha(Q_2)) & = & 0\,.
%\end{eqnarray}
They define the vertical lines in the phase diagram given by
(\ref{regime_a}), (\ref{regime_b}) and (\ref{regime_c}).
This type of phase diagram has been introduced first by Kagan et al.
\cite{kagan99} 
for the three leg ladder in the rung cluster approximation.
Note, that the constraints for ($N^{(p)}$, $Q_{\mbox{tot}}$)(\ref{e0_II})
%,(\ref{e0_III})
are taken into account explicitely on both sides of the inequalities
(\ref{ineq_4233}), (\ref{ineq_0422}), (\ref{ineq_0211}).
This procedure can be considered as an alternative to the usual
grand canonical one, where the charge conservation constraint (\ref{e0_II})
is eliminated by a reservoir with a fixed chemical potential.
%(\ref{chemical_pot}).

%The results obtained from plaquette clusters yield some indication for a charge
%transfer gap at $\rho=1/2$. Its width can be estimated from
%%From Fig. \ref{fig2} we expect a charge transfer gap at $\rho=1/2$.
%%Its width can be estimated from the energy differences:
%\begin{eqnarray}
%\Delta_1(2,\alpha) & = & \frac{1}{2}\Big(E(4)+2E(1)-3E(2)\Big)\quad\nonumber\\
% & & \mbox{for\quad}
%\frac{2}{\sqrt{21}}\leq\alpha\leq 2\label{Delta_1_2}\\
%\Delta_2(2,\alpha) & = & \frac{1}{2}\Big(E(4)-2E(2)\Big)\hspace{2.5cm}\nonumber\\
% & & \mbox{for\quad}
%2\leq\alpha\leq 4\sqrt{\frac{2}{3}}\,.\label{Delta_2_2}
%\end{eqnarray}
%The $\alpha$-dependence of $\Delta(2,\alpha)$ can be seen in Fig. \ref{fig4} 
%at the end of Sec. \ref{sec4} together with the corresponding results
%obtained from $4\times 4$ clusters. Finite-size effects are large and
%it is not possible to estimate whether or not a charge transfer gap
%will survive in the limit $L\times L$, $L\rightarrow\infty$.

%(for the $2\times 2$ and also for
%the $4\times 4$ system) can be seen in Fig. \ref{fig4} at the end of Sec.
%\ref{sec4}.





%Note, that in the upper right part of the phase diagram, the 
%emergence of phase separation is already visible. Owing to
%the inequalities (\ref{ineq_4233}), (\ref{ineq_0422}) and 
%(\ref{ineq_0211}) only plaquette 
%pairs with ``separated'' charges $(0,4)$, $(2,4)$ and $(0,2)$
%survive.

%In the next section, we compute the interaction of neighbouring
%plaquettes. The result will
%provide us with a second argument for phase separation.







%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Interaction between neighbouring plaquettes.
%Phase separation on large scales.\label{sec3}}

%The product ansatz (\ref{prod_ansatz}) with isolated plaquette ground states
%-- as shown in Fig. \ref{fig1} -- is highly degenerate. In zeroth
%order perturbation theory ($t''=0$) each distribution of the plaquette
%charges $Q_1$, $Q_2$ over the whole lattice leads to the same ground
%state energy per site (\ref{ge_Q1Q2}) in the thermodynamical limit.

%Let us denote the interaction energy between neighbouring plaquettes
%with charges $Q_1$, $Q_2$ by $W(Q_1,Q_2)$. If the difference:
%\begin{eqnarray}
%\Delta(Q_1,Q_2,\alpha) & = & W(Q_1,Q_1)+W(Q_2,Q_2)-2W(Q_1,Q_2)
%\label{difference}
%\end{eqnarray}
%is negative, the ground state prefers charge separation in the
%following sense:

%The numbers $N(Q_j,Q_j)$ of neighbouring plaquettes with the same charge
%$Q_j$ are maximal:
%\begin{eqnarray}
%\frac{N(Q_j,Q_j)}{N} & = & 2\frac{N(Q_j)}{N}\label{condition_1}
%\end{eqnarray}
%in the thermodynamical limit $N\rightarrow\infty$.
%$N(Q_j)$ are the numbers of plaquettes with charge $Q_j$ constrained by
%(\ref{e0_II}).% and (\ref{e0_III}).

%As a consequence of (\ref{condition_1}), the number of neighbouring
%plaquettes $N(Q_1,Q_2)$ with different charges $Q_1\neq Q_2$ are minimal
%and vanish
%\begin{eqnarray}
%\frac{N(Q_1,Q_2)}{N} & \rightarrow & 0\hspace{1.0cm}\mbox{for\quad}
%N\rightarrow\infty\,.
%\label{condition_2}
%\end{eqnarray}
%Neighbouring plaquettes with different charges define the phase
%boundary between the two phases with charge densities $\rho_j=Q_j/N,
%\,\, j=1,2$ respectively.

%If the difference (\ref{difference}) of interaction energies is
%positive, the ground state prefers a maximal number $N(Q_1,Q_2)$ of
%neighbouring plaquettes with different charges $Q_1$, $Q_2$, which leads
%to a more uniform distribution of the charges.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}[ht!]
%\centerline{\hspace{0.1cm}\includegraphics[width=5.0cm,angle=0]{Fig3.eps}
%\hspace{0.1cm}}
%\caption{Interaction between neighbouring plaquettes with charges $Q_1,Q_2$
%shown for a pair of nearest neighbour sites $x$ and $x'$.}
%\label{fig3n}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%Perturbation theory in the hopping parameter $t''$ for the interaction
%between neighbouring plaquettes was developped in Appendix B of Ref.
%\cite{fl04}. According to (\ref{element_h_plaq}) and Fig. \ref{fig3n}
%the relevant matrix elements are:
%
%\begin{eqnarray}
%&&W(Q_1,Q_2) =  \sum_x\langle Q_1,Q_2|h^{(x,x')}|Q_1,Q_2\rangle\nonumber\\
% & = & \alpha\sum_x\Big(\langle Q_1|{\bf S}(x)|Q_1\rangle
%\langle Q_2|{\bf S}(x')|Q_2\rangle
%-\frac{1}{4}\langle Q_1|n(x)|Q_1\rangle
%\langle Q_2|n(x')|Q_2\rangle\Big)\,.
%\label{relevant_ME}
%\end{eqnarray}
%%Note that the hopping term in (\ref{element_h_plaq}) can contribute only if both
%%charges $Q_1,Q_2$ differ by one unit: $Q_2=Q_1\pm 1$. We will not
%%consider this case in the following.

%If one of the two plaquettes has even charge ($Q_1=0,2,4$) and thereby 
%total spin zero [(\ref{eQ0_1}),(\ref{eQ2_1}),(\ref{eQ4_1})] the spin
%matrix elements
%\begin{eqnarray}
%\langle Q_1|{\bf S}_1|Q_1\rangle & = & 0\label{cl_spin}
%\end{eqnarray}
%vanish and the interaction energy is given by
%\begin{eqnarray}
%W(Q_1,Q_2) & = & -\frac{\alpha}{4}\sum_x\langle Q_1|n(x)|Q_1\rangle
%\langle Q_2|n(x')|Q_2\rangle
%= -\frac{\alpha}{32}Q_1Q_2\,. \label{condition_3a}
%\end{eqnarray}
%In this case the difference (\ref{difference}) turns out to be negative:
%\begin{eqnarray}
%\Delta(Q_1,Q_2,\alpha) & = & -\frac{\alpha}{4}\sum_x\Big(
%\langle Q_1|n(x)|Q_1\rangle -\langle Q_2|n(x)|Q_2\rangle\Big)^2 <0\,.
%\label{condition_3}
%\end{eqnarray}
%Inequality (\ref{condition_3}) implies, that we find phase separation 
%on a macroscopic
%scale, if both plaquette charges $(Q_1,Q_2)$ in the phase diagram 
%[Fig. \ref{fig2}(A)] are even.
%%\begin{eqnarray}
%%(Q_1,Q_2) & = & (0,4),\,(2,4)\nonumber %,\,(0,2)\,.\nonumber
%%\end{eqnarray}
 
%In the domains

%\begin{itemize}

%\item
%\begin{eqnarray}
%\alpha & > & 4\sqrt{\frac{2}{3}}=\alpha(0)\hspace{2.0cm} 0\leq\rho\leq 1\label{domain_1}
%\end{eqnarray}
%the plaquette charges take values $Q_1=0,Q_2=4$ and the ground state is characterized
%by a distribution of plaquette charges with maximal $N(4,4)=(\rho/2)N$.
%%\begin{eqnarray}
%%\frac{N(4,4)}{N} & = & 2\frac{N(4)}{N}=\frac{1}{2}\rho\,.\label{condition_4}
%%\end{eqnarray}
%Up to first order the ground state energy per site in the regime
%(\ref{condition_2}) is linear in $\rho$
%\begin{eqnarray}
%\varepsilon(\rho) & = & \frac{\rho}{4}\Big(E^{(p)}(4)+2t''W(4,4)\Big)\,.
%\end{eqnarray}
%%Let us next consider the domain

%\item
%\begin{eqnarray}
%\alpha(2)=\frac{2}{\sqrt{21}}<\alpha<\alpha(0)=4\sqrt{\frac{2}{3}} 
%\hspace{1cm} 1/2\leq\rho\leq 1\,.
%\label{domain_2}
%\end{eqnarray}
%the ground state is characterized by a distribution of plaquette
%charges $Q=4,2$ with maximal $N(4,4)=(\rho-1/2)N$ [$N(2,2)=(1-\rho)N$]
%%\begin{eqnarray}
%%\frac{N(4,4)}{N}=\rho-\frac{1}{2}\,, & & \,\,\frac{N(2,2)}{N}=1-\rho\,,\,\,
%%\frac{N(4,2)}{N}=0
%%\frac{N(4,4)}{N} & = & 2\frac{N(4)}{N}=\rho-\frac{1}{2}\nonumber\\
%%\frac{N(2,2)}{N} & = & 2\frac{N(2)}{N}=1-\rho\nonumber\\
%%\frac{N(4,2)}{N} & = & 0\hspace{2.0cm}\mbox{for\quad}N\rightarrow\infty
%%\end{eqnarray}
%%(for $N\rightarrow\infty$)
%and the ground state energy per site turns out to be
%\begin{eqnarray}
%\varepsilon(\rho) & = & \frac{1}{2}(1-\rho)\Big(E^{(p)}(2)+
%2t''W(2,2)\Big)\nonumber\\
% & & +\frac{1}{2}(\rho-1/2)\Big(E^{(p)}(4)+
%2t''W(4,4)\Big)\,.
%\label{eps_rho_W}
%\end{eqnarray}
%%In the domain

%%\item
%%\begin{eqnarray}
%%2<\alpha<4\sqrt{\frac{2}{3}} \hspace{2cm} 0\leq\rho\leq 1/2\,,
%%\label{domain_3}
%%\end{eqnarray}
%%the ground state is given by a distribution of plaquettes with charges
%%$Q_1=0$, $Q_2=2$ such that $N(2,2)=N\rho$ is maximal [$N(0,0)=N(1/2-\rho)$].
%%%\begin{eqnarray}
%%%\frac{N(2,2)}{N}=\rho\,, & & \,\,\frac{N(0,0)}{N}=\frac{1}{2}-\rho\,,\,\,\,
%%%\frac{N(2,0)}{N}=0
%%%\frac{N(2,2)}{N} & = & 2\frac{N(2)}{N}=\rho\nonumber\\
%%%\frac{N(0,0)}{N} & = & 2\frac{N(0)}{N}=1/2-\rho\nonumber\\
%%%\frac{N(2,0)}{N} & = & 0\,.
%%%\end{eqnarray}
%%In the regime (\ref{domain_3}), the resulting ground state energy per site reads:
%%\begin{eqnarray}
%%\varepsilon(\rho) & = & \frac{\rho}{2}\Big(E^{(p)}(2)+
%%2t''W(2,2)\Big)\,.\label{eps_rho_W_II}
%%\end{eqnarray}

%\end{itemize}

%It is remarkable to note, that in the regimes (\ref{domain_1}) and (\ref{domain_2}) 
%%and (\ref{domain_3}) 
%first order perturbation theory is ``diagonal'' in the sense,
%that it does not change the charge of neighbouring plaquettes (\ref{relevant_ME}).
%Both constituent plaquette charges $Q_1$, $Q_2$ are even and an exchange of
%neighbouring plaquette charges (cf. Fig. \ref{fig3n})
%\begin{eqnarray}
%(Q_1,Q_2) & \rightarrow & (Q_2,Q_1)\label{Q1Q2_exchange}
%\end{eqnarray}
%would demand the hopping of at least two electrons.
%This cannot occur in first order perturbation theory. %(\ref{relevant_ME})
%%treated so far. 
%In other words, the holes are confined in the hole cluster
%if $\alpha$ is large enough. However, for small $\alpha$ values (left part
%in Fig. \ref{fig2}) the charges $Q_1$, $Q_2$ which constitute the respective
%ground state in the $\rho-\alpha$ plane only differ by one unit. In this
%situation first order perturbation theory allows the exchange of the
%plaquette charges (\ref{Q1Q2_exchange}) due to the hopping matrix element
%\begin{eqnarray}
%\hspace{-0.5cm}W(Q_1,Q_2;Q_2,Q_1) & = & \sum_x\langle Q_1Q_2|h^{(xx')}|Q_2Q_1\rangle\,.
%\label{W_Q12_21}
%\end{eqnarray}
%These ``non-diagonal''matrix elements deconfine the holes, such that they
%can hop over the whole lattice. First order perturbation theory cannot be
%treated analytically in this case but leads to an effective Hamiltonian.
%The computation of its ground state can be achieved only with numerical
%methods.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Product ansatz for the ground state with large clusters\label{sec4}}

The considerations which led to the phase diagram in Fig. \ref{fig2}(A) based
on plaquette clusters can easily be extended to larger $L\times L=N_c$
clusters with $N_c$ sites and charges $Q=N_c,\ldots,0$. We start with
the ground state energies $E(Q,N_c,\alpha)$ and their dependence
on $\alpha$. 
The generalization of (\ref{ge_Q1Q2}) to a cluster
with $N_c$ sites and charges $Q_1<Q_2$ is straightforward
%abdollah-a1
by replacing $E^{(p)}(Q_1) \rightarrow E(Q_1,\alpha,N_c)$.
%\begin{eqnarray}
%\varepsilon(\rho,\alpha) & = & \frac{1}{Q_2-Q_1}\Big\{
%E(Q_1,\alpha,N_c)\Big(\frac{Q_2}{N_c}-\rho\Big)+
%E(Q_2,\alpha,N_c)\Big(\rho-\frac{Q_1}{N_c}\Big)\Big\}.
%\label{ge_Q1Q2_II}
%\end{eqnarray}
Again the ground state energy per site is linear in $\rho$ and the
chemical potential is constant:
\begin{eqnarray}
\mu(\rho,\alpha) & = & \frac{1}{Q_2-Q_1}\Big\{
E(Q_2,\alpha,N_c)-E(Q_1,\alpha,N_c)\Big\}\,.
\label{mu_rho}
\end{eqnarray}
Two domains ($Q_1,Q_2$) ($Q_2,Q_3$) with a common cluster charge $Q_2$
have a common boundary in the phase diagram at $\rho_2=Q_2/N_c$. At this
boundary the chemical potential is discontinuous with a jump
\begin{eqnarray}
\Delta(Q_2,\alpha) & = & \mu(\rho_2+0,\alpha)-\mu(\rho_2-0,\alpha)
\nonumber\\
 & = & \frac{E(Q_3)-E(Q_2)}{Q_3-Q_2}-\frac{E(Q_2)-E(Q_1)}{Q_2-Q_1}
\label{jump}
\end{eqnarray}
which vanishes for a specific value of $\alpha=\alpha(Q_2)$.
%abdollah-a1
%\begin{eqnarray}
%\Delta(Q_2,\alpha(Q_2)) & =  & 0\,.
%\label{Delta_Q2_0}
%\end{eqnarray}
The inequality
\begin{eqnarray}
\Delta(Q_2,\alpha) & <  & 0\quad\mbox{for\quad} \alpha>\alpha(Q_2)
\end{eqnarray}
means that the clusters with charge $Q_2$ and ground state energy
$E(Q_2,\alpha,N_c)$ can be substituted by clusters with charges
$Q_3$ and $Q_1$ and energies $E(Q_3,\alpha,N_c)$, $E(Q_1,\alpha,N_c)$.
Here the two domains ($Q_1,Q_2$) ($Q_2,Q_3$) merge 
%together:
%abdollah-a1
for $ \alpha>\alpha(Q_2)$
%\begin{eqnarray}
%(Q_1,Q_2), (Q_2,Q_3) & \rightarrow & (Q_1,Q_3)\quad
%\mbox{for\quad} \alpha>\alpha(Q_2)
%\end{eqnarray}
and the ground state is given by a cluster product ansatz with
charges ($Q_1,Q_3$).

Let us first consider the case [case ($A$)]
\begin{eqnarray}
Q_1=Q_2-1,\,\,Q_3=Q_2+1, \,\, & & Q_2\quad\mbox{odd},\,\,\alpha<\alpha(Q_2)\,.
\end{eqnarray}
Here the ground state is built up from clusters with charges:
\begin{eqnarray}
(Q_2-1,\,Q_2) & \hspace{0.5cm}\mbox{for}\hspace{0.5cm} & 
\frac{Q_2-1}{N_c}\leq\rho\leq\frac{Q_2}{N_c}
\label{domain1}\\
(Q_2,\,Q_2+1) & \hspace{0.5cm}\mbox{for}\hspace{0.5cm} & \frac{Q_2}{N_c}\leq\rho\leq\frac{Q_2+1}{N_c}\,.
\label{domain2}
\end{eqnarray}
The vanishing of the jump (\ref{jump}) in the chemical potential
\begin{eqnarray}
\Delta(Q_2,\alpha(Q_2)) & = & E(Q_2+1)+E(Q_2-1)-2E(Q_2)=0
\label{Delta_0}
\end{eqnarray}
defines the boundary $\alpha=\alpha(Q_2)$, where the two domains
(\ref{domain1}) and (\ref{domain2}) merge together:
\begin{eqnarray}
(Q_2-1,\,\,Q_2+1)\quad\mbox{for\quad}\frac{Q_2-1}{N_c}\leq\rho\leq\frac{Q_2+1}{N_c}\hspace{5mm}
\mbox{and}\,\,\alpha>\alpha(Q_2). 
\end{eqnarray}
The numerical evaluation of (\ref{Delta_0}) from the ground state energies
on a $4\times 4=16$ cluster -- with periodic boundary conditions -- yields
the couplings $\alpha(Q_2)$ listed in Table \ref{table_n1} (col. 1,2):

\begin{table}[t]
\begin{center}
\caption{Boundary couplings $\alpha(Q_2)$ [from left to right cases ($A$), ($B$), ($C$)] where
the jump (\ref{jump}) in the chemical potential vanishes for cases $(Q_1,Q_2,Q_3)$.}
\begin{tabular}{c|c||c|c||c|c}
 $(Q_1,Q_2,Q_3)$ &  $\alpha(Q_2)$  &  $(Q_1,Q_2,Q_3)$ &  $\alpha(Q_2)$  &
 $(Q_1,Q_2,Q_3)$ &  $\alpha(Q_2)$ \\ \hline
 $(14,15,16)$ & $<0.1$ & $(12,14,16)$ & $0.6$ & $(6,10,16)$ & $2.1..$\\
 $(12,13,14)$ & \hspace{0.4cm}$0.2$ & $(10,12,14)$ & $0.6$ & $(2,6,16)$ & $2.2..$\\
 $(10,11,12)$ & $<0.1$ & $(6,8,10)$ & $1.74$ & $(0,2,16)$ & $3.5..$\\
 $(8,9,10)$ & \hspace{0.4cm}$0.4$ & $(2,4,6)$ & $1.9$ &  & \\
 $(6,7,8)$ & $<0.1$ & & &  & \\
 $(4,5,6)$ & \hspace{0.4cm}$0.2$ & & &  & \\
 $(2,3,4)$ & $<0.1$ & & &  & \\
 $(0,1,2)$ & \hspace{0.4cm}$2.0$ & & &  & 
\end{tabular}
\label{table_n1}
\end{center}
\end{table}

%\begin{table}[t]
%\begin{center}
%\caption{Boundary couplings $\alpha(Q_2)$ for odd $Q_2$ where
% (\ref{jump})  vanishes, for $(Q_1,Q_2,Q_3)=(Q_2-1,Q_2,Q_2+1)$.}
%\begin{tabular}{c|c|c|c|c|c|c|c|c}
% $Q_2$ &  $15$ &
% $13$
% &  $11$ & 9 & 7 & 5 & 3 & 1
%\\ \hline
% $\alpha(Q_2)$ &  $<0.1$ &
% $0.2$
% & $<0.1$ & 0.4 & $<0.1$ & 0.2 & $<0.1$ & 2.0
%\end{tabular}
%\label{table1}
%\end{center}
%\end{table}

In the next step, we consider the cases [case ($B$)]:
\begin{eqnarray}
Q_1=Q_2-2,\,\,Q_3=Q_2+2,\hspace{5mm}
Q_2=14,12,8,4, \hspace{5mm}\alpha<\alpha(Q_2)\,.
\end{eqnarray}
Here the ground state is built up from clusters with charges
\begin{eqnarray}
(Q_2-2,\,Q_2) & \hspace{0.5cm}\mbox{for}\hspace{0.5cm} & \frac{Q_2-2}{N_c}\leq\rho\leq\frac{Q_2}{N_c}
\label{domain3}\\
(Q_2,\,Q_2+2) & \hspace{0.5cm}\mbox{for}\hspace{0.5cm} & \quad\frac{Q_2}{N_c}\leq\rho\leq\frac{Q_2+2}{N_c}\,.
\label{domain4}
\end{eqnarray}
The vanishing of the jump (\ref{jump}) in the chemical potential
\begin{eqnarray}
\Delta(Q_2,\alpha(Q_2)) & = & \frac{1}{2}\left[E(Q_2+2)+E(Q_2-2)-2E(Q_2)
\right]= 0
\label{Delta_02}
\end{eqnarray}
defines again the boundary $\alpha=\alpha(Q_2)$, where the two domains
(\ref{domain3}) and (\ref{domain4}) merge together:
\begin{equation}
(Q_2-2,\,\,Q_2+2)\quad\mbox{for\quad}\frac{Q_2-2}{N_c}\leq\rho\leq\frac{Q_2+2}{N_c}\hspace{5mm}
\mbox{and} \hspace{5mm}\alpha>\alpha(Q_2). 
\end{equation}
The numerical evaluation of (\ref{Delta_02}) from the ground state energies
on a $4\times 4=16$ cluster -- with periodic boundary conditions -- yields
the couplings $\alpha(Q_2)$ listed in Table \ref{table_n1} (col. 3,4):

%\begin{table}[t]
%\begin{center}
%\caption{Boundary couplings $\alpha(Q_2)$ for even $Q_2=14,12,8,4$ where 
% (\ref{jump}) vanishes, for $(Q_1,Q_2,Q_3)=(Q_2-2,Q_2,Q_2+2)$.}
%\begin{tabular}{c|c|c|c|c}
% $Q_2$ &  $14$ &
% $12$
% &  $8$ & 4 
%\\ \hline
% $\alpha(Q_2)$ &  $0.6$ &
% $0.6$
% & $1.74$ & 1.9 
%\end{tabular}
%\label{table2}
%\end{center}
%\end{table}



%\vspace{3.0cm}



We are left with clusters of charge $Q_2=10,6,2$ [case ($C$)]. 

They are eliminated successively by considering the jumps (\ref{jump}) in
the chemical potential for the cases in Table \ref{table_n1} (col. 5,6).
%\begin{table}[t]
%\begin{center}
%\caption{Successive elimination of charges $Q_2=10,6,2$ in the phase diagram
%(Fig. \ref{fig2}(B)) due to jumps (\ref{jump}) in the chemical potential.}
%\begin{tabular}{c|c|c}
% $(Q_1,Q_2,Q_3)$ & $\Delta(Q_2,\alpha)$ & $\alpha=\alpha(Q_2)$ \\ \hline
%(6,10,16) & $\frac{1}{12}\big[2E(16)+3E(6)-5E(10)\big]$ & $\alpha(10)=2.1\ldots$\\[5pt]\hline
%(2,6,16) & $\frac{1}{20}\big[2E(16)+5E(2)-7E(6)\big]$ & $\alpha(6)=2.2\ldots$\\[5pt]\hline
%(0,2,16) & $\frac{1}{14}\big[E(16)+7E(0)-8E(2)\big]$ & $\alpha(2)=3.5\ldots$\\[5pt]
%\end{tabular}
%\label{table4}
%\end{center}
%\end{table}
These jumps vanish at the couplings $\alpha=\alpha(Q_2)$ listed in the third 
column, such that the clusters with charges $Q_2$ can be eliminated for
$\alpha>\alpha(Q_2)$. The resulting phase diagram is shown in Fig. \ref{fig2}(B).
The pairs of integers in the rectangular domains denote the two cluster charges
which determine the ground state in the product ansatz.








%The jump in the
%chemical potential (\ref{Delta_Q2}) vanishes for

%\begin{itemize}

%\item[a)]

%For $Q_1=2$, $Q_2=6$, $Q_3=10$ the jump (\ref{jump}) in the chemical
%potential:
%\begin{eqnarray}
%\Delta(Q_2=6,\alpha(6)) & = & \frac{1}{4}\left[E(10)+E(2)-2E(6)
%\right]\nonumber\\
%\end{eqnarray}
%vanishes for
%\begin{eqnarray}
%\alpha=\alpha(6) & = & 2.2\,.
%\end{eqnarray}


%\item[b)]
%For $Q_1=2$, $Q_2=10$, $Q_3=16$ the jump (\ref{jump}) in the chemical
%potential:
%\begin{eqnarray}
%\Delta(Q_2=10,\alpha(10)) & = & \frac{E(16)-E(10)}{6}-\nonumber\\
% & & \frac{E(10)-E(2)}{8}
%\nonumber\\
%\end{eqnarray}
%vanishes for
%\begin{eqnarray}
%\alpha=\alpha(10) & = & 2.2\,.
%\end{eqnarray}

%\item[c)]
%For $Q_1=0$, $Q_2=2$, $Q_3=16$ the jump (\ref{jump}) in the chemical
%potential:
%\begin{eqnarray}
%\Delta(Q_2=2,\alpha(2)) & = & \frac{E(16)-E(2)}{14}-\nonumber\\
% & & \frac{E(2)-E(0)}{2}
%\nonumber\\
%\end{eqnarray}
%vanishes for
%\begin{eqnarray}
%\alpha=\alpha(2) & = & 3.5\,.
%\end{eqnarray}

%\end{itemize}


%Equation (\ref{Delta_Q2_2}) means, that the system for
%$\alpha>\alpha(Q_2=2)$ is completely separated into two phases
%with clusters of charge density $\rho=0$ and $\rho=1$, respectively.
%The resulting phase diagram is shown in Fig. \ref{fig3}. The pairs
%of integer ($Q_1,Q_2$) in the rectangular domains denote the two
%cluster charges which determine the ground state in the product
%ansatz.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}[ht!]
%\centerline{\hspace{0.0cm}\includegraphics[width=8.0cm,angle=0]{Fig4.ps}
%\hspace{0.0cm}}
%\caption{Schematic phase diagram obtained from $4\times 4$ clusters.
%The charges $Q_1,Q_2$ that constitute the respective ground states
%in the $\rho-\alpha$-plane are given.}
%\label{fig3}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let us comment the appropriate boundary conditions for the
$L\times L$ cluster in the variational product ansatz for the ground
state on the infinite lattice.
A priori we are free in our choice of the clusters and their boundary
conditions. In our opinion periodic boundary conditions are most
appropriate for the following reasons:

The variational product ansatz becomes exact in the limit of infinite
cluster ($L\rightarrow\infty$, $N_c=L^2\rightarrow\infty$). In this
case, the cluster energies $E(Q,\alpha,N_c)=N_c\varepsilon(\rho_1=Q/N_c,\alpha)$
are related to the ground state energies per site $\varepsilon(\rho_1=Q/N_c,\alpha)$
at fixed charge density. On finite clusters with $L\times L=N_c$ sites
the ground state energies are approximated most accurately with periodic
boundary conditions, since the interaction between the clusters is partly
taken into account by means of the periodic boundary terms. Of course --
in a perturbation expansion for the interaction between the clusters --
the periodic boundary terms have to be subtracted again in each order. 
%Indeed, these
%subtractions cancel in first order perturbation theory most of the
%interaction energies between the clusters.

%The calculation of the interaction energies between neighbouring
%$L\times L$ clusters with charges $Q_h$ (``hole'' clusters), $Q_e=L^2$
%(electron clusters) %and total spin 0
%can be easily extended from $L=2$ (cf. Fig. \ref{fig3n}) to larger %(even)
%$L$ values. Having in mind that in general $L$, links ($x,x'$) connect
%the neighbouring clusters, one can see immediately that the first line
%in (\ref{condition_3a}) and eq. (\ref{condition_3}) hold for general
%(even) $L$, if we extend the summation over all links ($x,x'$) and if
%the total cluster spin is zero [cf. (\ref{relevant_ME}), (\ref{cl_spin})].
%In particular, the differences (\ref{difference}), (\ref{condition_3})
%of interaction energies are negative -- irrespective of the type of
%boundary conditions .
%The subtraction of periodic boundary terms $p(Q_j)$ for neighbouring
%clusters with charges $Q_1$, $Q_2$
%\begin{eqnarray}
%W(Q_1,Q_2) & \rightarrow & W(Q_1,Q_2)-p(Q_1)-p(Q_2)\nonumber
%\end{eqnarray}
%does not affect the crucial difference $\Delta(Q_1,Q_2)$
%(\ref{difference}), which decides whether clusters with the same
%charge $Q_1=Q_2=L^2$ or $Q_1=Q_2=Q_h$ condensate. 
%Therefore, holes are confined in the hole clusters
%as long as $Q_h\leq L^2-2$ and first order perturbation theory is applicable.

%%%%%%%%%%%%%%%%%%%%%%%%%%


%From Fig. \ref{fig3} we read off two candidates for a charge transfer gap 
%at $\rho=\frac{5}{8},\frac{3}{8}$, which might survive in the
%thermodynamical limit.

%In our $4\times 4$ cluster approximation, the width of the charge
%transfer gap at $\rho=\frac{5}{8}$ is given by the energy differences
%\begin{eqnarray}
%\Delta_1(10,\alpha) & = & \frac{1}{6}\Big(E(16)+3E(8)-4E(10)\Big)\quad\nonumber\\
% & & \mbox{for\quad}
%0.6\leq\alpha\leq 1.74\label{Delta_1_10}\\
%\Delta_2(10,\alpha) & = & \frac{1}{12}\Big(2E(16)+3E(6)-5E(10)\Big)\hspace{0.5cm}\nonumber\\
% & & \mbox{for\quad}
%1.74\leq\alpha\leq 2.2\,.\label{Delta_2_10}
%\end{eqnarray}
%In the same way we find the width of the charge transfer gap at
%$\rho=\frac{3}{8}$ from the energy differences
%\begin{eqnarray}
%\Delta_1(6,\alpha) & = & \frac{1}{2}\Big(E(8)+E(4)-2E(6)\Big)\quad\nonumber\\
% & & \mbox{for\quad}
%0.6\leq\alpha\leq 1.7=\alpha(8)\,,\label{Delta_1_6}\\
%\Delta_2(6,\alpha) & = & \frac{1}{4}\Big(E(10)+2E(4)-3E(6)\Big)\hspace{0.5cm}\nonumber\\
% & & \mbox{for\quad}
%1.7\leq\alpha\leq 1.9=\alpha(4)\,,\label{Delta_2_6}\\
%\Delta_3(6,\alpha) & = & \frac{1}{4}\Big(E(10)+E(2)-2E(6)\Big)\hspace{0.5cm}\nonumber\\
% & & \mbox{for\quad}
%1.9\leq\alpha\leq 2.1=\alpha(10)\,,\label{Delta_3_6} \\
%\Delta_4(6,\alpha) & = & \frac{1}{20}\Big(2E(16)+5E(2)-7E(6)\Big)\hspace{0.5cm}\nonumber\\
% & & \mbox{for\quad}
%2.1\leq\alpha\leq 2.2=\alpha(6)\,.\label{Delta_4_6}
%\end{eqnarray}
%The $\alpha$-dependence of $\Delta(10,\alpha)$ and $\Delta(6,\alpha)$
%is shown in Fig. \ref{fig4}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}[ht!]
%\centerline{\hspace{0.0cm}\includegraphics[width=8.0cm,angle=0]{Fig4.ps}
%\hspace{0.0cm}}
%\caption{Variation of gap width $\Delta(Q_2,\alpha)$ %(\ref{Delta_Q2}) at
%at $\rho=Q_2/L^2$ for two $L\times L$ systems with $L=2$ [(\ref{Delta_1_2}),
%(\ref{Delta_2_2})] and $L=4$ [(\ref{Delta_1_10})-(\ref{Delta_4_6})].}
%\label{fig4}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{hole--hole correlators\label{sec5}}

%The phase diagrams in Fig. \ref{fig2} and Fig. \ref{fig3} contain
%the whole information on the ground state of the system provided
%that a product ansatz with two $L\times L$ clusters and charges
%$Q_1$, $Q_2$ is adequate. Since we know the cluster ground states 
%in each charge sector from the analytical
%calculation in Ref. \cite{fl04} for $L=2$ and our numerical
%calculation for $L=4$, we can also determine the hole--hole correlators:
%\begin{eqnarray}
%C_{(x,y)}(Q_1) & = & \langle n_h(0,0)n_h(x,y)\rangle -
%\langle n_h(0,0)\rangle\langle n_h(x,y)\rangle
%\label{h-h-c}
%\end{eqnarray}
%where $n_h(0,0)$, $n_h(x,y)$ count the number of holes at sites
%$(0,0)$ and $(x,y)$. 

%On the $4\times 4$ cluster with periodic boundary conditions
%5 independent correlators can be arranged according to the distance vector ($x,y$).
%%which are listed in Table \ref{table3}.
%%\begin{table}[ht!]
%%\begin{tabular}{c|c|c|c|c|c}
%%$x-y$ & $(1,0)$ & $(1,1)$ & $(2,0)$ & $(2,1)$ & $(2,2)$
%%\\ \hline
%%correlator & $C_1$ & $C_{1,1}$ & $C_2$ & $C_{2,1}$ & $C_{2,2}$
%%\end{tabular}
%%\caption{
%%}
%%\label{table3}
%%\end{table}
%Owing to the nearest neighbour couplings and the periodic boundary
%conditions, the $2D$ cluster $4\times 4=16$ is equivalent to the
%$4D$ cluster $2\times 2\times 2\times 2=16$ with an additional
%rotational symmetry which leads to the identity (Dagotto et al. \cite{dagotto90};
%Fabricius et al. \cite{fabricius91})
%\begin{eqnarray}
%C_{2,0} & = & C_{1,1}\,.
%\end{eqnarray}
%Let us start with the regime
%\begin{eqnarray}
%2.2 & \leq & \alpha\leq 3.5,\hspace{2cm} \frac{1}{8}\leq\rho\leq 1\,. 
%\end{eqnarray}
%According to the phase diagram in Fig. \ref{fig3} the ground state
%contains clusters with charges $Q_1=2$, $Q_2=16$. The hole--hole
%correlators (\ref{h-h-c}) in the $Q_1=2$ cluster turn out to be
%negative. Fig. \ref{fig5} shows the modulus of these correlators:
%$|C_{1,0}(\alpha)|$ decreases with $\alpha$, whereas $|C_{2,0}(\alpha)|$, $|C_{2,1}(\alpha)|$
%and $|C_{2,2}(\alpha)|$ increase with $\alpha$. 
%%We interprete
%%these increasing ``long''-range correlations as a hint to
%%condensation of holes for large $\alpha$-values.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}[ht!]
%\centerline{\hspace{0.0cm}\includegraphics[width=8.0cm,angle=0]{Fig5.ps}
%%\centerline{\hspace{0.0cm}\includegraphics[width=6.0cm,angle=-90]{FIG_6_norm_resc3.ps}
%\hspace{0.0cm}}
%\caption{Moduli of correlators $C_{1,0},C_{1,1},C_{2,0},C_{2,1},C_{2,2}$ for the $4\times 4$
%cluster (p.b.c.) with charge $Q=2$.}
%\label{fig5}
%$\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%Next let us look for the hole--hole correlators in the $\alpha$-regime
%\begin{eqnarray}
%0.6 & \leq & \alpha\leq 2.1,\hspace{2cm} \frac{5}{8}\leq\rho\leq 1\,. 
%\end{eqnarray}
%According to the phase diagram in Fig. \ref{fig3} the ground state contains 
%clusters with charges $Q_1=10$, $Q_2=16$. Again all correlators are negative.
%Fig. \ref{fig6} shows the moduli of the correlators in a $Q=10$ cluster.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}[ht!]
%\centerline{\hspace{0.0cm}\includegraphics[width=8.0cm,angle=0]{Fig6.ps}
%%\centerline{\hspace{0.0cm}\includegraphics[width=6.0cm,angle=-90]{FIG_7_norm_resc3.ps}
%\hspace{0.0cm}}
%\caption{Moduli of correlators $C_{1,0},C_{1,1},C_{2,0},C_{2,1},C_{2,2}$ for the $4\times 4$
%cluster (p.b.c.) with charge $Q=10$.}
%\label{fig6}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%The ``long''-range correlators $|C_{2,0}|=|C_{1,1}|$, $|C_{2,1}|$, $|C_{2,2}|$
%are small in comparison with the nearest neighbour correlator $|C_{1,0}|$
%for $0.6\leq\alpha\leq 1.5$. In this regime the system prefers formation
%of hole pairs on neighbouring sites.

%%%%%%%%%%%%%%%%%%

%On the $4\times 4$ cluster with periodic boundary
%conditions 5 correlators can be arranged:
%\begin{table}[ht!]
%\begin{tabular}{lll}
%nearest neighbour (NN) & $x-y\,=\,(1,0)$ & $\quad C_1$ 
%\\ %\hline
%next-nearest neighbour (1NNN) & $x-y\,=\,(1,1)$ & $\quad C_{1,1}$\\
%2nd-nearest neighbour (2NNN) & $x-y\,=\,(2,0)$ & $\quad C_{2}$\\
%``knight's move'' (3NNN) & $x-y\,=\,(2,1)$ & $\quad C_{2,1}$\\
%diagonal (4NNN) & $x-y\,=\,(2,2)$ & $\quad C_{2,2}$\\
%\end{tabular}
%%\caption{
%%}
%\label{table3}
%\end{table}
%Owing to the nearest neighbour couplings and the periodic
%boundary conditions, the $2D$ cluster $4\times 4=16$ is
%equivalent to the $4D$ cluster $2\times 2\times 2\times 2=16$
%with an additional rotational symmetry which leads to the
%identity
%\begin{eqnarray}
%C_2 & = & C_{1,1}\,.
%\end{eqnarray}
%Let us start with the regime
%\begin{eqnarray}
%2.2< & \alpha & < 3.5\,.
%\end{eqnarray}
%According to the phase diagram in Fig. \ref{fig3} the ground
%state contains clusters with charges $Q_1=2$, $Q_2=16$. The
%hole--hole correlators (\ref{h-h-c}) shown in Fig. \ref{fig5} for $Q_1=2$
%are negative; $C_1(\alpha)$ increases with $\alpha$ whereas
%$C_2(\alpha)=C_{1,1}(\alpha)$, $C_{2,1}(\alpha),C_{2,2}(\alpha)$
%decrease with $\alpha$. The long range correlation effects can
%be interpreted as a condensation of holes.


%Next let us look for the hole--hole correlators in the regime
%\begin{eqnarray}
%0.6< & \alpha & < 2.2\,.
%\end{eqnarray}
%According to the phase diagram in Fig. \ref{fig3} the ground
%state contains clusters with charges $Q_1=10$, $Q_2=16$.

%The hole--hole correlators for $Q=10$ are presented in Fig. 
%\ref{fig6}. All correlators are negative. The ``long'' range
%correlators $C_2=C_{1,1}$, $C_{2,1},C_{2,2}$
%decrease with $\alpha$ and turn out to be rather small now,
%in comparison with the nearest neighbour correlator $C_1$,
%which increases with $\alpha$.










%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Summary and discussion\label{sec6}}

In this paper we made an attempt to describe phase separation
in terms of a product cluster ansatz for the ground state of the
$2D$ $t-J$ model. 
%The clusters are of size $L\times L$ ($L=2,4$)
%and the ground state on these clusters are computed analytically
%(for $L=2$) and numerically (for $L=4$). 

The analytic results for $2\times 2$ plaquette clusters reveal
already the gross features as depicted in the phase diagram
in Fig. \ref{fig2}.
For $\rho$ and $\alpha$ large enough the ground state is built up
from two types of plaquettes $(Q_h,Q_e)=(0,4),(2,4)$. 
The hole clusters carry a charge $Q_h$ which increases with
decreasing $\alpha$:
%The first --
%called ``hole clusters'' -- carries the holes. Here the charge $Q_1$ is
\begin{eqnarray}
Q_h=0 \,\,\,\, &  & \hspace{5mm}\alpha(2)=4\sqrt{\frac{2}{3}}<\alpha 
\hspace{1.8cm}0\leq\rho\leq 1
 \label{regime_f1}\\
Q_h=2  \,\,\,\, &  & \hspace{5mm}\alpha(3)=\frac{2}{\sqrt{21}}\leq\alpha\leq \alpha(2)
\hspace{0.55cm}\frac{1}{2}\leq\rho\leq 1
\label{regime_f2}
\end{eqnarray}
where $\rho_1=Q_h/N$ denotes the charge density in the hole cluster.
The second type of plaquettes is completely occupied
with electrons $Q_e=4$.

In the regimes (\ref{regime_f1}) and (\ref{regime_f2}) the ground state
energy per site (\ref{ge_Q1Q2}) is linear in $\rho$ in the respective intervals.
%The interaction between neighbouring plaquettes -- as computed in first order
%perturbation theory in Section \ref{sec3} -- favours the condensation of
%plaquettes with maximal charge $Q_e=4$. This is the signature for
%phase separation.
The lower bounds in the $\alpha$-intervals listed in (\ref{regime_f1}), 
(\ref{regime_f2}) define two points on the boundary (\ref{large_rho})
for phase separation:
\begin{eqnarray}
\alpha_p(\rho_1=0)=4\sqrt{\frac{2}{3}}, \,\,\,\,\,\,& & \alpha_p(\rho_1=\frac{1}{2})=
\frac{2}{\sqrt{21}}\,.\label{R0}
\end{eqnarray}

The numerical results -- obtained from a product ansatz with $4\times 4$
clusters lead to the phase diagram in Fig. \ref{fig2}(B) similar to Fig. \ref{fig2}(A)
for $2\times 2$ clusters.
In each rectangular domain the cluster charges $Q_1,Q_2$ result from a
minimization of the ground state energy per site (\ref{ge_Q1Q2}) at fixed
charge density $\rho$ (in the infinite system). Phase separation is observed
in the upper part of Fig. \ref{fig2}(B)
with cluster charges $(Q_h,Q_e)=(0,16),(2,16),(6,16),(10,16)$
%, where the electron clusters
%have maximal charge $Q_e=16$. Holes occur in the clusters with
%charge $Q_1=Q_h<16$. 
The charge $Q_h$ of the hole clusters changes with the
spin exchange coupling $\alpha$
\begin{eqnarray}
Q_h=\,\,0 &  & \hspace{5mm}\alpha(2)=3.5<\alpha,\hspace{3.1cm} 0\leq\rho\leq 1
\hspace{0.4cm}\label{R1}\\
Q_h=\,\,2 &  & \hspace{5mm}\alpha(6)=2.2<\alpha<\alpha(2), \hspace{1.7cm}
\frac{1}{8}\leq\rho\leq 1\\
Q_h=\,\,6 &  & \hspace{5mm}\alpha(10)=2.1<\alpha<\alpha(6), \hspace{1.7cm}
\frac{3}{8}\leq\rho\leq 1\\
Q_h=10 &  & \hspace{5mm}\alpha(14)=0.6<\alpha<\alpha(10), \hspace{1.5cm}
\frac{5}{8}\leq\rho\leq 1\label{R4}
\end{eqnarray}

The ground state energy per site (\ref{ge_Q1Q2}) is linear in the $\rho$
intervals listed in (\ref{R1})-(\ref{R4}).
The lower bounds in the $\alpha$ intervals yield 4 points on the boundary
curve (\ref{large_rho}) for phase separation shown in Table \ref{table_n2}:

\begin{table}[ht!]
\begin{center}
\caption{Points on the boundary curve (\ref{large_rho}) for phase separation
derived from a $4\times 4$ cluster.}
\begin{tabular}{c|c|c|c|c}
%$\rho_1=\frac{Q_h}{16}$ & $0$ & $\frac{1}{8}$ & $\frac{3}{8}$ &
%$\frac{5}{8}$  \\ \hline
$\,\rho_1=Q_h/16\,$ & $\,0\,$ & $\,1/8\,$ & $\,3/8\,$ & $\,5/8\,$ \\ \hline
$\alpha_p(\rho_1)$ & 3.5 & 2.2 & 2.1 & 0.6
\end{tabular}
\label{table_n2}
\end{center}
\end{table}

The product ansatz (cf. (\ref{prod_ansatz}) for plaquettes) with isolated
cluster ground states is highly degenerate. Each distribution of electron
and hole clusters over the whole lattice leads to the same ground state
energy, if we neglect the interaction between neighbouring clusters 
(cf. Fig.\ref{fig3n}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[ht!]
\centerline{\hspace{0.1cm}\includegraphics[width=5.0cm,angle=0]{Fig3.eps}
\hspace{0.1cm}}
\caption{Interaction between neighbouring plaquettes with charges $Q_1,Q_2$
shown for a pair of nearest neighbour sites $x$ and $x'$.}
\label{fig3n}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let us denote the interaction energy between neighbouring clusters with
charges $Q_2=L^2$, $Q_1=Q_h$ by $W(Q_1,Q_2)$ (cf. Fig.\ref{fig3n}). If the
difference
\begin{eqnarray}
\Delta & = & W(L^2,L^2)+W(Q_h,Q_h)-2W(Q_h,L^2)<0\label{ineq_Delta}
\end{eqnarray}
is negative, the ground state prefers phase separation in the following
sense. The numbers $N(L^2,L^2)$, $N(Q_h,Q_h)$ of identical neighbouring
clusters is maximal
\begin{eqnarray}
\frac{N(L^2,L^2)}{N}=2\frac{N(L^2)}{N} & ,\quad &
\frac{N(Q_h,Q_h)}{N}=2\frac{N(Q_h)}{N}\quad\quad\mbox{for\quad}N\rightarrow\infty
\label{Cfg_1}
\end{eqnarray}
where $N(L^2)$ and $N(Q_h)$ are the numbers of electron and hole clusters,
respectively. As a consequence the number of neighbouring clusters $N(Q_h,L^2)$
with different charges is minimal:
\begin{eqnarray}
\frac{N(Q_h,L^2)}{N} & \rightarrow & 0\quad\quad\mbox{for\quad}N\rightarrow\infty\,.
\label{Cfg_2}
\end{eqnarray}
Within the product ansatz (\ref{prod_ansatz}) with plaquettes of charge
$Q_2=L^2$, $Q_1=Q_h$ the interaction energies turn out to be
\begin{eqnarray}
W(Q_1,Q_2) =  \sum_x\langle Q_1,Q_2|h^{(x,x')}|Q_1,Q_2\rangle
 & = & -\frac{\alpha}{32}Q_1Q_2,  \;\;\;\; (L=2),
\label{relevant_ME}
\end{eqnarray}
provided that the cluster charges are even and the total cluster spins are zero.
In this case the spin matrix elements
\begin{eqnarray}
\langle Q_1|{\bf S}_x|Q_1\rangle=\langle Q_2|{\bf S}_{x´}|Q_2\rangle & = & 0
\end{eqnarray}
and the hopping contributions arising in (\ref{element_h_plaq}) vanish
for $L^2-Q_h\geq 2$. Eq. (\ref{relevant_ME}) implies that the inequality
(\ref{ineq_Delta}) is valid and the ground state configuration (\ref{Cfg_1}),
(\ref{Cfg_2}) with two big clusters of charge density $\rho_2=1$, $\rho_1=Q_h/L^2$
is selected out. According to (\ref{R1})-(\ref{R4}) the necessary condition
$L^2-Q_h\geq 2$ is satisfied for $\alpha>0.6$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%abdollah-new

The phase diagrams in Fig. \ref{fig2}(A) and Fig. \ref{fig2}(B) contain
the whole information on the ground state of the system provided
that a product ansatz with two $L\times L$ clusters and charges
$Q_1$, $Q_2$ is adequate. Since we know the cluster ground states 
in each charge sector from the analytical
calculation in Ref. \cite{fl04} for $L=2$ and our numerical
calculation for $L=4$, we can also determine the hole--hole correlators:
\begin{eqnarray}
C_{(x,y)}(Q_1) & = & \langle n_h(0,0)n_h(x,y)\rangle -
\langle n_h(0,0)\rangle\langle n_h(x,y)\rangle
\label{h-h-c}
\end{eqnarray}
where $n_h(0,0)$, $n_h(x,y)$ count the number of holes at sites
$(0,0)$ and $(x,y)$. 

On the $4\times 4$ cluster with periodic boundary conditions
5 independent correlators can be arranged according to the distance vector ($x,y$).
%which are listed in Table \ref{table3}.
%\begin{table}[ht!]
%\begin{tabular}{c|c|c|c|c|c}
%$x-y$ & $(1,0)$ & $(1,1)$ & $(2,0)$ & $(2,1)$ & $(2,2)$
%\\ \hline
%correlator & $C_1$ & $C_{1,1}$ & $C_2$ & $C_{2,1}$ & $C_{2,2}$
%\end{tabular}
%\caption{
%}
%\label{table3}
%\end{table}
%Owing to the nearest neighbour couplings and the periodic boundary
%conditions, the $2D$ cluster $4\times 4=16$ is equivalent to the
%$4D$ cluster $2\times 2\times 2\times 2=16$ with an additional
%rotational symmetry which leads to the identity (Dagotto et al. \cite{dagotto90};
%Fabricius et al. \cite{fabricius91})
%\begin{eqnarray}
%C_{2,0} & = & C_{1,1}\,.
%\end{eqnarray}
%Let us start with the regime
Let us look at the following regime,
\begin{eqnarray}
2.2 & \leq & \alpha\leq 3.5,\hspace{2cm} \frac{1}{8}\leq\rho\leq 1\,. 
\end{eqnarray}
According to the phase diagram in Fig. \ref{fig2}(B) the ground state
contains clusters with charges $Q_1=2$, $Q_2=16$. The hole--hole
correlators (\ref{h-h-c}) in the $Q_1=2$ cluster turn out to be
negative.  However, the modulus of 
$|C_{1,0}(\alpha)|$ decreases with $\alpha$, whereas $|C_{2,0}(\alpha)|$, $|C_{2,1}(\alpha)|$
and $|C_{2,2}(\alpha)|$ increase with $\alpha$. 
We interprete
these increasing ``long''-range correlations as a hint to
condensation of holes for large $\alpha$-values.

Next let us look for the hole--hole correlators in the $\alpha$-regime
\begin{eqnarray}
0.6 & \leq & \alpha\leq 2.1,\hspace{2cm} \frac{5}{8}\leq\rho\leq 1\,. 
\end{eqnarray}
According to the phase diagram in Fig. \ref{fig2}(B) the ground state contains 
clusters with charges $Q_1=10$, $Q_2=16$. Again all correlators are negative.
The ``long''-range correlators $|C_{2,0}|=|C_{1,1}|$, $|C_{2,1}|$, $|C_{2,2}|$
are small in comparison with the nearest neighbour correlator $|C_{1,0}|$
for $0.6\leq\alpha\leq 1.5$. In this regime the system prefers formation
of hole pairs on neighbouring sites.

%end-abdollah-new
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%In the framework of the product cluster ansatz the mechanism for phase
%separation is explained as follows: ``Hole'' and ``electron''-clusters
%carry charges $Q_h<Q_e=L^2$ and cluster spin 0. 
%%First order perturbation 
%%theory does not allow the exchange of a neighbouring electron and hole
%%cluster, which would demand the hopping of $L^2-Q_h\geq 2$ electrons.
%%Therefore, holes are confined in the hole clusters and first order
%%perturbation theory prefers neighbouring hole clusters -- and thereby
%%as well neighbouring electron clusters. This leads to the factorization
%%(\ref{psiprod}) of the ground state into two big clusters with charge
%%density $\rho_1=Q_h/L^2$ and 1 respectively.
%%The argument rests on a perturbative treatment of the interaction
%%between neighbouring clusters and it is rather safe if the difference 
%%$L^2-Q_h$ is large enough, which happens for large $\alpha$ ($\alpha>0.6$).
%%
%% Aenderung:
%%
%First order perturbation theory in the hopping of one electron does not
%allow the exchange of a neighbouring electron and hole cluster if the
%charge difference $L^2-Q_h\geq 2$. Therefore, in this approximation
%holes are confined in the hole cluster. The approximation should be
%safe for large charge differences $L^2-Q_h$, which means for large
%$\alpha$-values as can be seen from (\ref{R1})-(\ref{R4}).
%Here the results (\ref{R0}) from $2\times 2$ and Table \ref{table_n2} from
%$4\times 4$ clusters coincide.

%First order perturbation theory yields an attractive interaction
%between clusters with the same charge suggesting a ground state with 
%two big clusters with charge density $\rho_1=Q_h/L^2$ and 1, respectively,
%as conjectured in (\ref{psiprod}).
%%
%% Ende des Einschubs
%%

On the other hand
for $\alpha<0.6$ and $\rho>0.75$, we observe rapid changes of the ground
state with $\rho$ and $\alpha$ as is indicated by the numerous small
rectangular domains in the left upper part in Fig. \ref{fig2}(B). In this
regime holes are no longer confined as can be seen directly from the
product ansatz with $2\times 2$ plaquettes and charges $Q_h=3$, $Q_e=4$
[cf. upper left part in Fig. \ref{fig2}(A)]. Here, the hopping matrix elements
(\ref{element_h_plaq}) are active already 
in first order perturbation theory
and allow for the exchange of hole ($Q_h=3$) and electron ($Q_e=4$)
clusters. In principle, the holes can now hop over the whole lattice
and destroy thereby phase separation. The distribution of the holes
can be determined only from a precise computation of the ground state
in the low doping $\delta$, low $\alpha$ regime.

%The controversary points of view on phase separation -- mentioned in 
%the introduction and in refs. \cite{emery90}-\cite{calandra98}
%and \cite{putikka92}-\cite{fehske91} -- reflect the fact,
%that an unambiguous determination of the rapidly changing ground state in
%the low $\delta$, low $\alpha$ regime meets severe problems. It is
%well-known, that the Fermi statistics for the electrons produce rapid
%changes in the ground state quantum numbers, e.g. the total ground state
%spin $S(\rho=Q/N,\alpha)$ for low doping $\delta=1-\rho$ and low $\alpha$.

%In the Nagaoka sector\cite{nagaoka66} with 1 hole ($Q=N-1$), the total
%spin $S(\rho=1-1/N,\alpha=0)=Q/2$ is maximal, i.e. the ground state is
%ferromagnetic. For $Q<N-1$, $S(\rho=Q/N,\alpha=0)$ changes with the charge
%$Q$ in an ``erratic'' way, which does not allow to extract the thermodynamical
%limit from finite cluster results.\cite{riera89} 

%Switching on the spin exchange coupling $\alpha$ one obtains a rapid transition
%of the ground state spin to zero (for $Q$ even, cf. ref. \cite{eisenberg02}).

%Phase separation has been investigated recently by Boninsegni\cite{boninsegni01,
%boninsegni02} in the bosonic version of the $t-J$ and $t-J_z$ model. Owing to the
%absence of the ``sign'' problem simulations can be done on rather large lattices
%($32\times 32$). Boninsegni found a uniform phase -- i.e. no phase separation
%in the bosonic version of the $t-J$ model for $0<\alpha<1.5$.

%The ground state spin $S(\rho,\alpha)$ in the bosonic version of the $t-J$ model
%\cite{fl05} has been found to be maximal $S(\rho,\alpha)=Q/2$ in a certain
%domain $0\leq\alpha\leq\alpha_F(\rho)$ of the $\rho-\alpha$ plane. For $\alpha>\alpha_F(\rho)$
%the ground state spin decreases step by step until it reaches the minimal value
%($S=0$ for $Q$ even, $S=1/2$ for $Q$ odd).

%From these findings we conclude that the question whether or not phase separation 
%survives in the experimentally accessible low doping ($\delta=1-\rho$), low
%$\alpha$--regime depends on the very details of the ground state in the
%thermodynamical limit. They can be studied in the $\rho-\alpha$ dependences 
%of the ground state quantum numbers, e.g. the total spin $S(\rho,\alpha)$.
%Within the framework of a hard core model -- like the $t-J$ model -- the 
%relative weight $\alpha=J/t$ of the hopping ($t$) and antiferromagnetic
%spin exchange coupling ($J$) is crucial. The latter favours phase separation
%and acts in the same manner for fermions and for bosons. On the other hand,
%the hopping term acts in a different way for fermions and for bosons.
%For bosons it favours ferromagnetism and therefore competes with the
%antiferromagnetic spin exchange. At the end this leads to a smaller domain
%in the $\rho-\alpha$ plane for phase separation. For fermions the hopping
%term generates the ``sign'' problem and leads to the rapid and irregular
%changes in the ground state quantum numbers like the total spin.                                                                                                    
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%It was demonstrated in [\onlinecite{fl05}], that in the bosonic version
%of the $t-J$ model the ground state spin is $S(\rho,\alpha)=Q/2$ %(**??**)
%is maximal in a certain domain $0<\alpha<\alpha_F(\rho)$ of the $\rho-\alpha$
%plane. Phase separation has been investigated recently by Boninsegni
%\cite{boninsegni01,boninsegni02} in the bosonic version of the $t-J$ and
%$t-J_z$ model. Owing to the absence of the ``sign'' problem simulations
%can be done on rather large lattices ($32\times 32$). Of course, we expect
%that the physical properties -- e.g. the boundary for the phase separation --
%change with the statistics. Boninsegni found a uniform phase -- i.e. no
%phase separation -- in the bosonic version of the $t-J$ model for $0<\alpha<1.5$.
%In the $t-J_z$ version the uniform phase shrinks with decreasing $\alpha$:
%at $\alpha=0.4$ $\rho\geq 0.75$, at $\alpha=0.2$ $\rho>0.85$.


%\vspace{2.0cm}



%In these domains, holes are confined in the hole clusters with even charge
%$Q_1$ and total spin zero, since first order perturbation theory for the interaction
%between neighbouring clusters does not allow the exchange of ``hole'' and
%``electron'' clusters.
%This is not the case for small $\alpha$-values shown in the left upper
%parts in Fig. \ref{fig2} and Fig. \ref{fig3}.
%E.g. in Fig. \ref{fig2} the ground state is built up from plaquettes with
%charges $Q_1=3$, $Q_2=4$ for $0\leq\alpha\leq\alpha_1=2/\sqrt{21}$,
%$3/4\leq\rho\leq 1$. Here the holes can hop between neighbouring plaquettes.




%%(1) For $\rho$ and $\alpha$ large enough the ground state is
%%built up from plaquettes with ``separated'' charges
%%\begin{eqnarray}
%%(Q_1,Q_2) & = & (0,4),\,\,(2,4)\,.
%%\label{Q1_Q2_Lr24}
%%\end{eqnarray}
%%(2) The interaction between neighbouring plaquettes --as computed
%%in first order perturbation theory (Section \ref{sec3})-- favours
%%the condensation of plaquettes with maximal charge $Q_1=4$ -- the
%%signature for phase separation.

%%%(3) There are indications for a charge transfer gap at $\rho=1/2$
%%%for $2/\sqrt{21}\leq\alpha\leq 4\sqrt{3/2}$ in zeroth order perturbation
%%%theory which decreases with $\alpha$ (Fig. \ref{fig4}).

%%%First order perturbation theory suppresses the width of the gap,
%%%such that it appears to close at a smaller $\alpha$ value
%%%($\alpha\simeq 2.4$).

%%The numerical results for a $4\times 4$ cluster with periodic 
%%boundary conditions (p.b.c.) lead to a phase diagram with a higher
%%resolution of the charge distribution in the ground state as shown
%%in Fig. \ref{fig3}.

%%By construction the ground state in the thermodynamical limit is
%%built up from a product ansatz with two types of $4\times 4$ clusters,
%%the first one with charge $Q_1$, the second one with $Q_2$. In each
%%rectangular domain of the phase diagram (Fig. \ref{fig3}), the cluster 
%%charges $Q_1,Q_2$ are chosen such that the ground state energy is
%%minimal. As a consequence one observes again for $\rho$ and $\alpha$
%%large enough ``separated'' cluster charges:
%%\begin{eqnarray}
%%(Q_1,Q_2) & = & (0,16),\,\,(2,16),\,\,(6,16),\,\,\,(10,16),\,\,(14,16)\,.
%%\nonumber\\ \label{Q1_Q2_L4}
%%\end{eqnarray}
%%%Moreover, there are indications  for a charge transfer gaps at
%%%$\rho=5/8, 3/8$ for $0.6\leq\alpha\leq 2.2$.
%%%Again the gap width (Fig. \ref{fig4}) decreases with $\alpha$.

%%%hier_neu
%%A comparison of Figs. \ref{fig2} and \ref{fig3}, obtained from
%%$L\times L$ clusters ($L=2,4$), reveals some striking
%%similarities and differences.

%%\begin{itemize}

%%\item[a)]
%%Complete phase separation with two clusters of charge density
%%$\rho_1=0$, $\rho_2=1$ occurs for
%%\begin{eqnarray}
%%L=2,\,\,\alpha>4\sqrt{2/3}, & & Q_1=0,\,\,Q_2=4\nonumber\\ 
%%L=4,\,\,\alpha>3.5,\hspace{0.6cm} & & Q_1=0,\,\,Q_2=16\,.\nonumber 
%%\end{eqnarray}
%%The chemical potential (\ref{mu_rho}) is constant in $\rho$
%%in the whole interval $0\leq\rho\leq 1$ and given by the ground
%%state energy per site of the $L\times L$ cluster with maximal
%%charge $Q=L^2$.

%%\item[b)]
%%The large domain in phase space with $2\times 2$ clusters and 
%%charges $Q=4,2$:
%%\begin{eqnarray}
%%\frac{1}{2}<\rho<1 & \quad & \frac{2}{\sqrt{21}}\leq\alpha\leq 4\sqrt{\frac{2}{3}}
%%\nonumber
%%\end{eqnarray}
%%is divided into three parts in the phase space of $4\times 4$ clusters
%%and charge densities $\frac{5}{8}<\rho<1$:
%%\begin{eqnarray}
%%\hspace{0.0cm}Q=16,10\hspace{-0.17cm} & \quad & \hspace{0.0cm}0.6\leq\alpha\leq\alpha(10)=2.1\nonumber\\
%%Q=16,6 & \quad & 2.1\leq\alpha\leq\alpha(6)\hspace{0.2cm}=2.2\nonumber\\
%%Q=16,2 & \quad & 2.2\leq\alpha\leq\alpha(2)\hspace{0.2cm}=3.5\nonumber\,.
%%\end{eqnarray}



%%%For $0.6\leq\alpha\leq 2.2$ the ground state contains clusters with
%%%charges $Q=16,10$.

%%%For $2.2\leq\alpha\leq 3.5$ the ground state contains clusters with
%%%charges $Q=16,2$.

%%%We suggest that the emergence of the second cluster with charge
%%%$Q_1=2$ for $L=2$ and $Q_1=10$ for $L=4$ originates from a charge
%%%transfer gap at $\rho_1=Q_1/L^2$.
%%%The position of $\rho_1$ is size dependent, but the charge transfer
%%%gap might survive in the thermodynamical limit.
%%%In Fig. \ref{fig4} we show the variation of the gap width (\ref{Delta_Q2})
%%%with $\alpha$ in the two cases $L=2$, $L=4$. Note that the gap
%%%width vanishes  for $\rho=5/8$ and $2.2\leq\alpha\leq 3.5$ in the
%%%phase diagram of the $4\times 4$ clusters.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%\begin{figure}[ht!]
%%%\centerline{\hspace{0.0cm}\includegraphics[width=8.0cm,angle=0]{Fig4.ps}
%%%\hspace{0.0cm}}
%%%\caption{Variation of gap width $\Delta(Q_2,\alpha)$ (\ref{Delta_Q2}) at
%%%$\rho=Q_2/L^2$ for two $L\times L$ systems with $L=2$, $L=4$.}
%%%\label{fig4}
%%%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%On the other hand, the gap width for $\rho=1/2$, $2\leq\alpha\leq 4\sqrt{2/3}$
%%%(solid curve in Fig. \ref{fig4}) does not vanish in the zeroth order
%%%perturbation theory for plaquette clusters. First order contributions
%%%--as they follow from (\ref{eps_rho_W}), (\ref{eps_rho_W_II}) and (\ref{W_Q1Q2})
%%%suppress the gap width by ($-t''\alpha/4$), such that its zero is shifted
%%%to $\alpha=2.4$ for $t''=1$.

%%%c) The jumps in the chemical potential (cf. Fig.\ref{fig4}) 

%%%at $\rho=1/2$
%%%for $L=2$; $\frac{2}{\sqrt{21}}\leq\alpha\leq 4\sqrt{\frac{2}{3}}$
%%%\,\,[(\ref{Delta_1_2}),(\ref{Delta_2_2})]

%%%at $\rho=\frac{5}{8},\frac{3}{8}$ for $L=4$; $0.6\leq\alpha\leq 2.2$
%%%\hspace{0.5cm}[(\ref{Delta_1_10})-(\ref{Delta_3_6})]
%%%can be interpreted as signatures for a charge transfer 
%%%gap, which might survive the thermodynamical limit.

%%\item[c)]
%Apart from these common gross features, the phase diagrams obtained
%from $2\times 2$ and $4\times 4$ clusters differ from each other, since
%the larger cluster size $4\times 4$ allows for a higher resolution of
%the charge distribution in the ground state. This is relevant in particular
%in the experimentally accessible region for phase separation with $\rho>0.75$, $\alpha<0.5$.
%Here we observe a rapid change of the ground state with $\rho$ and $\alpha$
%as is indicated by the numerous ``small'' rectangular domains in the left
%upper part of the phase diagram in Fig. \ref{fig3}. We think, that we meet 
%here a principal problem, which cannot be solved ``simply'' going to larger
%clusters -- irrespective of the numerical problems we encounter in trying
%to do so.

%%\end{itemize}


%It might happen that the rapid changes of the ground states do not allow
%to extract the ground state properties in the thermodynamical limit neither
%from finite clusters nor from extrapolations, which start from a regime
%($\alpha>0.5$, $\rho<0.75$), where the ground state appears to be more stable.

%%The instability of the ground state at $\alpha=0$ has been studied intensively
%%in the framework of the Hubbard model with infinite Coulomb repulsion
%%($U\rightarrow\infty$). It has been found, that the total spin $S(Q_2)$ of the
%%ground state changes in an ``erratic'' way with the total charge, which does
%%not allow to extract the thermodynamical limit from finite cluster results
%%except for the Nagaoka case [\onlinecite{nagaoka66}] 
%%with one hole, i.e. $Q=N-1$,
%%where $S(Q=N-1)=Q/2$ is maximal (i.e. ferromagnetic). If we switch on the
%%antiferromagnetic spin exchange coupling $\alpha$, the ground state (total)
%%spin for $N=Q-1$ changes rapidly from its maximal to its minimal value within
%%a tiny interval $0\leq\alpha < 0.1$.\cite{fl05}



%In this context
%the change of the total ground state spin $S(\rho=Q/N,\alpha)$ as function of
%the hole density $\delta=1-\rho$ and the exchange coupling $\alpha$ is of
%special interest. It has been studied intensively with the following results:

%\begin{enumerate}

%\item
%In the Nagaoka sector \cite{nagaoka66} with 1 hole ($Q=N-1$)
%\begin{eqnarray}
%S\Big(\rho=1-\frac{1}{N},\alpha=0\Big) & = & \frac{Q}{2}\nonumber
%\end{eqnarray}
%is maximal, i.e. the ground state is ferromagnetic.

%\item
%For $Q<N-1$, $S(Q,\alpha=0)$ changes with the total charge $Q$ in an
%''erratic'' way, which does not allow to extract the thermodynamical
%limit from finite cluster results (Riera, Young [\onlinecite{riera89}],
%Fehske et al. [\onlinecite{fehske91}]).
%It was shown in Ref. [\onlinecite{fl05}] that this property is intimately
%related to the Fermi statistics of electrons in a hard-core model 
%like the $t-J$ model studied here.

%\item
%On the other hand, a maximal ground state spin $S(\rho=Q/N,\alpha)=Q/2$ -- i.e.
%ferromagnetism -- is realized for all fillings $0<\rho\leq 1$ in a certain
%domain $0\leq\alpha\leq\alpha_F(\rho)$ of the phase space for hard-core
%bosons [\onlinecite{fl05}].

%\end{enumerate}


%Owing to the absence of the ``sign problem'', simulations of the bosonic version 
%of the $t-J$ and $t-J_z$ model can be done on rather large systems
%(up to $32\times 32$). They have been performed with the Greenfunction Monte
%Carlo Method (GFMC) by M. Boninsegni.\cite{boninsegni01,boninsegni02}
%He found a uniform phase -- i.e. no phase separation -- in the $t-J$ version
%for $\alpha<1.5$. In the $t-J_z$ version this uniform phase shrinks with
%decreasing $\alpha$: at $\alpha=0.4$ $\rho\ge 0.75$; $\alpha=0.2$ $\rho> 0.85$.

%Moreover, the simulations in the bosonic $t-J$, $t-J_z$ models did not provide
%any evidence for the formation of hole stripes. This is quite in contrast to
%the results of Ref. [\onlinecite{white98}], obtained with the fermionic $t-J$
%model on ladder systems.

%Our results on hole--hole correlators in Section \ref{sec5} indicate that
%formation of hole pairs on neighbouring sites is dominant in the regime
%$0.6\leq\alpha\leq 1.5$. For larger $\alpha$-values ($\alpha>1.5$) hole--hole
%correlators at larger distances become more and more important.


%%due to
%%the fact that the larger cluster size allows for a higher resolution
%%of the charge distribution in the ground state. This means in particular,
%%that the appropriate description of the charge (or hole) distribution
%%in the ground state for the physically interesting domain
%%($\rho\rightarrow 0.75$, $\alpha<0.5$ demands large clusters.




%%In Section \ref{sec5} we have computed the hole--hole  correlators
%%in a product ansatz with cluster charges (\ref{Q1_Q2_L4}). ``Long''
%%range correlators can be observed in a $4\times 4$ cluster with
%%charge $Q=2$ and $2.2\leq\alpha\leq 3.5$, indicating the condensation
%%of holes. In contrast, in a $4\times 4$ cluster with charge $Q=10$, 
%%$0.6\leq\alpha\leq 2.2$ hole--hole correlators at neighbouring sites
%%are dominant.





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\acknowledgments

%We are indebted to M. Karbach for a critical
%reading of the manuscript. 

%abdollah ' the follwoing is added
%A. L. would like to thank the physics department
%of Wuppertal university for their warm and nice hospitality 
%during his stay in Wuppertal.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%    Tabels  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{references}
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%\bibitem{vojta01}M. Vojta, A. H\"ubsch, R. M. Noack,
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%\bibitem{riera99}J. Riera, D. Poilblanc, E. Dagotto,
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%\bibitem{fle03}A. Fledderjohann, M. Karbach, K.-H. M\"utter, R. M. Wiessner,
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%\bibitem{lieb61}E. Lieb, T. Schultz, and D. Mattis, Annals of Phys. {\bf 16},
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%\bibitem{giagliardini98}P. Giagliardini, S. Haas, T. M. Rice,
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%Phys. Rev. Lett. {\bf 78}, 1984 (1997);
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%\bibitem{kagan99}M. Y. Kagan, S. Haas, T. M. Rice,
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%\bibitem{troyer96}M. Troyer, H. Tsunetsugu and T. M. Rice,
%Phys. Rev. B\,{\bf 53}, 251 (1996)
%properties of lightly doped t-J two-leg ladders



%\bibitem{oitmaa99}J. Oitmaa, C. J. Hamer, Z. Weihong, 
%Phys. Rev. B\,{\bf 60}, 16364 (1999)
%%series-expansion studies on the t-J two-leg ladder

%\bibitem{zheng02}W. Zheng, C. J. Hamer, J. Oitmaa, 
%Phys. Rev. B\,{\bf 65}, 165117 (2002)
%%series expansions for variable electron density

%\bibitem{kawaguchi02}A. Kawaguchi, A. Koga, K. Okunishi, N. Kawakami,
%cond-mat/0205635
%hole-doping effects on a frustrated spin ladder

%\bibitem{bulirsch64}R. Bulirsch, J. Stoer, Num. Math.\,{\bf 6}, 413 (1964)
%Fehlerabsch\"atzungen und Extrapolation mit rationalen Funktionen bei
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%\bibitem{shastry81}B. S. Shastry, B. Sutherland, Physica {\bf 108},
%B, 1308 (1981)


%\vspace{0.5cm}

%\bibitem{poilblanc03}D. Poilblanc, D. J. Scalapino, S. Capponi,
%Phys. Rev. Lett. {\bf 91}, 137203 (2003)
%superconducting gap for a two-leg t-J ladder




%\bibitem{eisenber02}E. Eisenberg, R. Berkovits, D. A. Huse,
%B. L. Altshuler,
%Phys. Rev. B\,{\bf 65}, 134437 (2002)
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Soderholm L 1988
{\it Phys. Rev. B.} {\bf 38} 11337 
%superconducting phase of La_2CuO_4+delta: a superconducting 
%composition resulting from phase separation

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Hammel P C, et. al 1990 {\it Phys. Rev. B.}{\bf 42} 6781

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\bibitem{cho93}
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\bibitem{emery90}Emery V J, Kivelson S A and Lin H Q 1990
{\it Phys. Rev. Lett.} {\bf 64} 475
%phase separation in the t-J model 

\bibitem{hellberg95}Hellberg C S and Manousakis E 1995
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%Two-dimensional t-J model at low electron density

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\bibitem{hellberg00}Hellberg C S and Manousakis E
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%\bibitem{hellberg00a}C. S. Hellberg, E. Manousakis,
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%\bibitem{fl05}Fledderjohann A, Langari A, M\"uller-Hartmann E and 
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%cond-mat/0111320
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%\bibitem{white98}S. R. White, D. J. Scalapino,
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%hierr





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%\bibitem{poilblanc95}D. Poilblanc,
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%\bibitem{riera02}J. A. Riera,
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%\vspace{0.5cm}

%\bibitem{gagliardini98}P. Gagliardini, S. Haas, T. M. Rice,
%Phys. Rev. B\,{\bf 58}, 9603 (1998)
%generalization of the Luttinger theorem for fermionic ladder systems

%\bibitem{eisenberg02}E. Eisenberg, E. H. Lieb,
%Phys. Rev. Lett.\,{\bf 89}, 220403 (2002)
%polarization of interacting bosons with spin







\vspace{0.5cm}

%\bibitem{fl03}A. Fledderjohann, A. Langari, K.-H. M\"utter,
%Eur. Phys. J. B\,{\bf 36}, 193 (2003)
%Charge density plateaux and insulating phases in the $t-J$
%model with ladder geometry



\end{thebibliography}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%        End Of Document      %%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



cond-mat0605229.extracted_bib
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%\bibitem{shastry81}B. S. Shastry, B. Sutherland, Physica {\bf 108},
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%\vspace{0.5cm}

%\bibitem{poilblanc03}D. Poilblanc, D. J. Scalapino, S. Capponi,
%Phys. Rev. Lett. {\bf 91}, 137203 (2003)
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%Two-dimensional t-J model at low electron density

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%Phase Separation at all Interaction Strengths in the t-J model

\bibitem{hellberg99}Hellberg C S and Manousakis E
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%stripes and the t-J model

\bibitem{hellberg00}Hellberg C S and Manousakis E
2000 {\it Phys. Rev. B.}{\bf 61} 11787
%green's-function Monte Carlo for lattice fermions:
%Application to the t-J Model

%\bibitem{hellberg00a}C. S. Hellberg, E. Manousakis,
%Phys. Rev. Lett. {\bf 84}, 3022 (2000)
%Hellberg and Manousakis Reply to the Comment by 
%Steven R. White and D. J. Scalapino.

\bibitem{putikka92}Putikka W O, Luchini  M U and Rice T M 1992
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%aspects of the phase diagram of the two-dimensional t-J model

\bibitem{putikka94}Putikka W O, Glenister  R L, Singh R R P and Tsunetsugu H 1994 {\it Phys. Rev. Lett.} {\bf 73} 170
%Indication of Spin-Charge Separation in the two-dimensional t-J model

\bibitem{rommer00}Rommer S, White S R and Scalapino D J 2000 {\it Phys. Rev. B.} {\bf 61} 13424
%phase separation in t-J ladders

\bibitem{valenti92}Valenti R and Gros C 1992
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\bibitem{dagotto92}Dagotto E, Moreo A, Ortolani F, Poilblanc D
and Riera J 1992
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%static and dynamical properties of doped hubbard clusters

%\bibitem{fabricius90}
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%hidden symmetries in the one-dimensional antiferromagnetic heisenberg model



\bibitem{dagotto98}Dagotto E, Martins G B, Riera J,
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%diagonalization in reduced hilbert spaces using a systematically
%improved basis: application to spin dynamics in lightly doped ladders

\bibitem{calandra98}Calandra M, Becca  F and Sorella S 1998
{\it Phys. Rev. Lett.} {\bf 81} 5185
%Charge Fluctuations close to Phase Separation in 
%the two-dimensional t-J model

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%{\it Phys. Rev. B.}{\bf 44} 8473
%hole dynamics in a strongly correlated two-dimensional
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\bibitem{coester58}Coester F 1958 {\it Nucl. Phys.} {\bf 7} 421;
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\bibitem{gros93}Gros C and Valenti R 1993 {\it Phys. Rev. B} {\bf 48} 418;
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\bibitem{potthoff03}Potthoff M 2003 {\it Eur. Phys. J. B} {\bf 32} 429;
%self-energy-functional approach to systems of correlated electrons
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%variational cluster approach to correlated electron systems in low dimensions

%\bibitem{note1}C. Dahnken, E. Arrigoni
%and W. Hanke, J. Low. Temp. Phys.\,{\bf 126}, 949 (2002)
%optical properties of high-Tc cuprates via a cluster-perturbation approach

\bibitem{fl04}Fledderjohann F, Langari A and M\"utter K H 2004
{\it J. Phys.: Condens. Matter} {\bf 16} 8571 
%Formation of clusters in the ground state of the $t-J$ model
%on a two leg ladder

\bibitem{nagaoka66}Nagaoka Y 1966
{\it Phys. Rev.} {\bf 147} 392 
%ferromagnetism in a narrow, almost half-filled s band

\bibitem{kagan99}Kagan M Y, Haas S and Rice T M 1999
Physica C {\bf 317} 185 
%phase diagram of three-leg ladders at strong coupling along the rungs

%\bibitem{dagotto90}Dagotto E, Joynt R, Moreo A, Bacci S and Gagliano E
%1990 {\it Phys. Rev. B} {\bf 41} 9049 
%strongly correlated electronic systems with one hole: dynamical properties

%\bibitem{fabricius91}
%Fabricius K, L\"ow U and M\"utter K H 1991 
%{\it Phys. Rev. B} {\bf 44} 9981 
%complete solution of the two-dimensional antiferromagnetic heisenberg
%model on small lattices

%\bibitem{riera89}Riera J A and  Young A P 1989,
%{\it Phys. Rev. B} {\bf 40} 5285 
%ferromagnetism in the one-band hubbard model

%\bibitem{eisenberg02}Eisenberg E and Lieb E H 2002
%{\it Phys. Rev. Lett.} {\bf 89} 220403
%polarization of interacting bosons with spin

%\bibitem{boninsegni01}Boninsegni M 2001
%{\it Phys. Rev. Lett.} {\bf 87} 087201
%phase separation in mixtures of hard core bosons

%\bibitem{boninsegni02}Boninsegni M 2002
%{\it Phys. Rev. B} {\bf 65} 134403 
%phase separation and stripes in a boson version of a doped quantum
%antiferromagnet

%\bibitem{fl05}Fledderjohann A, Langari A, M\"uller-Hartmann E and 
%M\"utter K H 2005
%{\it Eur. Phys. J. B} {\bf 43} 471
%Ferromagnetism in a hard-core boson model

%\bibitem{white02}S. R. White, I. Affleck, D. J. Scalapino,
%Phys. Rev. B\,{\bf 65}, 165122 (2002)
%cond-mat/0111320
%Friedel oscillations and charge density waves in chains and ladders

%\bibitem{white98}S. R. White, D. J. Scalapino,
%Phys. Rev. Lett.\,{\bf 81}, 3227 (1998);
%Phys. Rev. B\,{\bf 61}, 6320 (2000)

%hierr





%\bibitem{huse88}D. A. Huse,
%Phys. Rev. B\,{\bf 37}, 2380 (1988)
%ground-state staggered magnetization of two-dimensional quantum
%heisenberg antiferromagnets

%\bibitem{riera89}J. A. Riera, A. P. Young,
%Phys. Rev. B\,{\bf 39}, 9697(1989)
%binding of holes in one-band models of oxide superconductors

%\bibitem{barbieri90}A. Barbieri, J. A. Riera, A. P. Young,
%Phys. Rev. B\,{\bf 41}, 11697 (1990)
%stability of the saturated ferromagnetic state in the
%one-band Hubbard model

%\bibitem{shastry90}B. S. Shastry, H. R. Krishnamurthy, P. W. Anderson,
%Phys. Rev. B\,{\bf 41}, 2375 (1990)
%instability of the Nagaoka ferromagnetic state in the U=infty
%Hubbard model

%\bibitem{doucot89}B. Doucot, X. G. Wen,
%Phys. Rev. B\,{\bf 40}, 2719 (1989)
%instability of the Nagaoka state with more than one hole

%\bibitem{poilblanc95}D. Poilblanc,
%Phys. Rev. B\,{\bf 52}, 9201 (1995)
%phase diagram of the two-dimensional t-J model at low doping

%\bibitem{white02}S. R. White, I. Affleck, D. J. Scalapino,
%Phys. Rev. B\,{\bf 65}, 165122 (2002)
%friedel oscillations and charge density waves in chains and ladders

%\bibitem{boninsegni92}M. Boninsegni, E. Manousakis,
%Phys. Rev. B \,{\bf 46}, 560 (1992)
%green's-function monte carlo study of the t-J model




%\bibitem{eder97}R. Eder, Y. C. Chen, H. Q. Lin, Y. Ohta,
%C. T. Shih, T. K. Lee,
%Phys. Rev. B\,{\bf 55}, 12313 (1997)
%Systematic scaling in the low-energy excitations of the t-J model
%in one and two dimensions

%\bibitem{riera02}J. A. Riera,
%Phys. Rev. B\,{\bf 66}, 134523 (2002)
%Phase separation in effective hard-core boson and triplet
%models in one and two dimensions


%\vspace{0.5cm}

%\bibitem{gagliardini98}P. Gagliardini, S. Haas, T. M. Rice,
%Phys. Rev. B\,{\bf 58}, 9603 (1998)
%generalization of the Luttinger theorem for fermionic ladder systems

%\bibitem{eisenberg02}E. Eisenberg, E. H. Lieb,
%Phys. Rev. Lett.\,{\bf 89}, 220403 (2002)
%polarization of interacting bosons with spin







\vspace{0.5cm}

%\bibitem{fl03}A. Fledderjohann, A. Langari, K.-H. M\"utter,
%Eur. Phys. J. B\,{\bf 36}, 193 (2003)
%Charge density plateaux and insulating phases in the $t-J$
%model with ladder geometry



\end{thebibliography}