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\begin{document}
\null \vspace{30mm}
\noindent
{\Large {\bf Low-Temperature Phase Diagrams of Quantum Lattice
Systems. III. Examples.}}
\vspace{17mm}
\noindent
{\large By J\"urg Fr\"ohlich\footnote{e-mail: juerg@itp.phys.ethz.ch}
and Luc Rey-Bellet\footnote{Present adress:
Section de math\'ematiques, Universit\'e de Gen\`eve,\\ case postale 240, 1211
Gen\`eve 24, Switzerland. e-mail: rey@divsun.unige.ch}}\\
\vspace{5mm}
\noindent
Institut f\"ur Theoretische Physik, ETH-H\"onggerberg \\
8093 Z\"urich, Switzerland\\
\vspace{28mm}
\noindent{\small{\em Abstract.}
We use the low-temperature expansion and the extension of Pirogov-Sinai
theory developed in \cite{dff}, and the perturbation theory of
\cite{dfr} to describe the phase diagrams of two families of fermionic lattice
systems at low-temperature: the balanced model and a variant of the $t-J$
model.
\vspace{17mm}
\section{Introduction}
In this paper, we illustrate the
methods developed in two previous papers \cite{dff,dfr} to analyze the
low-temperature phase diagrams of
quantum lattice systems in terms of two classes of models, the balanced model
and the $t-J$ models (of interest in connection with high-temperature
superconductivity). In this paper, we describe our main results and discuss the
key ideas of the proofs, but we do not present all the details of our
calculations. Furthermore, we permit ourselves to describe various
conjectures. Methods similar to those in \cite{dff} have been developed
by Borgs, Kotecky and Ueltschi, \cite{bku}, for quantum spin systems.
\subsection{The models}
We consider two spin-$1/2$ models on a
two-dimensional lattice, (but our methods can be used in any dimension $d \geq
2$).
\medskip
\noindent
(i) {\it The balanced model}.
\noindent
The Hamiltonian of the model, for a finite subset $\Lambda$ of $\zed^{2}$, is
given by
\begin{eqnarray}
H_{bal}(t)\,&=&\,\sum_{\|x-y\|=1 \atop \langle xy \rangle \subset \Lambda}
\left(J\sigma_{x}^{(3)}\sigma_{y}^{(3)}\right)n_xn_y
\,+\,\sum_{\|x-y\|=\sqrt{2}\atop \langle xy \rangle \subset \Lambda}
\left(J'\sigma_{x}^{(3)}\sigma_{y}^{(3)}\right)n_xn_y \nonumber\\
&&\,+\,\sum_{x \in \Lambda}Un_{x\uparrow}n_{x\downarrow}
-\, \sum_{\|x-y\|=1 \atop \langle xy \rangle \subset
\Lambda}\sum_{\sigma\in \{\uparrow,\downarrow\}}
t\left(c_{x\sigma}^{\dagger}c_{y\sigma}\,+\, h.c.\right),
\label{hb}
\end{eqnarray}
where $\sigma_{x}^{(i)}$, $i=1,2,3$, are the Pauli matrices acting on the spin
space at site $x \in \Lambda$, $c_{x\sigma}^{\dagger}$ and $c_{x\sigma}$ are
the creation and annihilation operators for a particle with spin $\sigma$ at
site $x$, $n_{x\sigma}$ is the number operator for a particle with spin
$\sigma$ at site $x$, and $n_x=n_{x\uparrow}+n_{x\downarrow}$ is the total
particle number operator at site $x$. The Hilbert space at each lattice site
$x \in \Lambda$ is isomorphic to $\IC^{4}$.
As a basis for $\IC^{4}$, we choose $\{|\uparrow\downarrow\rangle$,
$|\uparrow\rangle$, $|\downarrow\rangle$, $|0\rangle\}$.
We shall consider the antiferromagnetic case, i.e., $J>0$ and
$J'>0$, and a large on-site repulsive interaction, $U\gg J$,
$U \gg J'$. The word "balanced" refers to the condition $J'=J/2$. At
half-filling, $H_{bal}(0)$ has an infinite number of
groundstates and does not satisfy the Peierls condition. The hopping term,
$H_{bal}(t) - H_{bal}(0)$, is treated as a perturbation. By using a variant
of perturbation theory developed in \cite{dfr}, we shall show that the hopping
term lifts the infinite degeneracy of the groundstate energy of $H_{bal}(0)$
in order $t^2$. Two N\'eel states will turn out to be the groundstates
of $H_{bal}(t)$, and long-range order survives at
finite temperature.
\medskip
\noindent
(ii) {\it A variant of the $t-J$ model}
\noindent
The Hamiltonian of this model is given by
\begin{eqnarray}
H_{t-J}(\lambda,t)&=&
\sum_{\|x-y\|=1 \atop \langle xy \rangle \subset
\Lambda}\left[\frac{1}{2}\left(J\sigma_{x}^{(3)}\sigma_{y}^{(3)}
+v\right)+\lambda\left(\sigma_{x}^{(1)}\sigma_{y}^{(1)}+
\sigma_{x}^{(2)}\sigma_{y}^{(2)}\right)\right]n_xn_y \nonumber \\
&&-\sum_{\|x-y\|=1 \atop\langle xy \rangle \subset \Lambda} \sum_{\sigma
\in \{\uparrow,\downarrow\}}t\left(c_{x\sigma}^{\dagger}c_{y\sigma}+
h.c.\right)+\sum_{\|x-y\|=\sqrt{2}\atop\langle xy \rangle \subset
\Lambda}v'n_xn_y-\sum_{x \in \Lambda}\mu n_x.
\label{htj}
\end{eqnarray}
We require an infinitely strong on-site repulsive interaction, i.e, we forbid
double occupancy. The Hilbert space each lattice site
$x \in \Lambda$ is isomorphic to $\IC^{3}$.
A convenient basis for
$\IC^{3}$ is $\{|\uparrow\rangle$, $|\downarrow\rangle$, $|0\rangle\}$.
We study the antiferromagnetic case, i.e, $J>0$. Since the unitary
transformation $\prod_{x\in\Lambda_{e}}\sigma_{x}^{(3)}$, where $\Lambda_{e}$
is the even sublattice of $\Lambda$, changes the sign of $\lambda$, we may
assume,
without loss of generality, that $\lambda>0$. We do not impose
any restriction on the values of $v$ and $v'$.
We study this model with the help of perturbative methods (see \cite{dfr})
which
are applicable only in the regime where $\lambda/J$ and a suitable
dimensionless version of $t$ is small.
This model describes the motion of holes in an antiferromagnetic background. It
is a variant of the $t-J$ model which may describe
some features of high-$T_c$ superconductors. The plain $t-J$ model
corresponds to the parameters values $v=-J, v'=0$ and can be derived from the
Hubbard model in the strong coupling limit. Although the
physically relevant parameter ranges are $\lambda \approx J/2$ and $t \approx
J$, it appears to be of interest to study the model in the limit of strong
anisotropy and weak hopping where rigorous results can be obtained, with
the hope that such results can be extrapolated to the physical domain. For
example these
results indicate in which domain of parameter space Fermi liquids or a
superconducting phase might appear.
\subsection{The methods}
For the convenience of the reader we briefly recall the results obtained in
\cite{dff,dfr}. The perturbation theory of \cite{dfr} will serve
to cast the Hamiltonians defined in \reff{hb} and \reff{htj} in a form that
enables us to apply the low-temperature expansion (Pirogov-Sinai) theory of
\cite{dff} to our models.
Standard references for the general formalism of quantum lattice systems
are \cite{sim,braro,rue}.
In this paper, we consider lattice gases of fermions and we require
a slight modification of the usual formalism. Indeed fermionic creation and
annihilation operators at different lattice sites do not commute but
anticommute. But we have in the sequel commutativity (or locality) conditions.
A quantum lattice system is defined by the following data:
\smallskip
\noindent
(i) To each $x \in \zed^{d}$ is associated an Hilbert space ${\cal H}_{x}$
which
is isomorphic, for all $x$, to a fixed, finite-dimensional Hilbert space
${\cal H}$. We choose an ordering on $\zed^{d}$, for example the spiral
ordering depicted in Figure \ref{order} for $d=2$ (and an analogous ordering
for
$ d \geq 3$).
\begin{figure}
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\caption{Spiral order in $\zed^2$}
\label{order}
\end{figure}
This
ordering is chosen to have the property that, for any finite set $X$, the set
$\overline{X} := \{z \in \zed^{\nu}, z \preceq X \}$ of lattice sites which
are smaller than $X$, or belong to $X$, is finite.
The Hilbert space associated to a finite subset
$X = \{x_1 \prec \cdots \prec x_{|X|}\} \subset \zed^{d}$ is the
ordered tensor product
\bd
{\cal H}_{X}= {\cal H}_{x_1} \otimes \cdots \otimes {\cal H}_{x_{|X|}}\,.
\ed
and we denote ${\cal L}({\cal H}_{X})$ the algebra of all bounded operators
on ${\cal H}_{X}$.
\smallskip
\noindent
(ii) For any finite subset $X \subset \zed^{d}$ two operators algebras acting
on ${\cal H}_{X}$ are given
\begin{itemize}
\item The (local) {\it field algebra} ${\cal F}_X \subset
{\cal L}({\cal H}_{\overline{X}})$,
\item The (local) {\it observable algebra} ${\cal A}_X \subseteq {\cal F}_X$,
\end{itemize}
which satisfy the following properties.
\smallskip
\noindent
(a) If $X \subset Y$ and $x \prec y$, for all $x \in X$ and all
$y \in Y \setminus X$, then there is a natural embedding of ${\cal F}_X$
into ${\cal F}_Y$ and, in the following, we write $B$ for both
$B \in {\cal F}_X$ and
$B \otimes {\bf 1}_{{\cal{H}}_{Y\backslash X}}$ in ${\cal F}_Y$.
\smallskip
\noindent
(b) For the infinite system, the (quasilocal) field and operator algebras are the
$C^{*}$-algebras given by
\bd
{\cal A}\,=\, \overline{\bigcup_{X \nearrow \zed^{\nu}} {\cal
A}_{X}}^{\;\;{\rm norm}}\;,\;\;\;
{\cal F}\,=\, \overline{\bigcup_{X \nearrow \zed^{\nu}} {\cal
F}_{X}}^{\;\;{\rm norm}}\;,
\ed
(the limit taken through a sequence of increasing subsets of $\zed^{d}$, where
increasing refers to the spiral ordering defined above).
\smallskip
\noindent
(c) The elements of the field and observable algebras satisfy the
commutativity condition:
If $X \cap Y = \emptyset$,
then for any $A \in {\cal F}_X$, $B \in {\cal A}_Y$
\begin{equation}
[A,B]\,=\,0\;.
\label{coco}
\end{equation}
For fermions, as in our examples, the field algebra, ${\cal F}_X$, is the
algebra generated by creation and annihlation operators at lattice sites $x
\in X$ and the observable algebra, ${\cal A}_X$, may be taken to the
subalgebra of ${\cal F}_X$
consisting of gauge-invariant operators. For example, for spin $1/2$ fermions,
we
take ${\cal H}_{x} \simeq \IC^{2}\otimes \IC^{2}$, for all $x \in \zed^{d}$,
with
the basis
\begin{eqnarray}
|0\rangle \,=\, \left(\begin{array}{c}0\\1\end{array}\right)
\otimes \left(\begin{array}{c} 0 \\ 1 \end{array}\right) \;\;\;&&
|\uparrow\rangle \,=\, \left(\begin{array}{c}1\\0\end{array}\right)
\otimes \left(\begin{array}{c} 0 \\ 1 \end{array}\right) \nonumber \\
|\downarrow\rangle \,=\,\left(\begin{array}{c}0\\1\end{array}\right)
\otimes \left(\begin{array}{c} 1 \\ 0 \end{array}\right)\;\;\;&&
|\uparrow\downarrow\rangle \,=\, \left(\begin{array}{c}1\\0\end{array}\right)
\otimes \left(\begin{array}{c} 1 \\ 0 \end{array}\right)\;. \label{basis}
\end{eqnarray}
Creation and annihilation operators are given in terms of a
Klein-Jordan-Wigner transformation
\begin{eqnarray}
c^{*}_{x\uparrow} = \otimes_{y \prec x} \left(\sigma^{3}_{y1}
\otimes \sigma^{3}_{y2}\right) \otimes \sigma^{+}_{x1} &&
c_{x\uparrow} = \left(c^{*}_{x\uparrow}\right)^{*} \nonumber \\
c^{*}_{x\downarrow} = \otimes_{y \prec x} \left(\sigma^{3}_{y1}
\otimes \sigma^{3}_{y2}\right) \otimes \sigma^{3}_{x1}\otimes
\sigma^{+}_{x2} &&
c_{x\downarrow} = \left(c^{*}_{x\downarrow}\right)^{*}\;, \nonumber
\label{kjw}
\end{eqnarray}
where $\sigma^{+}_{xi}, \sigma^{-}_{xi},\sigma^{3}_{xi}$, $x \in \zed^{d}$,
$i=1,2$ are the Pauli matrices acting at lattice site $x$ on the first or
second
factor of $\IC^{2}\otimes \IC^{2}$. This representation makes explicit
the non-local character of creation and annihilation operators:
$c^{*}_{x\sigma}$, $c_{x\sigma}$, $\sigma \in \{ \uparrow, \downarrow \}$
belong to ${\cal L}({\cal H}_{\overline x})$.
With this representation it is straightforward to check the commutativity
condition \reff{coco}.
The methods of \cite{dff} can applied to Hamiltonians
of the form
\bd
H\,=\,H_0({\underline \mu})\,+\, V, \label{ham} \nonumber
\ed
defined on the Hilbert space ${\cal H}_{{\bar \Lambda}}$, where
$\Lambda$ is a finite (arbitrarily large) subset of the lattice $\zed^{\nu}$.
We assume that
both operators, $H_0({\underline \mu})$ and $V$, can be written as sums of
local
operators which belong to the {\it observable algebras}:
\begin{eqnarray}
H_0({\underline \mu})&=&\sum_{X \subset \Lambda}\phi^{0}_{X}({\underline
\mu})\;, \hspace{1cm} \phi^{0}_{X}({\underline \mu}) \in {\cal A}_{X}
\nonumber\\
V &=&\sum_{X \subset \Lambda} V_X\;, \hspace{1.5cm} V_X \in {\cal A}_{X} \;.
\nonumber
\end{eqnarray}
We consider translation-invariant lattice systems, i.e. systems satisfying
\bd
\phi^{0}_{X}({\underline \mu}) \,=\,\phi^{0}_{X+a}({\underline \mu}),\;\;
V_X \,=\, V_{X+a}\;,
\ed
for all $X \subset \zed^{d}$, $a \in \zed^{d}$.
Furthermore, ${\underline \mu} \in \IR^{P-1}$ parametrizes "external fields"
(chemical potentials, magnetic field, ...).
Our assumptions on $H_0({\underline \mu})$ are the following: there
is a domain ${\cal O} \subset \IR^{P-1}$ such that the conditions
{\bf H1}-{\bf H4} below are satisfied.
\begin{itemize}
\item {\bf H1. $H_0({\underline \mu})$ is a finite-range classical
Hamiltonian}. \\
(a) The operators $\{\phi^{0}_{X}({\underline \mu})\}$ have finite range $r$,
i.e.,
\begin{equation}
r\,:=\, \max_{X:\Phi^{0}_X({\underline \mu}) \neq 0 \atop x,y \in M,\,\,1\leq
i \leq \nu}
|x_i - y_i|, \label{range}
\end{equation}
is finite.\\
(b) There exists a basis $\{e_{j_x}^{x}\}_{j_x \in I}$, where $I=\{1,\ldots ,
N\}$, $N={\rm dim}{\cal H}_{x}$, is a finite set independent of $x$, such
that, for all $X \subset \zed^{d}$ and all ${\underline \mu} \in {\cal O}$,
$\phi^{0}_{X}({\underline \mu})$ is diagonal in the tensor-product-basis
\begin{displaymath}
\{ \otimes_{x \in X} e_{j_x}^{x}\}_{j_x \in I}\;. \nonumber
\end{displaymath}
\end{itemize}
In our examples the basis $\{e_{j_x}^{x}\}_{j_x \in I}$ is given by
\reff{basis}.
The set of configurations on $\Lambda$, $\Omega_{\Lambda}$, is defined as the
set of all assignments $\{j_x\}_{x \in \Lambda}$ with $j_x \in I$. A
configuration $\omega_{\Lambda}$ is an element of $\Omega_{\Lambda}$ and, for
$X \subset \Lambda$, $\omega_{X}$ denotes the restriction of $\omega_{\Lambda}$
to the subset $X$. To each configuration $\omega_{X}$ one can associate
the state $\otimes_{x \in X} e_{j_x}^{x}$, with $\omega_{X} = \{j_x\}_{x \in
X}$, which will be denoted $e_{\omega_X}$.
Since the interactions $\phi^{0}_{X}$ are uniquely
determined by the matrix elements
\begin{eqnarray}
\langle e_{\omega_\Lambda}|\phi^{0}_{X}(\underline \mu)|e_{\omega_\Lambda}
\rangle\,
&=&\,\langle e_{\omega_X}|\phi^{0}_{X}(\underline \mu)|e_{\omega_X} \rangle\
\nonumber \\
&:=&\phi^{0}_{X}(\underline \mu)(\omega_{\Lambda}), \nonumber
\end{eqnarray}
we may view $\phi^{0}_{X}$ as a function on the set of
configurations.
Given a configuration $\omega \bydef \omega_{\zed^{\nu}}$ of the infinite
system $(\Lambda \nearrow \zed^{\nu})$, we let $e({\underline \mu})(\omega)$
denote the energy
density of $\omega$, defined by
\begin{equation}
e({\underline \mu})(\omega) \, = \, \overline{\lim}_{\Lambda \nearrow
\zed^{\nu}}
\frac{1}{|\Lambda|} \sum_{X \subset \Lambda} \langle e_{\omega_{\Lambda}} |
\phi^{0}_{X}({\underline \mu})| e_{\omega_{\Lambda}} \rangle \; .
\end{equation}
Note that the limit exists e.g. if $\omega$ is periodic.
Let ${\cal K}$ be the set of all of periodic states $\{ s_1 = e_{\omega_1},
\cdots s_P = e_{\omega_p}\}$ (where $\omega_1, \cdots, \omega_P$ are
periodic configurations) with the property that, for every $j=1, \cdots, P$,
$s_j$ is a groundstate of $H_0({\underline \mu})$, for some ${\underline \mu}
\in {\cal O}$, in the sense that $e_j({\underline \mu}) \bydef
e({\underline \mu})(\omega_j)$ is a minimum of
$e({\underline \mu})(\omega)$, $\omega \in \Omega$.
\begin{itemize}
\item {\bf H2. Peierls condition}. \\
The set, ${\cal K}$, of periodic groundstates of
$H_0({\underline \mu})$, with ${\underline \mu} \in {\cal O}$, is finite
and $H_0({\underline \mu})$ satisfies the Peierls condition
with a Peierls constant $\kappa$
independent of ${\underline \mu} \in {\cal O}$.
\end{itemize}
\noindent
The Peierls condition requires that there be a non-zero energy per unit
interface (``contour'') separating two periodic groundstates. A precise
statement and a
useful criterion (see \cite{hol}) are given in Appendix B.
Note that the dimension of ${\cal O}$, $P-1$, is assumed to be one less than
the number of groundstates.
For each value of ${\underline \mu}$, there is a set of periodic groundstates
of $H_0({\underline \mu})$
\bd
{\cal Q}^{(\infty,0)}({\underline \mu}) \,:=\, \{
s_j \,:\, e_j({\underline \mu}) = \min_{1 \leq k \leq P} \,
e_k({\underline \mu})\}.
\ed
The classical zero-temperature phase diagram is the family of manifolds
\bd
{\cal S}^{(\infty,0)}_{\{s_{p_1},\ldots,s_{p_k}\}} \,:=\,
\{{\underline \mu} \in {\cal O}\,:\, {\cal Q}^{(\infty,0)}({\underline \mu})=
\{s_{p_1},\ldots,s_{p_k}\}\},
\ed
with $1\le k\le P$; $s_{p_1},\ldots,s_{p_k}\in{\cal K}$. These manifolds
are called the {\em strata}\/ of the phase diagram.
\begin{itemize}
\item {\bf H3. Regularity of the phase diagram}.
The phase diagram is {\em regular}\/, i.e., the map
\bd
\underline \mu \; \longmapsto \;
\Bigl(e_1({\underline \mu})-\min_{k} e_k({\underline \mu}) \,,\,
\ldots\,,\,e_P({\underline \mu})-\min_{k} e_k({\underline \mu})\Bigr)
\ed
is a homeomorphism of ${\cal O}$ into the boundary of the positive octant in
$\IR^P$.
\end{itemize}
Note that this means that the stratum of maximum coexistence, ${\cal
S}^{(\infty,0)}_{{\cal K}}$, is a single point ($=$ the origin of the
$P$-octant), the
strata with $p-1$ groundstates are curves starting from it ($=$ the
coordinate
halfaxes), $\ldots$ , the strata with $P-k$ groundstates are
$k$-dimensional manifolds bounded by the strata with $P-k+1$ ground
states. (This geometry is also known as the {\em Gibbs phase rule}\/.)
\begin{itemize}
\item {\bf H4. Smoothness properties}. \\
The maps ${\underline \mu} \longmapsto \phi^{0}_{X}({\underline \mu})$
are differentiable, and their derivatives have uniformly bounded
norm on ${\cal O}$. Moreover the determinant of the matrix
\begin{equation}
\left({{\partial \over {\partial \mu_j}}
(e(\sigma_i) - e(\sigma_p))}
\right)_{1\le i,j \le p-1}
\end{equation}
is uniformly bounded away from zero throughout ${\cal O}$.
\end{itemize}
The perturbation Hamiltonian $V$ is assumed to satisfy the following
conditions.
\begin{itemize}
\item {\bf V1. Exponential decay}. \\
There exists $\varepsilon > 0$ and $\lambda< 1$ such that
\bd
\|V_B\|\,\leq\, \varepsilon \lambda^{g(B)},
\ed
where $g(B)$ is the cardinality of the smallest connected subset of the
lattice containing $B$.
\end{itemize}
For fermionic systems $V$ is assumed to be gauge-invariant. This property
is crucial both in proving statistical independence of the
weights of disjoint contours in a low-temperature expansion (see \cite{dff}),
and in our perturbation scheme (see Appendix A and \cite{dfr}).
The first part of the analysis in \cite{dff} yields a criterion to
determine the {\it stable phases} of $H$, for a fixed value of ${\underline
\mu}$, if $H_0({\underline \mu})$ is in the symmetric regime. The Hamiltonian
$H_0({\underline \mu})$ is said to be in the symmetric regime if, in a
neighbourhood of
${\underline \mu}$, $H_0$ has a single groundstate or, more generally,
finitely many groundstates related by some symmetry operation.
For a periodic state $s_j \in {\cal K}$, and finite regions
$\Gamma\supset\Lambda$, the Hamiltonian ``with boundary condition''
$s_j$ is the operator
\begin{equation}
H^{s_j}_{\Gamma}\;=\; \sum_{X \cap {\Lambda} \not= \emptyset} \,
P^{s_j}_{\Gamma\setminus\Lambda}{(\Phi^0_X({\underline \mu}) + V_X)}
P^{s_j}_{\Gamma\setminus\Lambda}\;,
\end{equation}
where $P^{s_j}_{\Gamma\setminus\Lambda}\bydef
|s_{j\Gamma\setminus\Lambda}\rangle \langle
s_{j\Gamma\setminus\Lambda}|$. The {\em Gibbs state}\/ in the
finite volume $\Lambda$ and boundary condition
$s_j$ and inverse temperature $\beta$ is the positive linear functional on
${\cal F}_\Lambda$ defined by
\bd
{\cal F}_\Lambda \ni A \;\mapsto\; \langle A \rangle^{j}_{\Lambda}\,=\,
\lim_{\Gamma \nearrow \zed^d}
{{\rm tr} A\, \exp\left({-\beta H_\Gamma^{s_j}}\right) \over
{\rm tr} \exp\left({-\beta H_\Gamma^{s_j}}\right)}\;.
\ed
The infinite-volume limits of these functionals determine the phase
diagram of the system.
%We denote by $Z^{j}_{\Lambda}$ the partition function for a finite subset
%$\Lambda$
%of $\zed^{\nu}$ with boundary condition $s_j$
%\bd
%Z^{j}_{\Lambda}\,=\, {\rm Tr}^{s_j}_{\Lambda} \left(\exp{-\beta H}\right),
%\ed
%and by $\langle A \rangle^{j}_{\Lambda}$ the finite-volume expectation value
%of a local observable $A \in {\cal A}_{\Lambda}$ with boundary conditions
%$s_j$
%\bd
%\langle A \rangle^{j}_{\Lambda}\,=\,\frac{1}{Z^{j}_{\Lambda}} {\rm
%Tr}^{s_j}_{\Lambda}
% \left( A \exp{-\beta H}\right).
%\ed
In \cite[Theorem 2.2]{dff} the following theorem has been proven:
\begin{theorem}(Low-temperature expansion in the symmetric regime.)
\noindent
Assume that, for a fixed value of ${\underline \mu}$, $H_0({\underline \mu})$
satisfies condition {\bf H1}, is in the symmetric
regime, and satisfies the Peierls
condition {\bf H2}. Assume, further, that the perturbation $V$ satisfies
condition
${\bf V1}$.
Then there are constants $ {\bar \kappa} =\OO(\kappa) $ and $\varepsilon_0 >
0$
such that,
for each
$\beta$, $\varepsilon$ and $\lambda$ in the region
\bd
\max \,(e^{-\beta {\bar \kappa}} , \frac{\varepsilon \lambda}{\kappa}) \,<\,
\varepsilon_0,
\ed
there exists a family of functions $f_{j}({\underline \mu})$,
$1\leq j \leq P$ (the truncated free energy densities), such that the
following holds: If, for some $ j \in \{1,\cdots,P\}$,
\begin{equation}
\,f_{j}({\underline \mu})\,=\,\min_{1\leq k \leq P}f_{k}({\underline \mu})
\label{stable}
\end{equation}
for some values of $\varepsilon$, $\beta$ and $\lambda$, then
\begin{enumerate}
\item
$f_{j}({\underline \mu})$ coincides with the true free energy of the system
\begin{equation}
f_{j}({\underline \mu})\,=\,-\frac{1}{\beta} \lim_{\Lambda \nearrow
\zed^{\nu}} \frac{1}{|\Lambda|}\log{Z^{j}_{\Lambda}} \;. \label{free}
\end{equation}
%and
%\bd
%f_j({\underline \mu})\,=\,e_j({\underline \mu})\,+\, \varepsilon_{\beta,t}
%\ed
%with
%\bd
%lim_{\beta \rightarrow \infty \atop t \rightarrow 0}\varepsilon_{\beta, t}\,
%=0.
%\ed
\item
The infinite-volume limit
\begin{equation}
\langle A \rangle^{j} \,:=\,\lim_{\Lambda \nearrow
\zed^{\nu}} \langle A \rangle^{j}_{\Lambda}, \label{quex}
\end{equation}
exists for any local observable $A$.
\item
\bd
|\, \langle A \rangle^{j} \,-\, \langle s_j|A|s_j \rangle\,| \leq \|A\| |X|
O(\varepsilon_0)
\ed
for any operator $A$ in ${\cal A}_{X}$.
\end{enumerate}
\end{theorem}
Theorem 1.1 shows that the long-range order of the groundstate survives at
finite temperature and under the addition of a sufficiently small quantum perturbation.
This theorem, combined with a high temperature expansion (see \cite{park}),
permits to prove the existence, at finite temperature, of phase transitions
between ordered phases and a disordered phase.
The second part of the analysis refers to the stability of the phase
diagram as a whole. It extends Pirogov-Sinai theory
\cite{pirsin,boim} to a certain class of quantum Hamiltonians.
The set of coexisting Gibbs states, ${\cal
Q}^{(\beta,\lambda)}(\underline\mu)$, is
defined similarly to the set ${\cal Q}^{(\infty,0)}(\underline\mu)$, replacing
groundstate energy densities by truncated free energy densities (see
\cite{dff}):
\bd
{\cal Q}^{(\beta,\lambda)}(\underline\mu)\,=\,\{ s_j \,:\, f_j({\underline
\mu}) =
\min_{1 \leq k \leq P} \,f_k({\underline \mu})\}.
\ed
One then defines
\bd
{\cal S}^{(\beta,\lambda)}_{\{s_{p_1},\ldots,s_{p_k}\}} \,:=\,
\{{\underline \mu} \in {\cal O}\,:\, {\cal Q}^{(\beta,\lambda)}({\underline
\mu})=
\{s_{p_1},\ldots,s_{p_k}\}\}.
\ed
The following theorem is proven in \cite[Theorem 2.3]{dff}:
\begin{theorem}(Pirogov-Sinai theory)
\noindent
Assume that the Hamiltonian $H_0({\underline \mu})+V$ satisfies conditions
{\bf H1} - {\bf H4} and {\bf V1}.
Then there are constants ${\bar \kappa} = \OO(\kappa)$ and $\varepsilon_0 > 0$
such that, for each $\beta$, $\varepsilon$ and $\lambda$ in the region
\bd
\max \,(e^{-\beta {\bar \kappa}} , \frac{\varepsilon \lambda}{\kappa} )\,<\,
\varepsilon_0,
\ed
there exists a non-empty open
set ${\cal O}_{\beta,\lambda}\in \IR^{P-1}$ such that:
\begin{itemize}
\item
The phase diagram
defined by the strata ${\cal O}_{\beta,\lambda} \cap
{\cal S}^{(\beta,\lambda)}_{\{s_{p_1},\ldots,s_{p_k}\}}$ is regular,
and these strata are differentiable manifolds.
\item
As $\varepsilon_0 \rightarrow 0$, the strata ${\cal O}_{\beta,\lambda} \cap
{\cal S}^{(\beta,\lambda)}_{\{s_{p_1},\ldots,s_{p_k}\}}$ tends to the
zero-temperature classical strata ${\cal O} \cap
{\cal S}^{(\infty,0)}_{\{s_{p_1},\ldots,s_{p_k}\}}$, pointwise in
${\underline \mu}$. In particular, the distance between the maximal-coexistence
manifolds ${\cal S}^{(\beta,\lambda)}_{{\cal K}}$ and
${\cal S}^{(\infty,0)}_{{\cal K}}$ is $\OO(\varepsilon_0)$.
\end{itemize}
\end{theorem}
This theorem describes the first order phase transitions (coexistence of
several
phases), when the parameters ${\underline \mu}$ vary. It shows that the phase
diagram of $H_0({\underline \mu}) + V$, at low-temperature and for a
sufficiently small perturbation $V$, is a smooth deformation of the phase
diagram
of $H_0({\underline \mu})$ at zero temperature.
In \cite{dfr} it is shown how to describe the low-temperature phase diagram of
certain quantum lattice systems for which conditions {\bf H2} and {\bf H3}
are {\it not}, a priori, satisfied. Indeed, it may happen that the
groundstates of the Hamiltonian $H_0({\underline \mu})$ are infinitely
degenerate (in the
thermodynamic limit), but that the perturbation reduces this
degeneracy to a finite one. We
consider Hamiltonians of the form
\bd
H(t)\,=\,H_0({\underline \mu})\,+\,tV \, , \nonumber
\ed
t being a perturbation parameter.
We assume that $H_0({\underline \mu})$ satisfies conditions {\bf H1} and
{\bf H4} and, in addition, we require the following hypotheses on
$H_0({\underline \mu})$ and $V$.
\begin{itemize}
\item{\bf P1. M-potential}.\\
$H_0({\underline \mu})$ can be written as
\begin{equation}
H_0({\underline \mu})\,=\,\sum_{M \subset \Lambda} \phi^0_M({\underline \mu}),
\label{mpot} \end{equation}
where $\{\phi^0_M({\underline \mu})\}$ is a translation-invariant, finite-range
m-potential (see \cite{hol}).
\end{itemize}
\noindent
More explicitly, condition {\bf P1} amounts to assuming the following
property.
There exists at least one configuration
$\omega$ of the infinite system minimizing
$\Phi^{0}_M$, for all $M$, i.e.,
\begin{equation}
\Phi^{0}_M(\omega) =
\min_{\atop\omega'}\,\Phi^{0}_M(\omega'),\,\,\,\,\,{\rm
for\,\,all}\,\, M \subset \zed^{d} \label{mini}
\end{equation}
The set of all configurations for which
(\ref{mini}) holds are groundstate configurations of $H_0({\underline
\mu})$. Condition {\bf P1} allows us to define local groundstates of the
restriction of $H_0({\underline \mu})$ to a subset of the lattice.
We assume that the perturbation $V$ satisfies, instead of {\bf V1}, the following
stronger condition.
\begin{itemize}
\item{\bf P2. Finite-Range}\\
$V$ is a finite-range translation-invariant perturbation Hamiltonian.
\end{itemize}
In \cite{dfr} we construct a unitary transformation, $U(t)$, such that
the transformed Hamiltonian,
\bd
{\widetilde H}(t)\,=\,U(t)H(t)U(t)^{-1}, \label{uniti}
\ed
can be cast in the form
\bd
{\widetilde H}(t)\,=\, H_0(t)\,+\, V(t), \label{cast} \nonumber
\ed
such that Pirogov-Sinai theory, see \cite{dff,dfr}, can be
applied to the pair $\bigl( H_0(t)\,,\,V(t) \bigr)$, with $H_0(t)$
playing the role of $H_0({\underline \mu})$ and $V(t)$ playing the role
of the quantum perturbation $V$. In our applications, $U(t)$ will have the form
\bd
U(t)\,=\, \exp{(tS)}\, ,
\ed
with $S = - S^{*}$ independent of $t$. The construction of $S$ is reviewed in
Appendix A. This choice of $U(t)$ corresponds to second order perturbation
theory.
If the accidental degeneracy of the groundstates of $H_0({\underline \mu})$ is
lifted by the
perturbation $tV$ in some finite order of perturbation theory one may
construct corresponding unitary transformations (see \cite{dfr}).
In order to apply Theorem 1.1
one has to check that the Hamiltonian $H_0(t)$ satisfies
conditions {\bf H1}, {\bf H2}. Condition {\bf H1} requires that
there be a tensor-product basis in which $ H_0(t)$ is diagonal.
Condition {\bf H2} is equivalent to saying that the perturbation lifts the
infinite degeneracy of the groundstates to a finite one and that the Peierls
condition holds with a ($t$-dependent) Peierls constant $\kappa$.
If these conditions hold the spectrum of $H_0(t)$ can be described as
follows. Above the groundstate there are ``low-energy'' states whose energies
are separated from the groundstate energy by a ($t$-dependent) gap $\kappa$
and ``high-energy'' states whose energies
are separated from the groundstate energy by a gap $\Delta$ of order $1$.
In order to apply Pirogov-Sinai theory (Theorem 1.2) one has to further check
condition {\bf H3}. That the phase diagram of $H_0({\underline \mu})$ is not
regular may mean, for example, that, on the hypersurfaces of the phase diagram
separating two regions
with different sets of groundstates, there are, besides those of the adjacent
regions, (infinitely) many other ones. We have to understand how the
degeneracy is lifted by the perturbation: It may happen that the phase
diagram of $H_0(t)$ becomes regular or that a new phase appears in a
small domain surrounding the hypersurface in question (as in the t-J model
treated below).
Concerning condition {\bf V1} the following lemma is proven in \cite{dfr}.
\begin{lemma} Assume that the Hamiltonian $H_0({\underline \mu}) + tV$
satisfies conditions {\bf H1}, {\bf P1} and {\bf P2}.
Then the unitary transformation $U(t)$ (see Appendix A) is such that,
for sufficiently small $t$,
the new perturbation $V(t)$ satisfies condition {\bf V1} (exponential decay).
\end{lemma}
For quantum lattice systems, the perturbation is usually not relatively
bounded,
and we cannot apply standard theorems of analytic perturbation theory
(as in e.g. \cite{kat}).
We prove that if the Hamiltonian satisfies conditions {\bf P1} and {\bf P2}
then the unitary transformation $U(t)$ can be written in terms of {\it
local} operators. The commutativity of operators localized in disjoint subsets
of the lattice (see \reff{coco}) ensures the exponential decay of the
interactions contributing to ${\widetilde H}(t)$.
Note that, when applying Theorems 1.1 or 1.2 to the Hamiltonian
${\widetilde H}(t)$, all relevant parameters $\kappa$, $\varepsilon$ and
$\lambda$
depend on $t$.
In subsequent sections we analyze concrete models.
\section{The balanced model}
We consider the Hamiltonian
\bd
H_{bal}(t)\,=\,H_{bal}(0)\,+\,tK, \nonumber
\ed
where
\begin{equation}
H_{bal}(0)\, = \,\sum_{\|x-y\|=1\atop \langle xy \rangle \subset \Lambda}
(J\sigma_{x}^{(3)}\sigma_{y}^{(3)})n_xn_y
\,+\,\sum_{\|x-y\|=\sqrt{2}\atop \langle xy \rangle \subset \Lambda}
(J'\sigma_{x}^{(3)}\sigma_{y}^{(3)})n_xn_y
\, + \,\sum_{x\in \Lambda}Un_{x\uparrow}n_{x\downarrow},
\end{equation}
and
\begin{equation}
K\, = \,\sum_{\|x-y\|=1,\atop \langle xy \rangle \subset \Lambda} K_X
= \,- \sum_{\|x-y\|=1,\atop \langle xy \rangle \subset \Lambda} \sum_{\sigma
\in \{\uparrow,\downarrow\}} \left(c_{x\sigma}^{\dagger}c_{y\sigma}\,+\,
c_{y\sigma}^{\dagger}c_{y\sigma}\right) \label{kax}\;.
\end{equation}
As announced in the introduction, we consider the antiferromagnetic case,
i.e., $J>0$ and
$J'>0$, with $J'=J/2$ and with a large on-site repulsive
interaction, $U\gg J$.
The Hamiltonian is defined on any finite subset $\Lambda$ of $\zed^{2}$.
For convenience, we take $\Lambda$ to be a large square.
In the sequel, we consider the model at half-filling, i.e., the
number of particles is equal to the number of sites,
$N=\sum_{x}n_{x}=|\Lambda|$.
The groundstates of $H_{bal}(0)$ are easily determined by regrouping the
interactions so as to express them in terms of an m-potential:
\bd
H_{bal}(0)=\sum_{B}\phi^0_{B},
\ed
where
\bd
\phi^0_{B}\,=\,\sum_{\|x-y\|=1 \atop \subset B} (\frac{J}{2}
\sigma_{x}^{(3)}\sigma_{y}^{(3)})n_xn_y\,+\,\sum_{\|x-y\|=\sqrt{2} \atop
\subset B}(J'\sigma_{x}^{(3)}\sigma_{y}^{(3)})n_xn_y\,+\,
\sum_{x \in B}\frac{U}{4}n_{x\uparrow}n_{x\downarrow} \nonumber
\ed
and $B$ is a two-by-two block. One verifies that
$\phi^0_B$ takes its minimal value for the
following configurations (up to rotations by $\pi/2$ and global spin flips).
\begin{displaymath}
\left(\begin{array}{cc}+&-\\-&+\end{array}\right) \hspace{0.5cm}
\left(\begin{array}{cc}+&-\\+&-\end{array}\right)
\end{displaymath}
The groundstates are the configurations with the property that, on all
two-by-two blocks, one encounters one of the above configurations. One can
alternatively describe the
groundstates as consecutive lines of N\'eel ordered spins where each line can
be flipped independently of the other ones. Their degeneracy is infinite in
the thermodynamic limit.
The groundstate energy density is
\bd
e_{bal}=-J. \nonumber
\ed
One can see that $H_{bal}(0)$ does not satisfy the Peierls condition: In
Fig.\ref{cont} we display two examples of configurations with one "island"
of one groundstate in another groundstate for which the Peierls condition does
not hold. The solid lines separate the
regions with different groundstates and the dashed lines indicate the parts of
the contour which contribute to the energy of the contour.
The energy of the configuration shown on the left of Fig.\ref{cont}
does not depend on the length of the contour but only on the number of corners
of the contour. The contour separating the two groundstates can be made
arbitrarily long without changing
its energy. The energy of the configuration shown on the right of
Fig.\ref{cont} is proportional to the vertical pieces of the contour. The
length
of the contour in the horizontal direction can be arbitrarily long without
changing
its energy.
\begin{figure}
\begin{picture}(300,60)(-100,-20)
\put(0,0){$\tiny \begin{array}{cccccccc}-&+&-&+&-&+&-&+\\ +&-&+&-&+&-&+&-\\
-&+&+&-&+&-&-&+\\ +&-&-&+&-&+&+&-\\ -&+&+&-&+&-&-&+\\ +&-&-&+&-&+&+&-\\
-&+&-&+&-&+&-&+\\ +&-&+&-&+&-&+&- \end{array}$}
\put(150,0){$\tiny \begin{array}{cccccccc}-&+&-&+&-&+&-&+\\ +&-&+&-&+&-&+&-\\
-&+&+&-&+&-&-&+\\ +&-&+&-&+&-&+&-\\ -&+&+&-&+&-&-&+\\ +&-&+&-&+&-&+&-\\
-&+&-&+&-&+&-&+\\ +&-&+&-&+&-&+&- \end{array}$}
\put(32,-10){\framebox(60,27.5)}
\put(182,-10){\framebox(60,27.5)}
\put(20,10){\dashbox{1}(25,14)}
\put(82.5,10){\dashbox{1}(25,14)}
\put(20,-17.5){\dashbox{1}(25,14)}
\put(82.5,-17.5){\dashbox{1}(25,14)}
\put(170,-17.5){\dashbox{1}(25,40)}
\put(232.5,-17.5){\dashbox{1}(25,40)}
\end{picture}
\caption{Examples of contours which do not satisfy the Peierls condition}
\label{cont}
\end{figure}
Next, we consider the unitary transformation $\exp{(tS)}$ defined in Appendix
A (see \reff{s11}). The operator $S$ can be written as a sum of local
operators
\begin{eqnarray}
S\,&=&\,\sum_{X=\langle xy \rangle}S_{B_X}\nonumber \\
&=&\,\sum_{X=\langle xy \rangle}{\rm ad}^{-1}{\bar H}_{0X}(K^{01}_{B_{X}}),
\nonumber
\end{eqnarray}
where $X = \langle xy \rangle$ is a pair of nearest neighbours, and $B_X$, the
support of the operator $S_{B_X}$, is the set of lattice sites at distance
$\le 1$ from $X$ (see (\ref{bex})). The local operators ${\bar H}_{0X}$ are
defined in (\ref{hbar}), and
\bd
K^{01}_{B_{X}} = P^{0}_{B_X}K_XP^{1}_{B_X} + P^{1}_{B_X}K_XP^{0}_{B_X}
\ed
where $K_X$ is given by (\ref{kax}) and $P^{0}_{B_X}$, $P^{1}_{B_X}$ are
local projection operators with support $B_X$ (see (\ref{proj1})).
The operators $P^{0}_{B_X}$ are the projections onto the subspaces of states
which, when restricted to $B_X$, are groundstates of $H_{bal}(0)$, and
$P^{1}_{B_X}$ are the projections onto the subspaces of states which, when
restricted to $B_X$, are excited states, the excitation
being localized in $X$.
The transformed Hamiltonian reads (see (\ref{ach}))
\begin{eqnarray}
{\widetilde H}_{bal}(t)\,&=&\exp{(tS)}\,H_{bal}(t)\,\exp{(-tS)}\nonumber \\
&=&\,H_0 (t)\,+\, V(t),\nonumber
\end{eqnarray}
with
\begin{equation}
H_0(t)\,=\,H_{bal}(0) + t\sum_{X=\langle xy \rangle} K^{00}_{B_X} +
\frac{t^2}{2} \sum_{X=\langle xy \rangle}
P^0_{B_X}{\rm ad}S_{1B_X}(K^{01}_{B_{X}})P^0_{B_X}\;, \label{asch}
\end{equation}
where $K^{00}_{B_X}$ is defined in (\ref{v00}), and ${\rm ad}A(B)$ denotes the
commutator, $AB-BA$.
The second term of (\ref{asch}) vanishes and the third one gives a
contribution to the groundstate energy given by
\begin{eqnarray}
&&\frac{t^2}{2}\sum_{X=\langle xy \rangle}\bigl[ {\rm ad}^{-1}{\bar
H}_{0X}(P^{0}_{B_X}K_XP^{1}_{B_X})\;P^{1}_{B_X}K_XP^{0}_{B_X}
- P^{0}_{B_X}K_XP^{1}_{B_X}\;{\rm ad}^{-1}{\bar
H}_{0X}(P^{1}_{B_X}K_XP^{0}_{B_X})\bigr] \nonumber \\
&&=\sum_{X=\langle xy \rangle} \frac{-t^2}{E_{1X}-E_0}P^{0}_{B_X} \,,\nonumber
\end{eqnarray}
where $E_{1X}\,=\,3J+U$ is the energy of the excitation caused by the hopping
of a single particle, and $E_0$ is the groundstate energy of $H_{bal}(0)$.
It is easy to see that the groundstates of $H_0(t)$ are N\'eel
states with an energy density
\bd
e_0(t)= -J-t^2\frac{4}{2J+U}, \nonumber
\ed
and that the Peierls condition holds for $H_0(t)$ with a constant
proportional to $t^2$. By Lemma 1.3 and Theorem 1.1, we have the following
result
\begin{proposition}
There exists a constant $t_0$ such that:
\begin{enumerate}
\item
For any $0 < t < t_0$, the groundstates of ${\widetilde H}_{bal}(t)$ at
half-filling
are small perturbations of the N\'eel states.
\item
There exists $\beta_0 = \beta_0(t) < \infty$ such that, for
$0 < t < t_0$ and $\beta > \beta_0(t)$, the long-range order of the groundstates
persists.
\end{enumerate}
\end{proposition}
Note that, in Proposition 2.1, $t$ must be {\it strictly positive}.
\section{The $t-J-\lambda -v-v'$ model}
In this section, we study the $t-J-\lambda -v-v'$ model, i.e., the family of
Hamiltonians
\bd
H_{t-J}(\lambda,t)\,=\,H_{t-J}(0,0)\,+\,\lambda V\,+\,tK \;,
\ed
where
\begin{eqnarray}
H_{t-J}(0,0)&=&\sum_{\|x-y\|=1\atop \langle xy \rangle \subset \Lambda}
\frac{1}{2}\left(J\sigma_{x}^{(3)}\sigma_{y}^{(3)}+v\right)n_xn_y
+\sum_{\|x-y\|=\sqrt{2}\atop \langle xy \rangle \subset
\Lambda}v'n_xn_y-\sum_{x\in\Lambda}\mu n_x \\
K&=& -\sum_{\|x-y\|=1 \atop \langle xy \rangle \subset \Lambda} \sum_{\sigma
\in
\{\uparrow,\downarrow\}} \left(c_{x\sigma}^{\dagger}c_{y\sigma} +
c_{y\sigma}^{\dagger}c_{x\sigma}\right)\label{kp} \\
V&=&\sum_{\|x-y\|=1\atop \langle xy \rangle \subset \Lambda}
\left(\sigma_{x}^{(1)}\sigma_{y}^{(1)}
+\sigma_{x}^{(2)}\sigma_{y}^{(2)}\right)n_xn_y. \label{vp}
\end{eqnarray}
The system described by this Hamiltonian is confined to an arbitrary finite
subset
$\Lambda$ of $\zed^{2}$ which we take, for convenience, to be a large
square.
This chapter is organized as follows. We first recall some results, derived by
Gruber and S\"ut\"o \cite{grusu}, on the Hamiltonian $H_{t-J}(0,0)$.
We then investigate the stable phases of $H_{t-J}(\lambda,t)$, for small $t$
and $\lambda$. For the regions of the phase diagram where the groundstates of
$H_{t-J}(0,0)$ are infinitely degenerate, we use the perturbation scheme of
\cite{dfr} (see Introduction and Appendix A) together with
the low-temperature expansion
of \cite{dff} (Theorem 1.1) to study the degeneracy breaking effect of the
perturbation. We determine the low-energy spectrum of $H_{t-J}(\lambda,t)$
to leading order in $t$ and $\lambda$,
(i.e., to order $t^2$ and $\lambda^2$). We are able to determine all stable
phases of the Hamiltonian $H_{t-J}(\lambda,t)$, except for one for which a
higher
order computation is needed. Finally, we investigate the regularity
properties of the Hamiltonian $H_{t-J}(\lambda,t)$ for small $t$ and
$\lambda$. We have several kinds of results (some of them only at a heuristic
level).
\medskip
\noindent
(i) For some parts of the phase diagram ($v0, t=0$, one can derive an effective (Ising-type) Hamiltonian. For
$t>0$,
the situation is more involved: for it can happen that the effective
Hamiltonian is {\it not} of finite-range. In our model, we find a phenomenon
that could be of some significance for
superconductivity: We find that, for $v=J, v'>0$, the effective
Hamiltonian is not of Ising-type: there are terms describing hopping of
two next-neighbour holes. Albanese and one of the authors have shown,
\cite{fral},
that the groundstates of two holes in an antiferromagnetic
background describe a (delocalized) bound pair. We identify here the region of
the
phase diagram where such states might occur. This problem as well as related
ones (see \cite{alda}) are, in our view, worth being further investigated.
\subsection{The classical J-v-v' model}
We first consider the classical limit of our Hamiltonian ($t=\lambda=0$)
\bd
H_{t-J}(0,0)=\sum_{\|x-y\|=1 \atop (xy) \subset \Lambda}
\frac{1}{2}\left(J\sigma_{x}^{(3)}\sigma_{y}^{(3)}+v\right)n_xn_y+
\sum_{\|x-y\|=\sqrt2 \atop (xy) \subset \Lambda} v'n_xn_y -
\sum_{x \in \Lambda} \mu n_x .
\ed
This model is equivalent to a spin-$1$ model. Is has several physical
interpretations, e.g., $He^{3}-He^{4}$ mixtures, ternary mixtures, or diluted
magnetic alloys. We recall some results, derived in \cite{grusu}, about
the phase diagram of this model at low-temperature.
\begin{figure}
\begin{picture}(250,150)(-120,0)
\put(100,40){\vector(1,0){110}}
\put(100,40){\vector(0,1){90}}
\put(60,60){\line(-2,-1){60}}
\multiput(60,60)(0,20){3}{\line(0,1){10}}
\put(190,30){$\mu$}
\put(110,130){$v'$}
\put(0,70){$\begin{array}{cc}\circ&\circ\\\circ&\circ\end{array}$}
\put(110,10){$\begin{array}{cc}+&-\\-&+\end{array}$}
\put(110,105){$\begin{array}{cc}\circ&\circ\\-&+\end{array}$}
\multiput(60,60)(20,5){7}{\line(4,1){10}}
\put(190,100){$\mu=\frac{3(v-J)}{2}+4v'$}
\put(0,20){$\mu=\frac{(v-J)}{2}+2v'$}
\put(20,120){$\mu=\frac{(v-J)}{2}$}
\end{picture}
\caption{Zero-temperature phase diagram of $H_{t-J}(0,0)$ for $v\subset
B}\frac{1}{4}\left(J\sigma_{x}^{(3)}\sigma_{y}^{(3)}
+v\right)n_xn_y+ \sum_{\|x-y\|=\sqrt{2}\atop \subset B}v'n_xn_y
-\sum_{x\in B}\frac{1}{4}\mu n_x.
\ed
On a square $B$ there are $3^{4}$ possible configurations. Since we consider
the case $J>0$, we can exclude all configurations on the square $B$ where
one pair of nearest neighbour
spins is parallel. This reduces the number of possible configurations to
$35$. The invariance of
the Hamiltonian under rotations by $\pi/2$ and under spin flip splits these
configurations into 6 groups listed below.
\begin{displaymath}
\omega_{B} \in \left\{
\left(\begin{array}{cc}+&-\\-&+\end{array}\right)
\left(\begin{array}{cc}+&-\\-&\circ\end{array}\right)
\left(\begin{array}{cc}\circ&\circ\\-&+\end{array}\right)
\left(\begin{array}{cc}\circ&-\\+&\circ\end{array}\right)
\left(\begin{array}{cc}\circ&\circ\\\circ&+\end{array}\right)
\left(\begin{array}{cc}\circ&\circ\\\circ&\circ\end{array}\right)\right\}
\end{displaymath}
The groundstates are obtained by constructing configurations which, when
restricted to an arbitrary square $B$, are among those 35 allowed
configurations. Note that the energy per site of any one of these
configurations, $\omega$, is equal to the value of $\phi_{B}(\omega)$. The
zero-temperature phase diagram of $H_{t-J}(0,0)$ is shown in Figs.
\ref{cpd1}, \ref{cpd2} and \ref{cpd3} for different
domains of the parameters.
On the phase boundaries of the phase diagram, the allowed configurations on
each square $B$ are those allowed on both sides of the boundaries, except when
$v=J$, where additional allowed configurations exist; see Fig.\ref{cpd2}.
\begin{figure}
\begin{picture}(250,180)(-100,-20)
\put(170,40){\vector(1,0){40}}
%\put(100,0){\line(0,1){40}}
\put(100,120){\vector(0,1){20}}
\put(100,40){\line(-2,-1){70}}
\multiput(100,40)(0,20){4}{\line(0,1){10}}
\put(190,30){$\mu$}
\put(110,130){$v'$}
\put(20,50){$\begin{array}{cc}\circ&\circ\\\circ&\circ\end{array}$}
\put(140,10){$\begin{array}{cc}+&-\\-&+\end{array}$}
\put(140,125){$\begin{array}{cc}\circ&\circ\\-&+\end{array}$}
\multiput(100,40)(20,5){5}{\line(4,1){10}}
\put(200,70){$\mu=4v'$}
\put(0,-10){$\mu=2v'$}
\put(80,105){\vector(1,0){20}}
\put(55,102.5){$\begin{array}{cc}\circ&\circ\\\circ&+\end{array}$}
\put(60,10){\vector(-1,0){20}}
\put(60,10){$\begin{array}{cc}\circ&-\\+&\circ\end{array}$}
\put(165,77.2){\vector(0,-1){20}}
\put(150,80){$\begin{array}{cc}+&-\\-&\circ\end{array}$}
\end{picture}
\caption{Zero-temperature phase diagram of $H_{t-J}(0,0)$ for $v=J$}
\label{cpd2}
\end{figure}
On the solid lines, the groundstates are those of both regions adjacent
to the line. Hence the phase diagram is regular on the solid
lines.
On the dashed lines, apart from the groundstates of both regions adjacent
to the line, there are infinitely many other groundstates. The phase diagram
is therefore not regular on the dashed lines.
\begin{figure}
\begin{picture}(350,200)(-40,50)
\put(310,100){\vector(1,0){40}}
%\put(100,30){\line(0,1){70}}
\put(100,180){\vector(0,1){20}}
\put(100,100){\line(-2,-1){70}}
\multiput(100,100)(0,20){4}{\line(0,1){10}}
\put(350,90){$\mu$}
\put(110,190){$v'$}
\multiput(100,100)(20,5){4}{\line(4,1){10}}
\multiput(180,120)(20,0){4}{\line(1,0){10}}
\multiput(180,120)(0,20){4}{\line(0,1){10}}
\multiput(260,100)(0,10){2}{\line(0,1){5}}
\multiput(260,120)(20,5){3}{\line(4,1){10}}
\multiput(260,100)(20,5){3}{\line(4,1){10}}
\put(260,100){\line(-2,-1){70}}
\put(40,160){$ \begin{array}{cc}\circ&\circ\\\circ&\circ\end{array}$}
\put(115,160){$ \begin{array}{cc}\circ&\circ\\\circ&+\end{array}$}
\put(210,160){$ \begin{array}{cc}\circ&\circ\\-&+\end{array}$}
\put(130,80){$ \begin{array}{cc}\circ&-\\+&\circ\end{array}$}
\put(270,70){$ \begin{array}{cc}+&-\\-&+\end{array}$}
\put(268,115){\tiny$\begin{array}{cc}+&-\\-&\circ\end{array}$}
\put(160,200){$\mu=\frac{(v-J)}{2}$}
\put(20,50){$\mu=2v'$}
\put(180,50){$\mu=2(v-J)+2v'$}
\put(320,145){$\mu=(v-J)+4v'$}
\put(320,110){$\mu=2(v-J)+4v'$}
\put(200,130){$\frac{(v-J)}{4}$}
\put(115,115){$\mu=4v'$}
\end{picture}
\caption{Zero-temperature phase diagram of $H_{t-J}(0,0)$ for $v>J$}
\label{cpd3}
\end{figure}
The low-temperature phase diagram of this model has been studied by Gruber
and S\"uto \cite{grusu}. They show how to extend the usual Peierls argument
and Pirogov-Sinai theory to some models with an infinite number of
groundstates.
Their key idea is to introduce an equivalence relation among groundstates
and to replace the concept of groundstates by the concept of equivalence
classes. Furthermore, they introduce some conditions replacing the Peierls
condition. A necessary condition for their Peierls argument to apply is that
the
model has a residual entropy. They prove e.g. that the ordering of the
groundstates described by
$\tiny\left(\begin{array}{cc}\circ&-\\+&\circ\end{array}\right)$ survives
at finite temperature, and that there are coexistence lines between three or
four phases which are small deformations of the solid lines shown in
Fig.\ref{cpd3}.
\subsection{The stable phases of the $t-J-\lambda-v-v'$ model}
In this section, we discuss the stability of phases under perturbations
in open domains of parameter space excluding the phase boundaries of
$H_{t-J}(0,0)$. The phase boundaries are discussed in Sections 3.3 and 3.4.
For the region of the phase diagram where the groundstates of
$H_{t-J}(0,0)$ are the N\'eel states or the empty lattice, one can directly
apply the low-temperature expansion of \cite{dff} (Theorem 1.1) and prove the
persistence of
long-range order for small values of $\lambda$ and $t$ and for low enough
temperatures.
For the other phases the degeneracy is infinite. An explicit description of
the degeneracy in different regions of the phase diagram follows below. Then
one has to apply the perturbation scheme described in Appendix A and compute
the groundstates of $H_0(\lambda,t)$.
The perturbations are given by
\bd
V=\sum_{X=\langle xy \rangle} V_X \nonumber \,, \;\;\;
K=\sum_{X=\langle xy \rangle} K_X \nonumber \;,
\ed
where $X=\langle xy \rangle$ is a pair of nearest neighbours, and $V_X$ and
$K_X$ are given in eqs. (\ref{vp}) and (\ref{kp}). The unitary
transformation is given by $U(\lambda,t)=\exp{(tS^{K}+\lambda S^{V})}$,
where
\begin{eqnarray}
S^K&=&\sum_X S^{K}_{B_X}=\sum_X
{\rm ad}^{-1}{\bar H}_{0X}\left(K^{01}_{B_{X}}\right) \\
S^V&=&\sum_X S^{V}_{B_X}=\sum_X
{\rm ad}^{-1}{\bar H}_{0X} \left(V^{01}_{B_{X}}\right).
\label{unit}
\end{eqnarray}
Here $B_X$ is the set of sites at distance $\le 1$ from $X$;
the operators ${\bar H}_{0X}$, $K^{01}_{B_{X}}$ and $V^{01}_{B_{X}}$ are
defined in
Appendix A (see (\ref{v01}), (\ref{hbar}) and (\ref{adah})).
The transformed Hamiltonian, ${\widetilde H}_{t-J}(\lambda,t)$, has the form
\begin{eqnarray}
{\widetilde H}_{t-J}(\lambda,t)\,&=&\, U(\lambda,t) H_{t-J}(\lambda,t)
U(\lambda,t)^{-1} \;, \nonumber \\
&=&\, H_0(\lambda,t) + V(\lambda,t) \nonumber
\end{eqnarray}
where
\begin{eqnarray}
H_0(\lambda,t)\,&=&\,H_{t-J}(0,0) + t\sum_{X=\langle xy \rangle}K^{00}_{B_X}
+ \lambda \sum_{X=\langle xy \rangle} V^{00}_{B_X} \nonumber \\
&&+\frac{t^2}{2}\sum_{X=\langle xy \rangle}P^0_{B_X}{\rm
ad}S^{K}_{B_X}(K^{01}_{B_X})P^0_{B_X} \nonumber \\
&&+ \frac{\lambda^2}{2}\sum_{X=\langle xy \rangle}P^0_{B_X}{\rm
ad}S^{V}_{B_X}(V^{01}_{B_X})P^0_{B_X} \nonumber \\
&&+\frac{\lambda t}{2}\sum_{X=\langle xy \rangle}P^0_{B_X}\left({\rm
ad}S^{K}_{B_X} (V^{01}_{B_X})+ {\rm
ad}S^{V}_{B_X}(K^{01}_{B_X})\right)P^0_{B_X} \;, \label{til}
\end{eqnarray}
and $K^{00}_{B_X}$ and $V^{00}_{B_X}$ are given by (\ref{v00}). The operator
$V(\lambda,t)$ can be written as a sum of local terms, which, by
Lemma 1.3, satisfy condition {\bf V1} (see Appendix A).
Next, we compute the groundstates of $H_0(\lambda,t)$ for the different
parts of the phase diagram. Note that $K^{00}_{B_X}$ and
the last two terms of (\ref{til}) vanish: there is no contribution of order
$t$ and of order $\lambda t$. The other
terms yield
\begin{eqnarray}
\frac{t^2}{2}P^0_{B_X}{\rm ad}S^{K}_{B_X}(K^{01}_{B_X})P^0_{B_X}&=&-t^2
\sum_{X}\frac{1}{E^K_{1X}-E_0}P^0_{B_X}\\
\frac{\lambda^2}{2}P^0_{B_X}{\rm ad}S^{V}_{B_X}(V^{01}_{B_X})P^0_{B_X}
&=&-\lambda^2 \sum_{X}\frac{1}{E^V_{1X}-E_0}P^0_{B_X} \;,
\end{eqnarray}
where $E_0$ is the groundstate energy of $H_{t-J}(0,0)$, $E^K_{1X}$ is the
energy of the excitation caused by the hopping of a particle, and $E^V_{1X}$ is
the energy of the excitation caused by flipping a pair of nearest neighbour
spins. We have the following results for the different regions of the phase
diagram (see Figs.\ref{cpd1}, \ref{cpd2} and \ref{cpd3}).
\medskip
\noindent
(i) $\omega_B \in
\left\{\tiny\left(\begin{array}{cc}+&-\\-&\circ\end{array}\right)\right\}$.
The
groundstates of $H_{t-J}(0,0)$ consist of lines of N\'eel-ordered spins
alternating with lines with particle density $1/2$. The degeneracy is
infinite since the positions of the holes in each line are independent of the
other lines.
Both kinds of terms, the magnetic terms and the hopping terms, lift the
degeneracy of
the groundstates. The magnetic fluctuations favour the arrangement of
holes ($1/4_d$) shown on the left of Fig.\ref{1/4}. The hopping terms
favour the same arrangement, for $v' < (v-J)/4$, and the arrangement
($1/4_v$) shown
on the right of Fig.\ref{1/4}, for $v' > (v-J)/4$. The corresponding
groundstate energy densities are given by
\begin{figure}
\begin{center}
$\begin{array}{ccccccccccccc}
-&+&-&+&-&+ &\hspace{1cm}&-&+&-&+&-&+ \\
+&\circ&+&\circ&+&\circ&&+&\circ&+&\circ&+&\circ \\
-&+&-&+&-&+ &&-&+&-&+&-&+ \\
\circ&-&\circ&-&\circ&- &&+&\circ&+&\circ&+&\circ \\
-&+&-&+&-&+ &&-&+&-&+&-&+ \\
+&\circ&+&\circ&+&\circ &&+&\circ&+&\circ&+&\circ
\end{array}$
\end{center}
\caption{Groundstates $(1/4_d)$ and $(1/4_v)$ of $H_0(\lambda,t)$, for
hole density $1/4$}\label{1/4}
\end{figure}
\begin{eqnarray}
e(1/4_{d})&=& \frac{(v-J)}{2}+v'-\frac{3\mu}{4} -\frac{7\lambda^{2}}{6J}
-t^2\left(\frac{1}{4v'+6J} + \frac{1}{v+5J}\right) \nonumber \\
e(1/4_{v})&=&\frac{(v-J)}{2}+v'-\frac{3\mu}{4}
-\frac{\lambda^{2}}{J} - t^2\left(\frac{2}{v+5J}\right). \nonumber
\end{eqnarray}
\smallskip
\noindent
(ii) $\omega_B \in \left\{\tiny
\left(\begin{array}{cc}\circ&\circ\\\circ&+\end{array}\right)\right\}$.
The
groundstates of $H_{t-J}(0,0)$ can be described as lines of holes alternating
with lines with particle density $1/2$. The degeneracy is infinite
since the positions of the particles in one line are independent of the other
lines and the orientations of the spins are arbitrary.
The magnetic terms do not lift the degeneracy of the groundstates.
The hopping terms favour antiferromagnetic
ordering between spins at distance $2$ within the lines.
The hopping terms favour the groundstates $(3/4_v)$ shown on the left of
Fig.\ref{3/4}, for $v' > (v-J)/4$,
and the groundstates $(3/4_d)$ shown on the right of Fig.\ref{3/4}, for $v' <
(v-J)/4$. The degeneracy
of the groundstates $(3/4_d)$ is still infinite. (The spins in every line
of density $1/2$ can be flipped, independently in different lines.)
We expect this
degeneracy to be lifted only by the terms of order $t^4$ of the perturbation
series.
The corresponding groundstate energy densities are given by
\begin{figure}
\begin{center}
$\begin{array}{ccccccccccccc}
\circ&\circ&\circ&\circ&\circ&\circ&
&\circ&\circ&\circ&\circ&\circ&\circ\\
+&\circ&-&\circ&+&\circ &\hspace{2cm}&\circ&+&\circ&-&\circ&+\\
\circ&\circ&\circ&\circ&\circ&\circ&
&\circ&\circ&\circ&\circ&\circ&\circ\\
-&\circ&+&\circ&-&\circ&&-&\circ&+&\circ&-&\circ\\
\circ&\circ&\circ&\circ&\circ&\circ&
&\circ&\circ&\circ&\circ&\circ&\circ\\
+&\circ&-&\circ&+&\circ &&\circ&+&\circ&-&\circ&+
\end{array}$
\end{center}
\caption{Groundstates $(3/4_v)$ and $(3/4_d)$ of $H_0(\lambda,t)$, for
hole density
$3/4$}\label{3/4}
\end{figure}
\begin{eqnarray}
e(3/4v)&=&\frac{(v-J)}{2}+v'-\frac{1}{4}\mu - t^2
\left(\frac{2}{v-J}\right) \nonumber \\
e(3/4d)&=&\frac{(v-J)}{2}+v'-\frac{1}{4}\mu - t^2 \left(\frac{1}{v-J} +
\frac{1}{2v'}\right). \nonumber
\end{eqnarray}
\smallskip
\noindent
(iii) $\omega_B \in \left\{\tiny
\left(\begin{array}{cc}\circ&\circ\\+&-\end{array}\right)\right\}$.
The groundstates of $H_{t-J}(0,0)$ are lines of holes alternating with lines
of N\'eel-ordered spins.
The magnetic terms do not lift the degeneracy of the groundstates. The
hopping terms do: they favour the antiferromagnetic ordering $(1/2_{af})$ of
the spins at distance two (Fig.\ref{1/2}). The groundstate energy density is
given by
\bd
e(1/2_{af})= \frac{(v-J)}{4}-\frac{\mu}{2} -\frac{\lambda^{2}}{J}
-\frac{2t^2}{8v' - (v-J)}. \nonumber
\ed
\smallskip
\begin{figure}
\begin{center}
$\begin{array}{cccccc}
\circ &\circ &\circ&\circ &\circ&\circ \\
+&-&+&-&+&- \\
\circ &\circ &\circ&\circ &\circ&\circ \\
-&+&-&+&-&+ \\
\circ &\circ &\circ&\circ &\circ&\circ \\
+&-&+&-&+&-
\end{array}$
\end{center}
\caption{Groundstates $(1/2_{af})$ of $H_0(\lambda,t)$, for hole
density $1/2$}\label{1/2}
\end{figure}
\noindent
(iv) $\omega_B \in \left\{\tiny
\left(\begin{array}{cc}\circ&-\\+&\circ\end{array}\right)\right\}$.
In the groundstates of $H_{t-J}(0,0)$ the holes and the particles are
arranged in a
checkerboard configuration, the orientation of the spins is arbitrary. We
discuss this case in \cite{dfr}. It is shown there that, for $v'>5J/3$, the
degeneracy of the groundstates becomes finite in order $t^4$; the spins are
aligned in one diagonal direction and N\'eel ordered in the other
diagonal direction.
\medskip
We now apply Theorem 1.1 (the low-temperature expansion) to the
Hamiltonian
\bd
{\widetilde H}_{t-J}(\lambda, t)\,=\,H_0(\lambda,t)\,+\,V(\lambda,t),
\ed
in the region of parameters, where $H_0(\lambda,t)$ has a finite
number of groundstates.
Lemma 1.3 ensures that $V(\lambda,t)$ satisfies condition {\bf
V1}. Condition {\bf H1} is clearly satisfied, since $H_0(\lambda,t)$
is diagonal in the same basis as $H_0$. To check the Peierls condition, we use
the criterion of Holzstynsky and Slawny, (see \cite{hol} and Appendix B). We
decompose the
Hamiltonian as follows,
\bd
H_0(\lambda,t)=\sum_{M \subset \Lambda} \phi_{M}(\lambda,t),
\ed
where $M$ is a four by four box containig 16 lattice sites, and
\begin{eqnarray}
&&\phi_{M}(\lambda,t) = \sum_{\|x-y\|=1 \atop \langle xy \rangle
\subset M} \frac{1}{24}\left(J\sigma_{x}^{(3)}\sigma_{y}^{(3)}
+v\right)n_xn_y + \sum_{\|x-y\|=\sqrt2 \atop \langle xy \rangle \subset M}
\frac{v'}{9}n_xn_y \nonumber \\
&& - \sum_{x} \frac{\mu}{16} n_x
+\frac{1}{2}\sum_{X=\langle xy\rangle \,:\atop B_{X} \subset M} P^0_{B_X}\left(
\frac{\lambda^2}{2} {\rm ad}S^{V}_{1B_{X}}(V^{01}_{B_{X}}) + \frac{t^2}{2}
{\rm ad}S^{K}_{1B_{X}}(K^{01}_{B_{X}})\right)P^0_{B_X}. \nonumber
\end{eqnarray}
Since the periodic groundstates found in this section (their degeneracy is
finite) have all period 4, it is easily seen that the terms $\{
\phi_M\}$ form an $m$-potential and hence, by the criterion quoted in
Appendix B, $ H_0(\lambda,t)$
satisfies the Peierls condition. Using Theorem 1.1 and Lemma 1.3, we obtain
the following result.
\begin{proposition}
For each choice of $\mu$, $v-J$ and $v'$ for which the groundstates of
$H_0(\lambda,t)$ are $(1/4_d)$, $(1/4_v)$, $(3/4_v)$ or $(1/2_{af})$
(see Figs \ref{1/4}, \ref{3/4} and \ref{1/2}) there are constants $t_0$ and
$\lambda_0$ depending on $\mu$, $v-J$ and $v'$ such that
\begin{enumerate}
\item For $0< \lambda < \lambda_0$ and $0 < t 0$, we conjecture that one can use
low-temperature expansion
methods to prove that, in a small neighbourhood of $\{ J=v \,, v'=0\,,
\mu=0\}$,
and for low-enough temperatures, the ordering described on the left of
Fig. \ref{dim} survives.
\begin{figure}
\begin{center}
$\begin{array}{cccccccc}
\circ \hspace{0.3cm} \circ& sdi &\circ \hspace{0.3cm} \circ&
&\circ \hspace{0.3cm} \circ& \circ & \circ \hspace{0.3cm} \circ& \circ \\
sdi &\circ \hspace{0.3cm} \circ& sdi &
&sdi&\circ&sdi&\circ \\
\circ \hspace{0.3cm} \circ& sdi &\circ \hspace{0.3cm} \circ&
\hspace{2cm}&\circ \hspace{0.3cm} \circ&\circ&\circ \hspace{0.3cm} \circ&\circ
\\
sdi &\circ \hspace{0.3cm} \circ& sdi &
&sdi&\circ&sdi&\circ \\
\circ\hspace{0.3cm} \circ& sdi &\circ \hspace{0.3cm} \circ&
&\circ \hspace{0.3cm}\circ&\circ&\circ \hspace{0.3cm} \circ&\circ \\
sdi &\circ \hspace{0.3cm} \circ& sdi &
&sdi&\circ&sdi&\circ \\
&&&sdi = \frac{1}{\sqrt{2}}\left(|\uparrow \downarrow \rangle -
|\downarrow \uparrow \rangle\right)&&&&
\end{array}$
\end{center}
\caption{Dimer configurations with hole densities $1/2$ and
$2/3$}\label{dim}
\end{figure}
\item
$\mu=0,v'>0$,
$\omega_{B} \in \left\{\tiny
\left(\begin{array}{cc}\circ&\circ\\-&+\end{array}\right)
\left(\begin{array}{cc}\circ&\circ\\\circ&+\end{array}\right)
\left(\begin{array}{cc}\circ&\circ\\\circ&\circ\end{array}\right)\right\}$.
The groundstates of $H_{t-J}(0,0)$ consist of segments of lines of particles of
arbitrary length isolated by walls of holes. For $\lambda \ll v'$, the
magnetic perturbation favours
the phases of dimers with hole density $2/3$ described in
Fig.\ref{dim}, since these are the only states whose energy has a
contribution proportional to $\lambda$. Their energy density is
\begin{equation}
e(2/3_{d})=\frac{(v-J)}{12}-\frac{\mu}{3}-\frac{\lambda}{3} + O(\lambda^2,t^2).
\label{23}\end{equation}
Note that the degeneracy of the groundstates remains infinite.
\item The line $\mu=4v'$ is discussed in Sect. 3.4.
\end{itemize}
\smallskip
\begin{figure}
\begin{picture}(250,180)(-110,-20)
\put(180,40){\vector(1,0){20}}
%\put(0,40){\line(1,0){95}}
\put(95,35){\dashbox{2}(10,10)}
%\put(100,0){\line(0,1){35}}
\put(100,120){\vector(0,1){20}}
\put(95,37.5){\line(-2,-1){70}}
\multiput(95,45)(0,10){8}{\line(0,1){5}}
\multiput(105,45)(0,10){8}{\line(0,1){5}}
\put(100,90){\vector(-1,0){20}}
\put(55,90){${\large 2/3_{d}}$}
\put(100,40){\vector(0,-1){20}}
\put(90,10){${\large 1/2_{d}}$}
\put(190,30){$\mu$}
\put(110,120){v'}
\put(30,60){$\begin{array}{cc}\circ&\circ\\\circ&\circ\end{array}$}
\put(140,10){$\begin{array}{cc}+&-\\-&+\end{array}$}
\put(135,60){{\large Quantum}}
\put(135,47.5){{\large Liquid ?}}
\put(140,90){${\large 1/2_{af}}$}
\multiput(105,41.25)(20,5){5}{\line(4,1){10}}
\end{picture}
\caption{Zero-temperature phase diagram of $H_0(\lambda,t)$ for $v=J$}
\label{qpd2}
\end{figure}
\noindent
(iii) $v>J$ (see Fig. \ref{cpd3}).
\begin{itemize}
\item
On the lines $\mu=0$, ($\mu=4v'$),
$\omega_{B}\in
\left\{\tiny\left(\begin{array}{cc}\circ&\circ\\\circ&+\end{array}\right)
\left(\begin{array}{cc}\circ&\circ\\\circ&\circ\end{array}\right)\right\}$
$ \left(\left\{\tiny\left(\begin{array}{cc}\circ&-\\+&\circ\end{array}\right)
\left(\begin{array}{cc}\circ&\circ\\\circ&+\end{array}\right)\right\},
{\rm resp.}\right)$.
The groundstates of $H_{t-J}(0,0)$ consist of isolated particles with hardcore
conditions. The magnetic perturbation does not lift the degeneracy of the
groundstates. This is a domain of the phase diagram where we do not
understand how the hopping
perturbation lifts the degeneracy.
\item
$\mu=(v-J)$, $v' \ge (v-J)/4$, $\omega_{B} \in \left\{\tiny
\left(\begin{array}{cc}\circ&\circ\\-&+\end{array}\right)
\left(\begin{array}{cc}\circ&\circ\\\circ&+\end{array}\right)\right\}$.
The magnetic perturbation again favours the phase of dimers with hole density
$2/3$ shown on the right of Fig.\ref{dim}, with energy density given in
(\ref{23}).
\item
$v'=(v-J)/4$,
$ \omega_{B} \in
\left\{\tiny\left(\begin{array}{cc}\circ&\circ\\-&+\end{array}\right)
\left(\begin{array}{cc}\circ&-\\+&\circ\end{array}\right)\right\}.$
The groundstates of $H_{t-J}(0,0)$ consist of consecutive lines with hole
density $1/2$, the position of the holes in one line being independent of the
position of the holes in the adjacent lines. It is not difficult to check that
the magnetic perturbation favours the groundstate of dimers with hole density
$1/2$, shown on the left of Fig.\ref{dim}, with energy density given in
(\ref{12}).
\end{itemize}
\begin{figure}
\begin{center}
\begin{picture}(320,180)(20,50)
\put(300,100){\vector(1,0){30}}
\put(100,100){\line(-2,-1){70}}
\put(100,180){\vector(0,1){20}}
\multiput(100,100)(0,20){4}{\line(0,1){10}}
\put(330,90){$\mu$}
\put(110,190){$v'$}
\multiput(100,100)(20,5){4}{\line(4,1){10}}
\multiput(100,120)(20,0){4}{\line(1,0){10}}
\multiput(260,120)(20,0){4}{\line(1,0){10}}
\put(177.5,117.5){\dashbox{2}(80,5)}
\put(177.5,122.5){\dashbox{2}(5,80)}
\put(257.5,100){\dashbox{2}(5,20)}
\multiput(260,120)(20,5){4}{\line(4,1){10}}
\multiput(260,100)(20,5){4}{\line(4,1){10}}
\put(260,100){\line(-2,-1){70}}
\put(260,110){\vector(-2,-1){20}}
\put(40,160){$ \begin{array}{cc}\circ&\circ\\\circ&\circ\end{array}$}
\put(210,128){${\large (1/2_{d})}$}
\put(185,160){${\large (2/3_{d})}$}
\put(210,100){${\large (1/3_{l})}$}
\put(120,160){${\large (3/4_{v})}$}
\put(105,110){${\large (3/4_{d})}$}
\put(265,110){${\large (1/4_{d})}$}
\put(310,125){${\large (1/4_{v})}$}
\put(240,160){${\large (1/2_{af})}$}
\put(140,80){${\large (1/2_{cb})}$}
\put(270,70){$ \begin{array}{cc}+&-\\-&+\end{array}$}
\end{picture}
\end{center}
\caption{Zero-temperature phase diagram of $ H_0(\lambda,t)$ for $v>J$}
\label{qpd3}
\end{figure}
\subsection{Effective Hamiltonians for configurations of holes in an
antiferromagnetic background}
In this section, we investigate some non-regular parts of the phase diagrams
shown in Figs.\ref{cpd2} and
\ref{cpd3}, where the groundstates of $H_{t-J}(0,0)$ can be described as a
``liquid'' of holes. The positions of the holes are constrained by an extended
hardcore condition, and the density of holes must belong to a certain interval.
\begin{enumerate}
\item For $v>J$, on the line $\mu=2(v-J)+4v'$, the groundstates correspond
to arbitrary
configurations of holes in the antiferromagnetic background without any
pairs of nearest neighbour (n.n.) and next nearest neighbour (n.n.n.) holes,
and the hole density is between $0$ and $1/4$.
\item For $v>J$, on the line $\mu=2(v-J)$, the groundstates correspond
to arbitrary
configurations of holes without any pairs of n.n., and the hole density
is between $0$ and $1/2$, for $v'=0$, and between $1/4$ and
$1/2$, for $v'>0$.
\item For $v=J$ (resp. $v>J$), on the line $\mu=4v'$ (resp. $\mu=(v-J)+4v'$),
the groundstates correspond
to arbitrary configurations of holes without any pair of
n.n.n., and the hole density is between $0$ and $1/2$, for $v=J$,
(between $1/4$ and $1/2$, for $v>J$, respectively).
\end{enumerate}
We shall consider only cases 1 and 2, for $\lambda > 0$, $t=0$ (case 3 can be
treated in a similar way), and case 3 for $t>0$.
We first consider the case $\lambda > 0, t=0$.
In order to find the groundstates of $H_0(\lambda,0)$, we
work in the canonical ensemble, i.e., we fix the number of holes and compute
their interaction by performing the unitary transformation (\ref{unit}). In a
sense, we eliminate ("trace out") the spin degrees of freedom and obtain
an effective
(Ising-type) Hamiltonian governing the arrangement of holes in the
antiferromagnetic background.
The energy, $E(\{x_{1}, \ldots , x_{n}\})$, of a configuration of holes
at sites $x_{1}, \ldots , x_{n}$ is normalized by
subtracting the energy of the half-filled lattice,
$E_{h.f.}=|\Lambda|\left((v-J)+2v'-4\lambda^2/3J\right)$. We set
\begin{eqnarray}
\varepsilon(\{x_{1}, \ldots , x_{n}\})&=&\varepsilon(x_{1}-x_{2}, \ldots,
x_{1}-x_{n}) \nonumber\\ & \equiv & E(\{x_{1}, \ldots , x_{n}\}) - E_{h.f.},
\nonumber
\end{eqnarray}
and recursively define {\em N-body potentials}
\begin{eqnarray}
&&{\cal U}_{N}(\{x_{1}, \ldots , x_{N}\})={\cal U}_{N}(x_{1}-x_{2}, \ldots,
x_{1}-x_{N}) \nonumber \\
&& = \varepsilon(\{x_{1}, \ldots , x_{N}\})
-\sum_{j=1}^{N-2} \sum_{\{i_{1},\ldots, i_{j}\}\subset \{1,\ldots
,N\}}{\cal U}_{N-j}(\{x_{1},
\ldots , x_{N}\} \backslash \{x_{i_{1}}, \ldots , x_{i_{j}}\})
-\sum_{i=1}^{N}
\varepsilon(x_{i}).\nonumber
\end{eqnarray}
All the energies and $N$-body potentials are invariant under
translations, rotations by $\pi/2$ and reflections.
It is now straightforward, but lengthy, to compute the effective Hamiltonians
in different regimes. They have the following form
\begin{eqnarray}
H_{eff}&=&\sum_{x}\varepsilon(x) n_{x}+\sum_{xy}{\cal U}_{2}(x-y) n_{x}n_{y} +
\sum_{xyz}{\cal U}_{3}(x-y,x-z) n_{x}n_{y}n_{z} \nonumber \\
&&+ \sum_{xyzw}{\cal U}_{4}(x-y,x-z,x-w)n_{w}n_{x}n_{y}n_{z}
\end{eqnarray}
where $n_x=0,1$ is the hole number operator at site $x$.
The hole selfenergy is given by
\begin{equation}
\varepsilon(x) \equiv \varepsilon = -2(v-J)-4v'+\frac{16\lambda^{2}}{15J}.
\label{se}
\end{equation}
The two-body potentials are given by
\begin{eqnarray}
{\cal U}_{2}(e_{1})&=& \frac{(v-J)}{2}, \label{2-1}\\
{\cal U}_{2}(e_{1}+e_{2})& = & v' +\frac{4\lambda^{2}}{15J},\label{2-2}\\
{\cal U}_{2}(2e_{1})&=& \frac{2\lambda^{2}}{15J}, \label{2-3}\\
{\cal U}_{2}(2e_{1}+e_{2})&=&- \frac{3\lambda^{2}}{15J},\label{2-4}\\
{\cal U}_{2}(3e_{1})&=& -\frac{1\lambda^{2}}{15J}, \label{2-5}
\end{eqnarray}
where $e_1$ and $e_2$ are the unit vectors of the lattice $\zed^{2}$.
The three-body and four-body potentials are
considered below, for different regimes.
Finding the groundstates of such Ising-type Hamiltonians is, a priori, not
easy. One way of finding them is to prove that one can express $H_{eff}$ in
terms of an $m$-potential. This is usually not possible. A more refined way,
(see \cite{ken}), is to find a decomposition $H_{eff}=\sum_{M}\phi_M$ of
the Hamiltonian and an auxiliary family of interactions $K_{M}$ such that
$\sum_{M}K_{M}=0$. Then one has to find configurations minimizing $\phi_M
+ K_M$. We expect that this method would yield rigorous results on
the groundstates of the Hamiltonians derived above, but we shall not prove
such results in this paper.
For the case of a n.n and a n.n.n hardcore condition, the 2-body potentials are
given by (\ref{2-3}), (\ref{2-4}) and (\ref{2-5}). The only other
non-vanishing many-body potential is the 3-body potential; see
Fig.\ref{3b}. In Fig.\ref{n.n.n.n.}, we show the conjectured groundstates for
hole density $1/5$ and $1/4$.
\begin{figure}
\begin{center}
\begin{tabular}{llll}
$\bullet$&$\circ$&$\bullet$&$-\frac{\lambda^{2}}{15J}$ \\
$\bullet$&$\bullet$&$\bullet$& \\
$\circ$&$\bullet$&$\circ$&
\end{tabular}
\end{center}
\caption{Three-body potential in the presence of n.n. and n.n.n.
hardcore condition}
\label{3b}
\end{figure}
\begin{figure}
\begin{center}
$\begin{array}{ccccccccccccc}
-&+&-&+&\circ&+& \hspace{2cm} &\circ&+&\circ&+&\circ&+ \\
+&-&\circ&-&+&-& &+&-&+&-&+&- \\
\circ&+&-&+&-&\circ& &-&\circ&-&\circ&-&\circ \\
+&-&+&\circ&+&-& &+&-&+&-&+&- \\
-&\circ&-&+&-&+& &\circ&+&\circ&+&\circ&+ \\
+&-&+&-&\circ&-& &+&-&+&-&+&-
\end{array}$
\end{center}
\caption{Groundstates of $H_{eff}$ with n.n. and n.n.n. hardcore conditions for
densities $1/5$ and $1/4$} \label{n.n.n.n.}
\end{figure}
The corresponding energy densities are
\begin{eqnarray}
e(1/5)&=&\frac{3(v-J)}{5}+\frac{6v'}{5}-\frac{4\mu}{5}
-\frac{6\lambda^{2}}{5J} \label{e5}\\
e(1/4_{d})&=& \frac{(v-J)}{2}+v'-\frac{3\mu}{4} -\frac{7\lambda^{2}}{6J}.
\label{e4}
\end{eqnarray}
In the corresponding part of the phase diagram
\bd
v > J,\,\,\, \mu=2(v-J)+4v'-2\lambda^2/3J,
\ed
it is easy to check that the N\'eel states,
the groundstates with hole densities
$1/4$ and $1/5$ (Fig.\ref{n.n.n.n.}) are degenerate. This leads to the
following conjecture on the groundstates of $ H_0(\lambda,0)$:
On the line $\mu=2(v-J)+4v'-2\lambda^2/3J$ the
groundstates consist of parallel lines of lattice sites which are either
completely occupied by holes or completely occupied by particles, the slope of
the lines with respect to the lattice is $(2,1)$, $(2,-1)$, $(1,2)$ or
$(-1,2)$. The degeneracy is still infinite in this part of the phase diagram,
and, to this order, the phase diagram is not regular.
\begin{figure}
\begin{center}
\begin{tabular}{lllllllllllllll}
$\bullet$&$\circ$&$\bullet$&$\bullet$&$-\frac{\lambda^{2}}{15J}$\hspace{0.5cm}
&$\bullet$&$\bullet$&$\bullet$&$\bullet$&
$+\frac{2\lambda^{2}}{15J}$\hspace{0.5cm}
&$\bullet$&$\bullet$&$\bullet$&$\bullet$&$-\frac{\lambda^{2}}{15J}$ \\
$\bullet$&$\bullet$&$\bullet$&$\bullet$& &$\bullet$&$\circ$&$\bullet$&
$\bullet$& &$\bullet$&$\bullet$&$\circ$&$\bullet$& \\
$\circ$&$\bullet$&$\circ$&$\bullet$&(3b.1) &$\circ$&$\bullet$&$\circ$&
$\bullet$&(3b.2) &$\circ$&$\bullet$&$\bullet$&$\circ$&(3b.3) \\
\end{tabular}
\vspace{0.5cm}
\begin{tabular}{lllll}
$\bullet$&$\circ$&$\bullet$&$\bullet$&$-\frac{2\lambda^{2}}{15J}$ \\
$\bullet$&$\bullet$&$\circ$&$\bullet$& \\
$\circ$&$\bullet$&$\bullet$&$\bullet$&(3b.4) \\
\end{tabular}
\vspace{0.5cm}
\begin{tabular}{lllllllllllllll}
$\bullet$&$\bullet$&$\circ$&$\bullet$&$-\frac{2\lambda^{2}}{15J}$ \hspace{0.5cm}
&$\bullet$&$\circ$&$\bullet$&$\bullet$
&$+\frac{4\lambda^{2}}{15J}$\hspace{0.5cm}
&$\bullet$&$\circ$&$\bullet$&$\bullet$&$-\frac{2\lambda^{2}}{15J}$ \\
$\circ$&$\bullet$&$\bullet$&$\circ$& &$\circ$&$\bullet$&$\circ$&
$\bullet$& &$\circ$&$\bullet$&$\bullet$&$\circ$&\\
$\bullet$&$\circ$&$\bullet$&$\bullet$&(4b.1) &$\bullet$&$\circ$&$\bullet$&
$\bullet$&(4b.2) &$\bullet$&$\circ$&$\bullet$&$\bullet$&(4b.3)\\
\end{tabular}
\end{center}
\caption{3-body and 4-body potentials for n.n. hardcore condition} \label{3b4b}
\end{figure}
For the case of a n.n. hardcore condition, the 2-body potentials are
given by (\ref{2-2}), (\ref{2-3}), (\ref{2-4}) and (\ref{2-5}). The
non-vanishing 3-body
potentials are given by (3b.1), (3b.2) (3b.3)and (3b.4) in Fig.\ref{3b4b}, and
the non-vanishing 4-body potentials are given by (4b.1), (4b.2) and (4b.3) in
Fig.\ref{3b4b}.
We show, in Fig.\ref{n.n.}, the conjectured groundstates for density $1/3$.
The energy density is
\bd
e(1/3_l)=\frac{(v-J)}{3}+ v' - \frac{2\mu}{3}
-\frac{4\lambda^{2}}{3J}.
\ed
We conjecture that a new phase corresponding to the groundstates with density
$1/3$ (Fig.\ref{n.n.}) appears when $\mu$ belongs to the interval
\bd
\mu \in (2(v-J)-8\lambda^2/J, 2(v-J)+\lambda^2/J),
\ed
around the line $\mu = 2(v-J)$. Since
the groundstates are infinitely degenerate in this phase, one cannot predict
anything about its stability.
\begin{figure}
\begin{center}
$\begin{array}{cccccc}
\circ&$+$&-&\circ&$+$&- \\
$+$&\circ&$+$&-&\circ&$+$ \\
-&$+$&\circ&$+$&-&\circ \\
\circ&-&$+$&\circ&$+$&- \\
-&\circ&-&$+$&\circ&$+$ \\
$+$&-&\circ&-&$+$&\circ
\end{array}$
\end{center}
\caption{Groundstates of $H_{eff}$ with n.n. hardcore conditions for
densities $1/3$} \label{n.n.}
\end{figure}
\smallskip
\noindent
{\bf Possible existence of a Fermi liquid of holes}
\smallskip
\noindent
For the case $t\not= 0$, the situation is more complicated. Then the effective
Hamiltonian is, in general, {\it not} of finite range. Furthermore,
for $v=J$ and $\mu=4v'$, it is {\it not} of Ising-type: in order
$t^2$, there are terms which correspond to the hopping of a pair of nearest
neighbour holes: If $\langle xy \rangle$ denotes a pair nearest neighbour with
$y = x + e_j$, $j=1,2$
\bd
T_{hopp}\,=\,\sum_{\langle xy \rangle} -\frac{t^2}{J} (c_{x+e_j}^{\dagger}
c_{x} c^{\dagger}_{y+e_j} c_{y}\,+\,
c_{y - e_j}^{\dagger} c_{y} c^{\dagger}_{x-e_j} c_{x})
\ed
where
$c_{x}$ and $c_{x}^{\dagger}$ are the annihilation and creation operator for
holes
at site $x$ obeying (spinless) fermion anticommutation relations.
This is a region of the phase diagram where we conjecture that the model is
{\it not} a insulator (see \cite{fral} and \cite{alda} for related results).
As one of the key results of this paper we regard this identification of
regions in parameter space, where Fermi liquids of holes or superconductivity
(see \cite{fral}) might appear.
\subsection{Conclusions}
The phase diagram of the $t-J-v-v'$ model, for small $\lambda$ and $t$ and at
low
temperature, shows a surprisingly rich stucture.
The methods developed in \cite{dff} and \cite{dfr} permit us to investigate the
stability of finitely degenerate phases of $H_{t-J}(0,0)$ under small
perturbations and the degeneracy-breaking effect of the perturbations
(Proposition 3.1, Section 3.2). They enable us to show that the phase diagram
of
$H_{t-J}(\lambda,t)$ may be regular even in regions, where
the phase diagram of $H_{t-J}(0,0)$
is not (Proposition 3.2, Section 3.3). They also suggest the appearance of
several new phases, among them phases of dimers and, possibly, Fermi liquids
or superconducting phases (Section 3.4).
\begin{appendix}
\section{The Unitary conjugations}
We consider Hamiltonians, $H(t)$, of the form $H_0+tV$ satisfying
conditions
{\bf H1}, {\bf P1} and {\bf P2} specified in the introduction.
We briefly recall here the construction of the unitary transformation
$U(t)$ used throughout the paper and refer to \cite{dfr} for proofs and
for extension of the methods to higher orders.
By condition {\bf P1}, $H_0$ can be expressed in terms of
finite-range, translation invariant {\it{m}} - potentials (see \cite{hol} and
(\ref{mpot})).
We define the set of groundstate configurations of $H_0$ associated with a
subset $Y$ of $\Lambda$ as follows:
\bd
\Omega^{0}_{Y}= \{\omega_{Y} : \Phi^{0}_X(\omega_{Y}) = \min_{\atop\omega'_{X}}
\Phi^{0}_X(\omega'_{X}) \,\, {\rm for}\, {\rm any} \, X\subseteq Y\}
\ed
and let
\bd
{\cal{H}}_{Y}^{0} := \{\psi\in{\cal{H}}_{Y}:\psi = \sum_
{\omega_{Y} \in \Omega_{Y}^{0}}
c_{\omega_{Y}} e_{\omega_{Y}}\}
\ed
be the subspace of ${\cal{H}}_{Y}$ spanned by the local groundstates in $Y$.
We define $P_{Y}^0$ to be the projection operator onto the subspace
${\cal{H}}_{Y}^{0}$.
Let $X$ be a finite subset of the lattice (which may be thought of as the
support
of an excitation). For $x\in X $, we define
\bd
W_x\,:=\,\left\{y \in \Lambda\,:\,|y_i - x_i| \leq r, \,{\rm for}\,1\leq i \leq
\nu\right\},
\ed
\begin{equation}
B_X\,:=\,\bigcup_{x\in X} W_x, \label{bex}
\end{equation}
$r$ being the range of the {\it{m}}-potential, (eq.(\ref {range})).
We introduce some special projection operators on
${\cal{H}}_{B_{X}}$:
Let $P^{0}_{B_X}$ be the orthogonal projection onto the subspace ${\cal
H}^{0}_{B_X}$.
Let $P^{1}_{B_X}$ be the orthogonal projection onto the space of states with
excitations localized in $X\subset B_{X}$. It is defined as
\begin{equation}
P^{1}_{B_X} := P^{0}_{B_X \setminus X} - P^{0}_{B_X}.\label{proj1}
\end{equation}
Two excitations with support in $X$ and $X'$ are said to be connected if
$X'\cap B_X \not= \emptyset$. If they are disconnected, our choice of $B_X$
implies that their unperturbed energies are additive.
We further define
\bd
P^{2}_{B_X} := {\bf 1}_{B_X} - P^{1}_{B_X} - P^{0}_{B_X},
\ed
where ${\bf 1}_{B_X}$ is the identity operator on ${\cal H}_{B_X}$.
We decompose the perturbation $V$ with respect to the partition of unity
\begin{equation}
{\bf 1}_{B_X} = P^{0}_{B_X} + P^{1}_{B_X} + P^{2}_{B_X}, \label{pau}
\end{equation} and
since we restrict our attention to an analysis of the perturbation of the
groundstates of $H_0$ by the term $tV$, it is convenient to
write the perturbation $V=\sum_{X}V_X$ as
\bd
V= V^{00}+ V^{01} +V^R,
\ed
where $V^{00}$ is the operator defined by
\begin{equation}
V^{00} \bydef \sum_{X} P^{0}_{B_X} V_X P^{0}_{B_X}
\bydef \sum_{X} V^{00}_{B_X}, \label{v00}
\end{equation}
and $V^{01}$ is the ``off-diagonal'' operator:
\begin{equation}
V^{01}\bydef \sum_X P^{0}_{B_X}V_XP^{1}_{B_X}+P^{1}_{B_X}V_XP^{0}_{B_X}
\bydef \sum_X V^{01}_{B_X}. \label{v01}
\end{equation}
Here $P^{1}_{B_X}V_XP^{0}_{B_X}$ is an operator in ${\cal{A}}_{B_X}$ that
vanishes unless it acts on some local groundstate; every term in
$V^{01}_{B_X}$ connects a local groundstate in ${\cal H}^{0}_{B_X}$
with an excited state localized in $X$.
The operator $V_R$ is given by
\begin{equation}
V^{R} \bydef \sum_{X} (P^{1}_{B_X}V_XP^{1}_{B_X} + P^{2}_{B_X}V_XP^{2}_{B_X})
\bydef \sum_{X} V^{R}_{B_X}. \nonumber
\end{equation}
We denote by ${\rm ad}A(B)$ the commutator $AB-BA$. We define
\bd
{\cal{O}}_{B_X} := \{A^{01}_{B_{X}}:=
P^{0}_{B_X}A_XP^{1}_{B_X}+P^{1}_{B_X}A_XP^{0}_{B_X} : A_X \in {\cal{A}}_X \}.
\ed
${\cal{O}}_{B_X}$ contains only "off-diagonal" operators (with respect to the
partition of unity (\ref{pau})).
We define
\begin{equation}
{\bar H}_{0X}\,=\,\sum_{M \cap X \not= \emptyset} \phi^0_M \;,\label{hbar}
\end{equation}
and, on ${\cal{O}}_{B_X}$, we define
\begin{equation}
{\rm ad}^{-1}{\bar H}_{0X}\left(A^{01}_{B_{X}}\right)\,=\int dP_E
\frac{A^{01}_{B_{X}}}{E-E'}dP_{E'}, \label{adah}
\end{equation}
where $\{P_E\}$ are the spectral projections of ${\bar H}_{0X}$.
It is easy to see that
\begin{equation}
{\rm ad}{\bar H}_{0X}\left({\rm ad}^{-1}{\bar H}_{0X}
\left(A^{01}_{B_{X}}\right)\right)\,=\,A^{01}_{B_{X}}.
\end{equation}
In \cite{dfr} it is shown that, due to our choice of $B_X$, we have that
\begin{equation}
{\rm ad} H_0 \left( {\rm ad}^{-1}{\bar H}_{0X} \left( A^{01}_{B_{X}} \right)
\right) \,=\, A^{01}_{B_{X}}\,, \label{crux}
\end{equation}
and that the operator norm of ${\rm ad}^{-1}{\bar H}_{0X}$ on ${\cal{O}}_{B_X}$
is bounded as follows:
\bd
|| {\rm ad}^{-1}{\bar H}_{0X} \mid_{{\cal{O}}_{B_X}}|| \le \frac{1}{\Delta}
\ed
where $\Delta$ is the gap between the groundstate energy of
${\bar H}_{0X}$ and the rest of its spectrum.
A unitary transformation, $U(t)$, is defined as
\bd
U(t)\,=\,e^{tS},
\ed
where
\begin{equation}
S=\sum_X S_{B_{X}}\label{s11},
\end{equation}
and
\begin{eqnarray}
S_{B_{X}}&=& {\rm ad}^{-1}{\bar H}_{0X} \left(V^{01}_{B_{X}}\right)
\nonumber\\
&=& {\rm ad}^{-1}{\bar H}_{0X} \left(
P^{0}_{B_X}V_XP^{1}_{B_X}+P^{1}_{B_X}V_XP^{0}_{B_X}\right). \label{s1b}
\end{eqnarray}
In \cite{dfr} it is proven that the transformed Hamiltonian,
\bd
{\widetilde H}(t)\,=\,U(t)H(t)U(t)^{-1}
\ed
can be cast in the following form
\bd
{\widetilde H}(t)\,=\, H_0(t) + V(t),
\ed
where $H_0(t)$ is given by
\begin{equation}
H_0 + t \sum_{X} V^{00}_{B_X} + \frac{t^2}{2} \sum_{X,X': \atop X' \cap B_X
\ne \emptyset }
P^0_{B_{X \cup X'}}{\rm ad}S_{B_X}(V^{01}_{B_{X'}})P^0_{B_{X \cup X'}}.
\label{ach}
\end{equation}
The proof is based on using the Lie-Schwinger series, the idendity \reff{crux}
and the commutativity of operators with disjoint support (see \reff{coco}).
The new perturbation, $ V(t)$, is given (see Lemma 1.3) by a sum of
exponentially decaying interactions satisfying condition {\bf V1}.
{\it To order} $t^2$, the
groundstate energy of $H(t)$, is given by the lowest eigenvalue of $H_0(t)$
and is obtained by diagonalizing $H_0(t)$.
One may apply Theorems 1.1 and 1.2 to the pair $(H_0(t),V(t))$, provided the
new unperturbed part $H_0(t)$ satisfies Conditions {\bf H1} - {\bf H4} of the
introduction.
\section{The Peierls condition}
We assume that $H_0$ is a finite-range Hamiltonian satisfying
condition {\bf H1},
\bd
H_0 = \sum_{Y\subset \Lambda} \Phi_{0Y}.
\ed
Let ${\cal K} := \{s_i\}_{i=1}^k $ be the set of periodic groundstate
configurations of $H_0$. In order to state the Peierls condition, we
introduce the notion of {\it contours}. We define sampling plaquettes
$P(x)$ as
\bd
P(x) := \{y \in \Lambda : \mid x_i - y_i \mid \le a \,\,{\rm for}\,\,
1\le i\le \nu \}\,.
\ed
The constant $a$ is chosen to be larger than the period of each
of the groundstate configurations of $H_0$ and the range of the
interactions $\Phi_{0Y}$. A contour is constructed
from sampling plaquettes on which the configuration does not coincide with
any of the
groundstate configurations of $H_0$. The defect set, $\partial
\omega$, of a configuration $\omega$ on the lattice $\Lambda$ is defined as
\bd
\partial \omega = \bigcup_{x \in \Lambda} \{P(x) :\omega_{P(x)} \not=
(s_i)_{P(x)}, \,\,\,{\rm for}\,\,\,{\rm all}\, 1\le i \le k \} \;.
\ed
A contour of a configuration $\omega$ is a pair $\gamma = (M,
\omega_{M})$, where $M$ is a maximally connected component of the defect set
$\partial \omega$. The set $M$ is the support of $\gamma$. Two contours
$\gamma$, $\gamma'$
are disjoint if no pair of plaquettes, $P \in \gamma$, $P' \in \gamma'$,
intersect.
Due to our choice of the size of the sampling plaquettes we have
a one-to-one correspondence between configurations and families of
contours. This is seen by associating to each configuration of the lattice,
the corresponding
set of disjoint contours, with the restriction that the interiors and exteriors
of nested contours match. Families of contours corresponding to a
configuration are said to be compatible. The energy of a configuration
is then expressible in
terms of contour energies. To each contour $\gamma$, there corresponds a
unique configuration
$\omega^{\gamma}$ that has $\gamma$ as its only contour. The configuration in
any
connected component of the set $\Lambda \setminus M$ coincides with one
of the groundstates. If the energy of the groundstate is normalized to zero
then the energy of the configuration $\omega^{\gamma}$ is given by
\begin{eqnarray}
H_0(\omega^{\gamma}) &=& \sum_{Y \cap \Lambda \ne \emptyset} \frac{\mid
Y \cap M\mid}{\mid Y \mid} \Phi_{0Y}(\omega^{\gamma}) \nonumber\\
&=:& E(\gamma). \nonumber
\end{eqnarray}
If a configuration $\omega$
corresponds to a family of compatible contours
\bd
\{\gamma_1,\ldots , \gamma_n\},
\ed
then its energy is given by the sum of the
energies of its contours, i.e.,
\bd
H_0(\omega)\,=\,\sum_{i=1}^{n}E(\gamma_i)
\ed
because of our choice of the size of the sampling plaquettes.
This allows us to rewrite the partition function
$Z(\Lambda)$ of the system in $\Lambda$ in terms of an ensemble of
contours, with no other than excluded-volume interaction.
and use cluster expansion methods to study the system at low-temperatures.
The {\it Peierls condition} demands that the energy of a contour be
proportional to the total number of lattice sites in the contour. More
precisely, it requires that there exist a positive constant $\kappa$, such
that the energy
$E(\gamma)$, of a contour $\gamma$, satisfies
\bd
E(\gamma) > \kappa \mid \gamma \mid,
\ed
$\mid \gamma \mid$ being the number of sites in $\gamma$.
%\end{enumerate}
A useful criterion for the Peierls condition to hold has been proven in
\cite{hol}.
\\
\\
\noindent
CRITERION :{\it If the Hamiltonian $H_0$ is expressible in terms of
a finite-range {\it m}-potential and has a
finite number of groundstates then the groundstates are necessarily periodic,
and the Peierls condition is satisfied.}
\\
\end{appendix}
\section*{Acknowledgements}
We thank C. Albanese, N. Datta and R. Fernandez for many useful discussions.
\begin{thebibliography}{99}
\bibitem{dff}
N.~Datta, R.~Fernandez, J.~Fr\"ohlich.
\newblock Low-temperature phase diagrams of quantum lattice systems.
I. Stability for quantum perturbations of classical systems with
finitely-many groundstates.
\newblock To be published in {\em J. Stat. Phys.}
\bibitem{dfr}
N.~Datta, R.~Fernandez, J.~Fr\"ohlich, L.~Rey-Bellet.
\newblock Low-temperature phase diagrams of quantum lattice
systems. II. Convergent perturbation expansions and stability in systems with
infinite degeneracy.
\newblock ETH-preprint.
\bibitem{bku}
C.~Borgs, R.~Kotecky, D.~Ueltschi.
\newblock Low-temperature phase diagrams for quantum perturbations of
classical
spin systems.
\newblock Preprint.
\bibitem{sim}
B.~Simon.
\newblock {\em The Statistical Mechanics of Lattice Gases}.
\newblock Princeton Univ.\ Press, Princeton, N.J., 1993.
\bibitem{braro}
O.~Bratteli and D.~W. Robinson.
\newblock {\em Operator Algebras and Quantum Statistical Mechanics I and II}.
\newblock Springer-Verlag, New York-Heidelberg-Berlin, 1987.
\bibitem{rue}
D.~Ruelle.
\newblock {\em Statistical Mechanics: Rigorous Results}.
\newblock W.~A.~Benjamin Inc., New York, 1969.
\bibitem{hol}
W.~Holsztynsky, J.~Slawny.
\newblock Peierls condition and number of groundstates.
\newblock {\em Comm. Math. Phys.} 61: 177, (1978)
\bibitem{park}
Y.~M.~Park.
\newblock The cluster expansion for classical and quantum lattice systems.
\newblock {\em J. Stat. Phys.}, 27:553, 1981.
\bibitem{pirsin}
S.~A. Pirogov, Ya.~G. Sinai.
\newblock Phase diagrams of classical lattice systems.
\newblock {\em Theor. Math. Phys.}, 25:1185--1192, 1976; 26:39--49, 1976.
\bibitem{boim}
C.~Borgs, J.Z.~Imbrie.
\newblock A unified approach to phase diagrams in field theory and statistical
mechanics.
\newblock {\em Comm. Math. Phys.}, 123:305, 1989.
\bibitem{kat}
T~Kato:
\newblock Perturbation theory for linear operators.
\newblock Springer Verlag, 1966.
\bibitem{grusu}
C.~Gruber, A.~S\"{u}t\"{o}.
\newblock Phase diagrams of lattice systems with residual entropy.
\newblock {\em J. Stat. Phys.}, 52: 113, 1988.
\bibitem{alban} C.~Albanese.
\newblock On the spectrum of the Heisenberg Hamiltonian.
\newblock {\em J. Stat.Phys.} 55: 297 (1989).
\bibitem{rey}
L.~Rey-Bellet.
\newblock Diploma-Thesis. ETH 1994.
\bibitem{kiemli}
S.~A. Kivelson, V.~J. Emery, H.~P. Lin.
\newblock Doped antiferromagnets in the weak-hopping limit.
\newblock {\em Phys Rev. B.}, 42: 6523, 1990.
\bibitem{gruje}
C.~Gruber, J.~Jedrzejewski , P.~Lemberger.
\newblock Groundstates of the spinless Falicov-Kimball model.
\newblock {\em J. Stat. Phys.}, 66: 913, 1991.
\bibitem{ken}
T.~Kennedy.
\newblock Some rigorous results on the groundstates of the Falicov-Kimball
model.
\newblock Submitted to {\em Rev. Math. Phys.}
\bibitem{fral}
C.~Albanese, J.~Fr\"{o}hlich.
\newblock Dynamical binding of holes in doped antiferromagnets.
\newblock {\em Helv. Phys. Acta}, to appear.
\bibitem{alda}
C.~Albanese, N.~Datta.
\newblock Quantum criticality, Mott transition and sign problem for a model of
lattice fermions.
\newblock {\em Comm. Math. Phys.}, 167:571, 1995.
\end{thebibliography}
\end{document}