\def\version{May 1995}
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\rightline{Preprint}
\rightline{\version}
\bigskip
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\topmatter
\title
Low Temperature Phase Diagrams for Quantum
Perturbations of Classical Spin Systems
\endtitle
\leftheadtext\nofrills
{\headerfont C.Borgs, R. Koteck\'y, D. Ueltschi}
\rightheadtext\nofrills
{\headerfont Quantum Perturbations of Classical Spin Systems}
\author
C\. Borgs \footnotemark"${}^{1,\dag}$",
R\. Koteck\'y\footnotemark"${}^{2,\ddag}$" and
and D\. Ueltschi\footnotemark"${}^{3}$"
\endauthor
\footnotetext"${}^{\dag}$"{Heisenberg Fellow, on leave from
Institut f\"ur Theoretische Physik, Freie Universit\"at Berlin}
\footnotetext"${}^{\ddag}$"{On leave of absence from Center for
Theoretical Study, Charles University, Praha. Partly
\phantom{iupp}supported by
the grants GA\v CR 202/93/0449 and GAUK 376 }
\affil
{\eightit
${}^1$School of Mathematics, Institute for Advanced Study, Princeton\\
${}^2$Centre de Physique Th\'eorique, CNRS, Marseille \\
${}^3$Institut de Physique Th\'eorique, EPF Lausanne}
\endaffil
\address Christian Borgs \hfill\newline
Freie Universit\"at Berlin,
Institut f\"ur Theoretische Physik
\hfill\newline
Arnimallee 14, 14195 Berlin
\endaddress
\email borgs\@dirac.physik.fuberlin.de \endemail
\address
Roman Koteck\'y
\hfill\newline
Center for Theoretical Study, Charles
University,\hfill\newline
Jilsk\'a 1, 110 00 Praha 1,
Czech Republic
\hfill\newline
\phantom{18.}and
\hfill\newline
Department of
Theoretical Physics, Charles University,\hfill\newline
V~Hole\v sovi\v
ck\'ach~2, 180~00~Praha~8, Czech Republic \endaddress
\email kotecky\@aci.cvut.cz \endemail
\address
Daniel Ueltschi
\hfill\newline
Institut de Physique Th\'eorique, EPFL,\hfill\newline
CH1015 Lausanne,
Switzerland
\endaddress
\email ueltschi\@eldpa.epfl.ch \endemail
%\keywords
%\endkeywords
\abstract
We consider a quantum spin system with Hamiltonian
$$
H=H^{(0)}+\lambda V,
$$
where $H^{(0)}$ is diagonal in a basis $\ket s=\bigotimes_x\ket{s_x}$
which may be labeled by the configurations $s=\{s_x\}$ of a suitable
classical spin system on $\Bbb Z^d$,
$$
H^{(0)}\ket s=H^{(0)}(s)\ket s.
$$
We assume that $H^{(0)}(s)$ is a finite range Hamiltonian with finitely many
ground states and a suitable Peierls condition for excitations, while $V$ is
a finite range or exponentially decaying quantum perturbation.
Mapping the $d$ dimensional quantum system onto a {\it classical} contour
system on a $d+1$ dimensional lattice, we use standard PirogovSinai theory
to show that the low temperature phase diagram of the quantum spin system is
a small perturbation of the zero temperature phase diagram of the classical
Hamiltonian $H^{(0)}$, provided $\lambda$ is sufficiently small. Our method
can be applied to bosonic systems without substantial change. The extension
to fermionic systems will be discussed in a subsequent paper.
\endabstract
\endtopmatter
\document
\head{1. Introduction}
\endhead
\subhead{1.1. General ideas}
\endsubhead
Many models of classical statistical mechanics provide examples of
firstorder phase transitions and phase coexistence at low temperatures.
It became clear already from the first proof of such a transition for the
Ising model by the Peierls argument \cite{Pei36, Gri64, Dob65} that
a convenient tool for the study of phase coexistence and firstorder phase
transitions is a representation in terms of configurations of geometrical
objects  contours. This has been systematically developed
in PirogovSinai theory \cite{PS75, Sin82}, see also \cite{KP84, Zah84,
BI89}, which allows one to prove these phenomena for a wide class of models,
with or without symmetry assumptions on the coexisting phases.
For quantum spin systems, the theory of firstorder phase transitions and
phase coexistence is much less developed. While several papers deal
with this problem in the presence of a symmetry relating the two phases,
\cite{Gin69 , Ken85}, no general theory is known
which provides a systematic
approach to quantum spin systems once the symmetry constraint
is relaxed.
In this paper we propose to develop such a theory for low temperature quantum
spin systems which are small perturbations of suitable classical systems.
To be more precise, we assume that the Hamiltonian of the system is of the
form
$$
H=H^{(0)}+\lambda V
\tag 1.1
$$
where $H^{(0)}$ is diagonal in a basis
$\ket s=\bigotimes_x\ket{s_x}$ which may be labelled
by the configurations $s=\{s_x\}$ of a classical spin system with finite
single spin space
$S=\{1,\cdots,S\}$ and
Hamiltonian
$$
\bra{s} H^{(0)}\ket{s}=H^{(0)}(s),
\tag 1.2
$$
while $V$ is a local or exponentially decaying quantum perturbation,
$$
V=\sum_A V_A,
\tag 1.3
$$
where the sum goes over connected sets $A$ and $V_A$ is an arbitrary
operator on ${\cal H}_A=\bigotimes_{x\in A} {\cal H}_x$, except for the
constraint that its norm $\V_A\$ is exponentially decaying with the size
of $A$.
Assuming that the classical system has a contour representation
in $d$ dimensions that allows to apply the methods of PirogovSinai theory
for sufficiently low temperatures, we propose to study the quantum
perturbation of this system by mapping it into a suitable contour system in
$d+1$ dimensions, which can again be analysed by the methods of Pirogov
Sinai theory.
Actually, our method is very similar to the method developed in \cite{Bor88},
were this strategy was used to develop weak coupling cluster expansions for
lattice gauge theories with discrete gauge group and continuous time.
Our approach differs, however, from that used by Ginibre \cite{Gin69}
and Kennedy \cite{Ken85}. In order to explain the main difference, let us first
recall their method. It is based on the idea of developping the density
matrix $e^{\beta H}$ of the model (1.1) arround the unperturbed matrix
$e^{\beta H^{(0)}}$ using Trotter's formula
$$
e^{\beta H}=\lim_{n\to\infty}
\Bigl(e^{(\beta/n) H^{(0)}}
\bigl(1\frac{\beta}n V\bigr)
\Bigr)^n.
\tag 1.4
$$
While the leading term of this expansion gives the partition function of the
classical spin system at temperature $\beta$, the expansion of $W$ according
to (1.3) will introduce transitions between classical contours at various
times, leading therefore to a representation in terms of ``quantum
contours'' on $\Bbb Z^d\times [0,\beta]$.
In the symmetric case considered in \cite{Ken85} and \cite{Gin69},
these quantum contours could be controlled with affordable effort using
standard methods. The asymmetric situation considered here, however,
obviously requires significant modifications involving something like a
``Quantum PirogovSinai'' theory. Such an approach
is currently being pursued by Datta, Fernandez and Fr\"ohlich
\cite{DFF95}, leading to results very similar to those
presented in this paper.
Here, we follow an alternative approach, motivated by \cite{Bor88}.
The main idea is not to consider contours on $\Bbb Z^d\times [0,\beta]$,
but to use a suitable blocked approach, which allows to map a
$d$dimensional quantum system onto a classical contour system on the
$(d+1)$dimensional block lattice. As a consequence, our results make it
possible to apply directly the usual PirogovSinai theory to quantum spin
systems as well, thereby allowing to analyze questions concerning the low
temperature phase structure, finite size scaling, analyticity properties,
etc. using the well developped machinery of standard PirogovSinai theory.
\subhead{1.2. Contour representations of quantum lattice models}
\endsubhead
In the remaining part of this introduction, we present the main ideas of our
approach. In the first step, we rewrite the partition function
$Z=\Tr e^{\beta H}$ of the quantum spin system as
$$
Z=\Tr T^M\quad\text{where}\quad T=e^{\tilde\beta H}
\quad\text{and}\quad\beta=M\tilde\beta
\tag 1.5
$$
with an integer $M$ to be chosen later. We then expand the partition
function $Z$ of the quantum system around the partition function
$Z^\class=\Tr e^{\beta H^{(0)}}$ of the classical spin system using the
Duhamel formula
(a reference concerning the Duhamel formula is e.g\. \cite{SS76})
for the transfer matrix $T=e^{\tilde\beta H}$.
Introducing the family ${\cal A}_0$ of all sets $A$
contributing to (1.3), the Duhamel expansion
gives
$$
T=e^{\tilde\beta H^{(0)} \tilde\beta \lambda\sum_{A\in{\cal A}_0}V_A}=
\sum_{\bold n}
\Bigl[
\prod_{A\in{\cal A}_0}
\frac{(\lambda)^{n_A}}{n_A!}
\int_0^{\tilde\beta}d\tau_A^1\dots d\tau_A^{n_A}
\Bigr]
T(\boldsymbol \tau,\bold n),
\tag1.6
$$
where $\bold n$ is an multiindex, $\bold n: \Cal A_0 \to \{0,
1, 2, \dots \}$, and $T(\boldsymbol
\tau,\bold n)$ is obtained from
$T^{(0)}=e^{\tilde\beta H}$ by ``inserting'' the operator
$V_A$ at the times $\tau_A^1,\dots \tau_A^{n_A}$, see Section 2
for the precise definition.
Next, we resum (1.6) to obtain the expansion
$$
T=\sum_{B} T(B),
\tag 1.7a
$$
where
$$
T(B)=
\sum_{\scriptstyle{\cal A=\{A_1,\dots,A_k\}}\atop
\scriptstyle{\cup_i A_i=B}}
\tilde T(\Cal A),
\tag 1.7b
$$
with
$$
\tilde T(\Cal A)
=
\sum _{\boldkey n: \supp \boldkey n= \Cal A}
\Bigl[
\prod_{A\in{\cal A}_0}
\frac{(\lambda)^{n_A}}{n_A!}\int_0^{\tilde\beta}d\tau_A^1\dots
d\tau_A^{n_A}
\Bigr]
T(\boldsymbol \tau,\boldkey n).
\tag 1.7c
$$
Using the basis $\ket s$ to rewrite (1.5) as
$$
Z=\sum_{s^{(1)},\dots, s^{(M)}}
\bra{s^{(1)}}T\ket{s^{(2)}}\dots \bra{s^{(M)}}T\ket{s^{(1)}}
\tag 1.8
$$
and inserting the formula (1.7) to expand $T$ around
$T^{(0)}$, we obtain
$$
Z=\sum_{\scriptstyle s^{(1)},\dots, s^{(M)}\atop
\scriptstyle B^{(1)},\dots, B^{(M)}}
\prod_{t=1}^M
\bra{s^{(t1)}}\tilde T(B^{(t)})\ket{s^{(t)}},
\tag 1.9
$$
where we have identified $s^{(0)}$ and $s^{(M)}$.
At this point, the quantum spin system is easily mapped to a classical
contour system in $d+1$ dimensions. Before doing so, let us discuss the
expansion (1.9) from a more heuristic point of view. Starting with the
leading term in $B^{(1)},\dots, B^{(M)}$, namely
the term where all $B^{(t)}$ are empty, the matrices $T(B^{(t)})$ reduce to
the unperturbed transfer matrix $T^{(0)}$, which implies that only the term
with $s^{(1)}\equiv s^{(2)}\equiv \dots\equiv s^{(M)}$
contributes to the sum over $s^{(1)},\dots, s^{(M)}$, giving the partition
function of the classical spin system at the inverse temperature $\beta$.
A non empty set $B^{(t)}$, on the other hand, corresponds to the insertion
of one or several operators $V_A$, $A\subset B^{(t)}$, inducing transitions
between different classical states $s^{(t1)}$ and $s^{(t)}$. It should be
noted, however, that for a fixed set of $B^{(t)}$'s
only those
spin configurations $s^{(1)}\equiv s^{(2)}\equiv \dots\equiv s^{(M)}$
contribute to (1.9), for which $s^{(t1)}$ and $s^{(t)}$ are identical on
all points $x$ for which $x\notin B^{(t)}$.
In order rewrite (1.9) in terms of contours, we assign contours to
``configurations'' specified by $s^{(1)},\dots, s^{(M)}$ and
$B^{(1)},\dots, B^{(M)}$
Namely, we
introduce elementary cubes as
the unit closed cubes $C\subset \Bbb R^{d+1}$ with centers $(x, t)$ where
$t\in\{1,2,\dots,M\}$ and $x \in\Bbb Z^d$. We say that an elementary cube
$C$ with center $(x,t)$, $x\in\Bbb Z^d$, lies in the $t$'th time slice;
we say that it is in a quantum excited state if $x\in B^{(t)}$, while we
say it is in a classical state if this is not the case. Consider now a cube
$C$ in the $t$'th time slice which is in a classical state.
Then $s^{(t1)}$ and $s^{(t)}$ must assume the same value $s_x\in S$ on the
corresponding point $x\in\Bbb Z^d$, and we say that the cube $C$ is in the
(classical) state $s_x$.
In order to explain our definition of contours, let us assume for the purpose
of this introduction that $H^{(0)}(s)$ is the Hamiltonian of a classical
spin system with nearest neighbor interactions, and that the contours of the
classical system
correspond
to bonds $\langle xy\rangle$ for which
$s_x\neq s_y$ (see Section 2 for the more general case).
We then say that a cube $C$ is part of the ground state region $W_m$,
$m\in S$, if it is in the classical state $m$, and if all neighboring cubes
in the same time slice are in the classical state $m$ as well.
All cubes which are not part of a ground state region are called excited.
As usual, we
define a contour as a connected component of the set of
excited cubes.
Resumming all terms in (1.9) which lead to the same set of ground state
regions and contours, we finally obtain $Z$ as a sum over sets
$\{Y_1,\dots,Y_n\}$ of nonoverlapping contours.
Given our definition of excited cubes, it is an easy exercise to show that
the weight of each such configuration factors into a product of contour
activities $\rho(Y_i)$ and ground state terms $e^{\tilde\beta e_m W_m}$,
where $e_m$ is the classical groundstate energy of the ground state $m$,
while $W_m$ is the number of cubes which are in the ground state $m$.
This gives the representation
$$
Z=\sum_{\{Y_1,\dots,Y_n\}}
\prod_i \rho(Y_i)
\prod_m e^{\tilde\beta e_m W_m},
\tag 1.10
$$
which is exactly of the same form as
the contour
representation of a classical
spin system.
We can therefore apply standard PirogovSinai theory to analyze the quantum
spin system considered here, provided we can prove a bound of the form
$$
\rho(Y)\leq e^{\gamma Y}e^{\tilde\beta e_0Y}
\tag 1.11
$$
where $\gamma$ is a sufficiently large constant, and $e_0=\min_m e_m$.
Observing that cubes in a quantum excitation are supressed by a small factor
proportional to $\lambda\tilde\beta$, while excited classical cubes are
exponentially suppressed by a ``classical'' contour energy proportional to
$\tilde\beta$, such a bound can easily be proven, see Section 4 for the
details.
Notice that the weights $\rho(Y)$ are in general complex. The version of
the PirogovSinai theory to be used thus must deal with
this fact. Actually, such a case has been discussed in \cite{BI89}
and we are
closely following their approach.
A novel feature
of the models considered here
stems from the fact that the resulting classical model
resides in a finite slab of thickness proportional to
$\beta$; actually we should talk of a cylinder because of the periodic
boundary conditions.
It is therefore not possible to directly apply standard
PirogovSinai theory. Actually, this gives rise to an
interesting problem in both quantum and classical spin systems:
dimensional crossover for first order phase transitions.
We will not discuss the physics of dimensional crossover
in this paper, but the technical modifications
needed to deal with finite temperature quantum spin systems
do actually
provide the necessary framework to deal with
this problem as well.
The organization of this paper is as follows:
In the next section, we state our main
assumptions and results.
In Section 3, we derive the contour representation (1.10),
proving in particular the needed factorization of contour activities.
In Section 4 we
prove the exponential decay of the contour activities.
Section 5 is devoted to the discussion of the resulting contour model
including the discussion of modifications to
PirogovSinai theory on a
finite slab.
In Section 6 we discuss expectation values of local observables,
and in Section 7 we combine the results of the preceeding
section to prove the theorems stated in Section 2.
Details of the necessary modifications to PirogovSinai theory
on a finite slab are deferred to an appendix.
We close this introduction with a discussion of possible
extensions. We recall that we assumed that $H^{(0)}$
is diagonal in a basis $\ket{s}=\otimes_x\ket{s_x}$,
where $s_x$ lies in a finite spin space $S$.
While this is a natural setting for quantum spin systems
as, e\.g\. the anisotropic quantum Heisenberg model,
it is not for the discussion of bosonic or fermionic
lattice gases. In this situation, a typical choice for
$H^{(0)}$ would be an operator which is diagonal in
the usual bosonic or fermionic Fock representation
where basis vectors are characterized by eigenvalues
of the corresponding number operators $n_x$.
While the correponding classical system still has
a finite state space
($n_x=n_{\sssize\uparrow}+n_{\ssize\downarrow}=0,1,2$)
in the fermionic case,
bosons now give rise to a state space
which contains infinitely many classical states per site.
In most application, however, this is not a serious problem,
because the hamiltonian $H^{(0)}$ supresses high values
of $n_x$ (or, in the usual field representation,
high values of the boson field $\phi_x$). Our methods and
results are therefore applicable to bosonic lattice gases
without major modifications.
For fermions, on the other hand, the antisymmetrization
of the wave function leads to sign problems in the
contour representation (1.10) which have to be dealt with
carefully. While this is {\it a priori} not obvious
at all, it turns out, however, that fermion signs do not
spoil the factorization properties needed to apply
PirogovSinai theory (see \cite{BK95}, where the methods
developed here are used to prove the existence of staggered
charge order in the
narrow band extended Hubbard model).
\head{2. Definition of the model, statements of results}
\endhead
\subhead{2.1. Assumptions on the classical model}
\endsubhead
We start this section by stating the precise assumptions for the classical
model. We consider a classical spin system with finite spin space
$S=\{1,\dots,S\}$, spin configurations $s:\Bbb Z^d\to S$, $x\mapsto s_x$,
and finite range Hamiltonian $H^{(0)}(s)$ with translation invariant
interactions, depending on a vector parameter $\mu\in \Cal U$,
where $\Cal U$ is an open subset of $\Bbb R^\nu$.
We assume that $H^{(0)}(s)$ is given in the form
$$
H^{(0)}(s)=\sum_{x}\Phi_x(s),
\tag 2.1
$$
where $\Phi_x(s)\in\Bbb R$ depends on $s$ only via the spins $s_y$
for which $y\in U(x)=\{y\in\Bbb Z^d\mid\dist(x,y)\leq R_0\}$, where $R_0$ is
a finite number.
In our notation we supress the dependence of $H^{(0)}$ on $\Phi_x$ and
$\mu$.
As usually, a configuration $g$ which minimises the Hamiltonian (2.1)
is called a ground state configuration.
For the purpose of this paper, we will assume that the number of ground
states of the Hamiltonian (2.1) is finite, and that all of them
are periodic.
More precisely, we will assume that there is a finite number of
periodic configurations $g^{(1)},\dots,g^{(r)}$, with (specific) energies
$$
e_m=e_m(\mu)
=\lim_{\Lambda\to\Bbb
Z^d}\frac{1}{\Lambda}\sum_{x\in\Lambda}\Phi_x(g^{(m)}),
\tag 2.2
$$
such that for each $\mu\in\Cal U$, the set of ground states
$G(\mu)$ is a subset of $\{g^{(1)},\dots,g^{(r)}\}$.
Obviously, $G(\mu)$ is given by those configurations $g^{(m)}$ for which
$e_m(\mu)$ is equal to the ``ground state energy''
$$
e_0=e_0(\mu)=\min_{m}e_m(\mu).
\tag 2.3
$$
Note that we may assume, without loss of generality, that $\Phi_x(g^{(m)})$
is independent of the point $x$ for all ground state configurations
$g^{(m)}$, because this condition can always be achieved by averaging
$\Phi_x(s)$ in (2.1) over the minimal common period of
$g^{(1)},\dots,g^{(r)}$.
Our goal will be to prove that the low temperature phase diagram of the
quantum model is a small perturbation of the classical ground state diagram
provided the quantum perturbation is sufficiently small.
In order to formulate and prove this statement, we need some assumptions on
the structure of the ground state diagram.
Here we assume that for some value of $\mu_0\in\Cal U$ all
states in $\{g^{(1)},\dots,g^{(r)}\}$ are ground states,
$$
e_m(\mu_0)=e_0(\mu_0)\qquad\text{for all}\qquad m=1,\dots,r,
\tag 2.4
$$
that $e_m(\mu)$ are $C^1$ functions in $\Cal U$, and that
the
matrix of derivatives
$$
E=
\Bigl(\frac{\partial e_m(\mu)}{\partial\mu_i}\Bigr)
\tag 2.5
$$
has rank $r1$ for all $\mu\in\Cal U$,
with uniform bounds on the inverse of the corresponding
submatrices.
We remark that this condition implies that the zero temperature phase diagram
has the usual structure of a $\nu(r1)$ dimensional coexistence surface
$S_0$ where all states $g^{(m)}$ are ground states, $r$ different
$\nu(r1)1$ dimensional surfaces $S_n$ ending in $S_0$ where all states
but the state $g^{(m)}$ are ground states, ...
Next, we formulate a suitable Peierls condition.
In order to present it, we introduce, for a given configuration $s$,
the notion of excited sites $x\in\Bbb Z^d$.
We say that
a site $x$ is in the
{\it state}
$g^{(m)}$ if the configuration
$s$ coincides with the configuration $g^{(m)}$ on $U(x)$, i.e. on all sites
$y$ for which
$\dist(x,y)\leq R_0$;
a site is {\it excited}, if it is not in any of the
states $g^{(1)},\dots,g^{(r)}$.
Given this notation, the Peierls assumption used in this paper is that there
exists a constant $\gamma_0>0$, independent of $\mu$, such that
$$
\Phi_x(s)\geq e_0(\mu)+\gamma_0
\qquad
\text{for all excited sites $x$ of
all configurations $s$}.
\tag 2.6
$$
Finally, we assume that the derivatives of $\phi_x$ are uniformly
bounded in $\Cal U$. More explicitely, we assume that there is
a constant $C_0<\infty$, such that
$$
\Bigl
\frac {\partial}{\partial\mu_i}
\phi_x(s)
\Bigr
\leq
C_0
\tag 2.7
$$
for all
$i=1, \dots , \nu$,
$\mu\in\Cal U$,
$x\in\Bbb Z^d$,
and all configurations
$s$.
\remark{Remark}
Given the assumptions stated in this subsection,
standard PirogovSinai theory implies that the
low temperature phase diagram of the classical
model has the same topological structure as the
corresponding zero temperature phase diagram
(see above).
\endremark
\subhead{2.2. Assumptions on the quantum perturbation}
\endsubhead
As pointed out in the introduction, we propose to develop a theory
which allows to control low temperature quantum spin systems that are small
perturbations of the classical system introduced above. We consider quantum
spin systems with Hamiltonians of the form
$$
H=H^{(0)}+\lambda V
\tag 2.8
$$
where $H^{(0)}$ is diagonal in a basis
$\ket{s}=\bigotimes_x\ket{s_x}$ that may be labelled by the
configurations $s=\{s_x\}$ of the classical spin system,
$$
H^{(0)}\ket{s}=H^{(0)}(s)\ket{s}.
\tag 2.9
$$
We assume that $V$ is of the form
$$
V=\sum_A V_A,
\tag 2.10
$$
where the sum goes over connected sets $A$ and $V_A$ is
a selfadjoint
operator on ${\cal H}_A=\bigotimes_{x\in A} {\cal H}_x$.
In addition to translation invariance,
we assume that $V_A$ and its derivatives,
$\frac {\partial}{\partial\mu_i}V_A$,
$i=1,\dots,\nu$, are bounded operators,
with a suitable decay constraint
on the corresponding operator norms
$\V_A\$ and
$\displaystyle
\Big\\frac {\partial}{\partial\mu_i}V_A\Bigr\$.
In order to formulate this constraint,
we introduce the Sobolev norm
$$
V_{\gamma}
=
\sum_{A: x\in A}
\biggl(
\V_A\
+
\sum_{i=1}^\nu
\Big\\frac {\partial}{\partial\mu_i}V_A\Bigr\
\biggr)
e^{\gammaA}
\tag 2.11
$$
where $A$ is the number of points in $A$.
Given this definition, our assumption on
the decay of $V$ is the assumption that
$$
V_{\gamma_Q}<\infty
\tag 2.12
$$
for a sufficiently large constant
$\gamma_Q$.
\remark{Remarks}
\item{i)}
For a finite range perturbation, where
$V_A=0$ if the diameter of $A$ exceeds
the range $R_Q$ of the interaction, the assumption
(2.12) is automatically fulfilled for arbitrary large
$\gamma_Q<\infty$.
\item{ii)} If the quantum perturbation $V$ is of infinite range,
we need that
$\V_A\$ and
$\displaystyle \Big\\frac {\partial}{\partial\mu_i}V_A\Bigr\$
decay exponentially in the size $A$ of $A$.
Assuming exponential decay with a sufficiently large
decay constant $\gamma$, and observing
that the number of connected sets $A$
of size $s$ that contain a given point $x\in\Bbb Z^d$
is bounded by $(2d)^{2s}$, the condition (2.12)
can be satisfied provided $\gamma>\gamma_Q+2\log(2d)$.
\item{iii)}
Strictly speaking, an exponentially decaying pair
potential,
$V=\sum_{x,y} V_{x,y}$ where the norm
$ V_{x,y} $ decays exponentially with the
distance between $x$ and $y$, is not in the class
considered in this section
because $V$ is not given as a sum over {\it connected}
sets $A$. It is obvious, however, that such a potential
can be {\it rewritten} in the required form, by artificially
connecting the two points $x$ and $y$ by a nearest neighbor
path. In (2.11), this effectively replaces the size of the
set $A=\{x,y\}$ by its $\ell_1$diameter $\sum_i x_iy_i$.
\endremark
\subhead{2.3. Finite volume states for the quantum system}
\endsubhead
In order to discuss the phase diagram of the quantum spin system, we will
consider suitable finite volume states
$\langle\boldsymbol\cdot\rangle_{q,\Lambda}$ which are
analogues
of the
classical states with boundary condition $q$, where $q=1,\dots,r$. We first
introduce, for any configuration $s$ and any finite set $A$, the vector
$\ket{s_A}=\bigotimes_{x\in A}\ket{s_x}$. Given a finite set
$\Lambda\subset\Bbb Z^d$, we then define suitable finite volume
Hamiltonians $H_{\Lambda}^{(0)}$ and $H_{\Lambda}$ on the Hilbert space
$\Cal H_{\bar{\Lambda}}= \bigotimes_{x\in\bar{\Lambda}}\Cal H_x$, where
$\bar{\Lambda}=\cup_{x\in\Lambda}U(x)$. Namely, we introduce operators
$$
H_x^{(0)}=\sum_{s_{\bar{\Lambda}}} \Phi_x(s_{\bar{\Lambda}})
\ket{s_{\bar{\Lambda}}}\bra{s_{\bar{\Lambda}}},
\tag 2.13
$$
$$
H_{\Lambda}^{(0)}=\sum_{x\in\Lambda} H^{(0)}_x,
\tag 2.14
$$
and
$$
H_\Lambda=H_{\Lambda}^{(0)}+
\lambda\sum_{A\subset\Lambda}V_A.
\tag 2.15
$$
The Hamilton operator with boundary conditions
$q$ is then defined as the ``partial expectation value''
$$
H_{q,\Lambda}=\bra{g^{(q)}_{\partial\Lambda}}H_\Lambda
\ket{g^{(q)}_{\partial\Lambda}},
\tag 2.16
$$
where $\partial \Lambda$ is the set $\bar\Lambda\setminus{\Lambda}$.
More precisely, $H_{q,\Lambda}$ is an operator on $\Cal H_\Lambda$ whose
matrix elements are
$$
\bra{s_\Lambda}H_{q,\Lambda}\ket{s'_\Lambda} =
\bra{s_\Lambda}\otimes
\bra{g^{(q)}_{\bar\Lambda\setminus\Lambda}} H_\Lambda
\ket{g^{(q)}_{\bar\Lambda\setminus\Lambda}} \otimes
\ket{s'_\Lambda}.
\tag 2.17
$$
Given the Hamiltonian with boundary conditions $q$,
we introduce the quantum state
$\langle \boldsymbol\cdot\rangle_{q,\Lambda}$ as
$$
\langle \boldsymbol\cdot\rangle_{q,\Lambda}=
{1\over Z_{q,\Lambda}}
\Tr_{\Cal H_\Lambda}(\boldsymbol\cdot \,e^{\beta H_{q,\Lambda}}),
\tag 2.18
$$
where
$$
Z_{q,\Lambda}=\Tr_{\Cal H_\Lambda}e^{\beta H_{q,\Lambda}}.
\tag 2.19
$$
Note that $H_{\Lambda}^{(0)}$ and $H_\Lambda$ are operators
on $\Cal H_{\bar\Lambda}$, while $H_{q,\Lambda}$ and it's analogues,
$$
H_{q,x}^{(0)}=
\bra{g^{(q)}_{\partial\Lambda}} H_{x}^{(0)}
\ket{g^{(q)}_{\partial\Lambda}}
\,
\tag 2.20
$$
and
$$
H_{q,\Lambda}^{(0)}=
\bra{g^{(q)}_{\partial\Lambda}}H_{\Lambda}^{(0)}
\ket{g^{(q)}_{\partial\Lambda}},
\tag 2.21
$$
are operators on $\Cal H_\Lambda$.
\remark{Remark}
Following Ginibre
\cite{Gin69},
it might seem more natural to implement
the boundary conditions with the help of suitable projection
operators $P_{\partial\Lambda}^{(q)}$. Here, this would amount to
defining
$
P_{\partial\Lambda}^{(q)}=\ket{g^{(q)}_{\partial\Lambda}}
\bra{g^{(q)}_{\partial\Lambda}}.
$
With the help of this projection operator, one would then define
$$
Z_{m,\Lambda} =
\Tr P_{\partial\Lambda}^{(q)} e^{\beta H_\Lambda}
$$
and similarly for the finite volume states (2.18).
Observing that
$$
\Tr_{\Cal H_{\Lambda}} W e^{\beta H_{q,\Lambda}}
= \Tr_{\Cal H_{\bar\Lambda}} W
P_{\partial\Lambda}^{(q)} e^{\beta H_\Lambda}
$$
for all operators $W$ on $\Cal H_\Lambda$, these two implementations of
boundary conditions are actually equivalent.
\endremark
\subhead{2.4. Statement of results}
\endsubhead
In order to state our results in the form of a theorem, we
recall that a local observable is an operator which is a
selfadjoint bounded operator on $\Cal H_\Lambda$ for some
finite set $\Lambda$. We also introduce, for each
$x$ in $\Bbb Z^d$ and any local obserbable $\ob$, the translate
$t_x(\ob)$. Defining finally
$\Lambda(L)$ as the box
$$
\Lambda(L)=\{x\in\Bbb Z^d \mid x_i\leq L
\quad\text{for all}\quad i=1,\dots,d\},
\tag 2.22
$$
our main results are stated in the following two
theorems.
\proclaim{Theorem 2.1} Let $d\geq 2$ and let $H^{(0)}$
be a Hamiltonian obeying the assumptions of Section {2.1}.
Then there are constants $0<\beta_0<\infty$ and $0<\gamma_Q<\infty$,
such that for all quantum perturbations $V$ obeying the assumptions
of Section {2.2}, all $\beta\geq\beta_0$ and all $\lambda\in\Bbb C$ with
$$
\lambda
\leq
\lambda_0:=
{1\over{e\beta_0 V_{\gamma_Q}}}
\,
\tag 2.23
$$
there are constants $\xi_q$ and continuously differentible functions
$f_q(\mu)$, $q=1,\dots,r$, such that the following
statements hold true whenever
$$
a_q(\beta,\lambda,\mu):=\Re f_q(\mu)\min_m\Re f_m(\mu)=0
\,.
\tag 2.24
$$
\item{i)} The infinite volume free energy corresponding to
$Z_{q,\Lambda(L)}$ exists and is equal to $f_q$:
$$
f_q=\frac 1\beta \lim_{L\to\infty}
\frac 1{\Lambda(L)}\log Z_{q,\Lambda(L)}
\tag 2.25
$$
\item{ii)} The infinite volume limit
$$
\langle \ob\rangle_q=\lim_{L\to\infty}\langle \ob\rangle_{q,\Lambda(L)}
\tag 2.26
$$
exists for all local observables $\ob$.
\item{iii)} For all local observables $\ob$ and $\obs$,
there exists a constant $C_{\ob,\obs}<\infty$, such that
$$
\bigl\langle \ob t_x(\obs)\rangle_q
 \langle \ob\rangle_q\langle t_x(\obs)\rangle_q
\bigr
\leq
C_{\ob,\obs} e^{x/\xi_q}.
\tag 2.27
$$
\item{iv)} The projection operators
$$
P^{(q)}_{U(x)}=
\ket{g_{U(x)}^{(q)}}\bra{g_{U(x)}^{(q)}}
\tag 2.28
$$
onto the ``classical states'' $g^{(q)}_{U(x)}$ obey the bounds
$$
\bigl
\langle P^{(q)}_{U(x)}\rangle_q
 1
\bigr < \frac 12
\tag 2.29
$$
and
$$
\bigl
\langle P^{(m)}_{U(x)}\rangle_q
\bigr < \frac 12
\tag 2.30
$$
for all $m\neq q$.
\item{v)} There exists a point $\tilde\mu_0\in\Cal U$
such that
$a_m(\tilde\mu_0)=0$ for all $m=1,\dots,r$.
For all $\mu\in\Cal U$, the matrix of derivatives
$$
F=
\Bigl(\frac{\partial \Re f_m(\mu)}{\partial\mu_i}\Bigr)
\tag 2.31
$$
has rank $r1$, and the
inverse of the corresponding
submatrix is uniformly bounded in $\Cal U$.
\endproclaim
\remark{Remarks}
i) Following the usual terminology of PirogovSinai theory, we call a
phase with $a_q=0$ {\it stable}. By the inverse function theorem,
statement v) of the Theorem implies that the phase diagram of the quantum
system has the
same structure as the zero temperature phase diagram of the
classical sytem, with a $\nu(r1)$ di\men\sional coexistence surface
$\tilde S_0$ where all states are stable, $r$ different
$\nu(r1)1$ di\men\sional surfaces $\tilde S_n$ ending in $\tilde S_0$
where all states
but the state $m$ are stable, $\cdots$.
ii) Choosing $\beta$ sufficiently large and
$\lambda$ sufficiently small, the bounds (2.29) and (2.30) can be
made arbitrary sharp. In this sense, the quantum states
$\langle\cdot\rangle_q$ are small perturbations of the corresponding
classical state whenever $q$ is stable.
iii) While Theorem 2.1 is stated (and proven) for general
complex $\lambda$, the physical situation corresponds,
of course, to real values of $\lambda$, as required by
the selfadjointness of the Hamiltonian $H$.
As we will see in Section 5, the ``metastable free energies''
$f_q$ are real in this case, making the
real part in (2.24) and (2.31) superfluous.
\endremark
\medskip
In order to state the next theorem, we define states
with periodic boundary conditons on
$\Lambda(L)$. To this end, we consider the
torus
$
\displaystyle
\Lambda_\per(L)=
\bigl(
\Bbb Z/(2L+1)\Bbb Z
\bigr)^d
$
and the corresponding Hamiltonian
$$
H_{\per,\Lambda(L)}=
\sum_{x\in\Lambda_\per(L)} H_x^{(0)}
+
\lambda\sum_{A\subset \Lambda_\per(L)}
V_A
\,,
\tag 2.32
$$
where the second sum goes over all
subsets $A\subset \Lambda_\per(L)$
which are do not wind around the torus
$\Lambda_\per(L)$.
With these definitions, we then introduce
the quantum state with periodic boundary
conditions as
$$
\langle \boldsymbol\cdot\rangle_{\per,\Lambda(L)}=
{1\over Z_{\per,\Lambda(L)}}
\Tr_{\Cal H_\Lambda(L)}(\boldsymbol\cdot \,e^{\beta H_{\per,\Lambda(L)}}),
\tag 2.33
$$
where
$$
Z_{\per,\Lambda(L)}=\Tr_{\Cal H_\Lambda(L)}e^{\beta H_{\per,\Lambda(L)}}.
\tag 2.34
$$
\proclaim{Theorem 2.2}
Let $H^{(0)}$, $V$, $\beta$ and $\lambda$ as in
Theorem {2.1}. Assume in addition that
$\lambda$ is real.
Then the infinite volume state with periodic boundary conditions,
$$
\langle \ob \rangle_\per
=\lim_{L\to\infty}\langle \ob \rangle_{\per,\Lambda(L)}
\tag 2.35
$$
exists for all local observables
$\ob$,
and is a convex combination (with equal weights)
of the stable states,
$$
\langle \ob\rangle_\per
=
\sum_{q\in Q(\mu)}
\frac 1{Q(\mu)} \langle \ob \rangle_q.
\tag 2.36
$$
Here
$$
Q(\mu)
=
\{q\in\{1,\dots,r\}\mid a_q(\mu)=0\}.
\tag 2.37
$$
\endproclaim
\head{3. Derivation of the contour representation}
\endhead
As explained in the introduction,
we start with the Duhamel expansion
for the transfer matrix.
In this section, we will consider a fixed finite volume
$\Lambda=\Lambda(L)=\{x\in\Bbb Z^d \mid x_i\leq L$
for all $i=1,\dots,d\}$, and a fixed value $q\in\{1,\dots,r\}$ for the
boundary condition;
further, we are not explicitely specifying this in our notation.
Introducing the transfer matrices
$$
T^{(0)}=e^{\tilde\beta H_{q,\Lambda}^{(0)}}
\tag 3.1
$$
and
$$
T=e^{\tilde\beta H_{q,\Lambda}},
\tag 3.2
$$
we rewrite the partition function $Z_{q,\Lambda}$ as
$$
Z_{q,\Lambda}=\Tr T^M,
\tag 3.3
$$
where $\tilde\beta$ and $M\in\Bbb N$ are related to the inverse temperature
$\beta$ by the equality
$$
\beta=M \tilde\beta.
\tag 3.4
$$
The Duhamel expansion (or Dyson series) for the operator $T$ yields
$$
T=\sum _{\boldkey n}
\Bigl[
\prod_{A\in{\cal A}_0}
\frac{(\lambda)^{n_A}}{n_A!}\int_0^{\tilde\beta}d\tau_A^1\dots
d\tau_A^{n_A}
\Bigr]
T(\boldsymbol \tau,\boldkey n).
\tag 3.5
$$
Here, $\Cal A_0$ is the family of all sets $A$ contributing to the sum
(2.15), $\boldkey n$ is an multiindex $ \boldkey n\: \Cal A_0\to
\{0,1,\dots,\}$ with finite $n=\sum_{A\in\Cal A_0}n_A$,
$\boldsymbol \tau=\{\tau_A^1,\dots ,\tau_A^{n_A}, A\in\Cal A_0\}\in
[0,\tilde\beta]^n$, and the operator
$T(\boldsymbol \tau,\boldkey n)$ is obtained from $T^{(0)}$ by ``inserting''
the operator $V_A$ at the times $\tau_A^1,\dots ,\tau_A^{n_A}$.
Formally, it can be defined as follows.
For a given $\boldkey n$ and $\boldsymbol \tau$, let
$\supp \boldkey n\equiv\Cal A=\{A_1,\dots,A_k\}$ be the set of
all $A\in\Cal A_0$ with $n_A \neq 0$, $n_i=n_{A_i}$, and $V_i=V_{A_i}$.
Let
$$
(s_1,\dots,s_n)
=\pi(\tau_{A_1}^1,\dots, \tau_{A_1}^{n_1},\dots,
\tau_{A_k}^1,\dots, \tau_{A_k}^{n_k})
$$
be a permutation of the times $\boldsymbol \tau$
such that $s_1\leq s_2\leq\dots\leq s_n$,
and set
$$
(\tilde V_1,\dots,\tilde V_n)=
\pi({V_1},\dots, {V_1},\dots,{V_k},\dots, {V_k}),
$$
where on the righthand side each $V_i$
appears exactly $n_i$ times.
Then $T(\boldsymbol \tau,\boldkey n)$ is defined by
$$
T(\boldsymbol \tau,\boldkey n)=
e^{s_1 H_{q,\Lambda}^{(0)}}
\tilde V_1 e^{(s_2s_1) H_{q,\Lambda}^{(0)}}
\tilde V_2 \dots
e^{(s_ns_{n1}) H_{q,\Lambda}^{(0)}} \tilde V_n
e^{(\tilde\beta  s_n) H_{q,\Lambda}^{(0)}}.
\tag 3.6
$$
Next, we resum (3.5) to obtain the expansion
$$
T=\sum_{B\subset \Lambda} T(B),
\tag 3.7
$$
where
$$
T(B)=\sum_{\scriptstyle{\cal A=\{A_1,\dots,A_k\}}
\atop \scriptstyle{\cup_i A_i= B}}
\tilde T(\Cal A),
\tag 3.8
$$
with
$$
\tilde T(\Cal A)=
\sum _{\boldkey n: \supp \boldkey n= \Cal A}
\Bigl[
\prod_{A\in{\cal A}}
\frac{(\lambda)^{n_A}}{n_A!}\int_0^{\tilde\beta}d\tau_A^1\dots
d\tau_A^{n_A}
\Bigr]
T(\boldsymbol \tau,\boldkey n).
\tag 3.9
$$
Before continuing with the expansion for the partition
function as sketched in the introduction, we discuss the factorization
properties of the operator $T(B)$.
Given a subset ${\Lambda^\prime}$ of $\Lambda$, we introduce the operators
$T_{\Lambda^\prime}(\boldsymbol \tau,\boldkey n)$,
$\tilde T_{\Lambda^\prime}(\Cal A)$, and
$T_{\Lambda^\prime}(B)$ that are obtained from
$T(\boldsymbol \tau,\boldkey n)$, $\tilde T(\Cal A)$, and
$T(B)$, respectively, by replacing
$H^{(0)}_{q,\Lambda}=\sum_{x\in\Lambda}H^{(0)}_{q,x}$ by the operator
$\sum_{x\in\Lambda^\prime}H^{(0)}_{q,x}$. Using the fact that
$$
[H^{(0)}_{q,x},H^{(0)}_{q,y}]=0
\qquad\text{for all}\qquad
x,y\in\Lambda,
\tag 3.10
$$
while
$$
[H^{(0)}_{q,x},V_A]=0
\qquad\text{if}\qquad
\dist(x,A)>R_0,
\tag 3.11
$$
one immediately obtains that
$$
T(B)=T^{(0)}_{\Lambda\setminus\bar B}\, T_{\bar B}(B) =
T_{\bar B}(B)\, T^{(0)}_{\Lambda\setminus\bar B},
\tag 3.12
$$
where $\bar B$ is the set
$$
\bar B=\{x\in\Lambda\,\,\dist(x,B)\leq R_0\}.
\tag 3.13
$$
Let us now consider a set $B$ that can be decomposed as $B=B_1\cup B_2$,
with $\bar B_1\cap \bar B_2=\emptyset$.
Then
$$
T_{\bar B}(B)=T_{\bar B_1}(B_1) T_{\bar B_2}(B_2)
=T_{\bar B_2}(B_2) T_{\bar B_1}(B_1),
\tag 3.14
$$
due to (3.10), (3.11), and the fact that
$$
[V_A,V_{A^\prime}]=0
\qquad\text{if}\qquad
A\cap A^\prime=\emptyset.
\tag 3.15
$$
For $B\subset \Lambda$,
we therefore get the decompositions
$$
T_{\bar B}(B) =\prod_{i=1}^k T_{\bar B_i}(B_i)
\tag 3.16
$$
and
$$
T(B)= T^{(0)}_{\Lambda\setminus\bar B}\,
\prod_{i=1}^k T_{\bar B_i}(B_i)
\tag 3.17
$$
provided $B=\cup_{i=1}^k B_i$, where $\bar B_1,\dots, \bar B_k$
are pairwise disjoint.
Deviating a little bit from the strategy explained in the introduction,
we further expand the transfer matrix $T$.
Using (3.17), we observe that
$$
\multline
\bra{s_B} \otimes \bra{s_{\Lambda\setminus B}}
T(B)
\ket{s'_{\Lambda\setminus B}}\otimes \ket{s'_B}
=\\=
\delta_{s_{\Lambda\setminus B},s'_{\Lambda\setminus B}}
e^{\tilde\beta\sum_{x\in \Lambda\setminus \bar B}
\Phi_x(s_{U(x)})}
\bra{s_B} \otimes \bra{s_{\Lambda\setminus B}}
T_{\bar B}(B)
\ket{s'_{\Lambda\setminus B}}\otimes \ket{s'_B}.
\endmultline
\tag 3.18
$$
Introducing thus the operator
$T_B(s_{\overline{\partial B}})$ on $\Cal H_B$ as the partial expectation
value
$$
T_B(s_{\overline{\partial B}})=
\bra{s_{\Lambda\setminus B}}T_{\bar B}(B)\ket{s_{\Lambda\setminus B}},
\tag 3.19
$$
we get
$$
T(B)=\sum_{s_{\Lambda\setminus B}}
e^{\tilde\beta\sum_{x\in\Lambda\setminus\bar B}\Phi_x(s_{U(x)})}
\bigl(\ket{s_{\Lambda\setminus B}}\bra{ s_{\Lambda\setminus B}}
\otimes T_B(s_{\overline{\partial B}})
\bigr)
\tag 3.20
$$
As before,
$U(x)=\{y\in\Bbb Z^d\,\,\dist(x,y)\leq R_0\}$,
while $\overline{\partial B}$
is the set
$$
\overline{\partial B}=\{x\notin B\,\,\dist(x,B)\leq 2R_0\}.
\tag 3.21
$$
Considering ``configurations'' $\Sigma=(B,s_{\Lambda\setminus B})$
on $\Lambda$ specyfiyng the
set $B$ as well as the configuration
$s_{\Lambda\setminus B}$ outside it, we can combine (3.7) and (3.20)
to get the expansion
$$
T=\sum_{\Sigma} K(\Sigma).
\tag 3.22
$$
Here the operators $K(\Sigma)$ (on $\Cal H_\Lambda$) are defined by
$$
K(\Sigma)\equiv K(B,s_{\Lambda\setminus B})=
e^{\tilde\beta\sum_{x\in\Lambda\setminus\bar B}\Phi_x(s_{U(x)})}
\Bigl(s_{\Lambda\setminus B}\rangle\langle s_{\Lambda\setminus B}
\otimes T_B(s_{\overline{\partial B}})
\Bigr).
\tag 3.23
$$
\remark{Remarks}
i) The operator $T_B(s_{\overline{\partial B}})$
inherits from $T_{\bar B}(B)$ the factorization property (3.16). Namely,
$$
T_{B}(s_{\overline{\partial B}})=
\bigotimes_{i=1}^n T_{B_i}(s_{\overline{\partial B_i}})
\tag 3.24
$$
provided $B=\cup_{i=1}^nB_i$ with $\bar B_1,\dots, \bar B_n$
pairwise disjoint.
ii) For $x$ and $B$ near to the boundary of $\Lambda$, the spin
configurations $s_{U(x)}$ and $s_{\overline{\partial B}}$
appearing in $\Phi_x$ and $T_B(s_{\overline{\partial B}})$
involve spins $s_y$ with $y\notin\Lambda$.
A more precise notation would therefore involve the spin configuration
$s_{\Lambda\setminus B}\cup g^{(q)}_{\partial \Lambda}$ restricted to the
sets ${U(x)}$ and ${\overline{\partial B}}$, respectively.
\endremark
Next, we combine the representation (3.22)
for the transfer matrix $T$ with the formula (3.3) to rewrite
$Z_{q,\Lambda}$ as
$$
Z_{q,\Lambda}=\sum_{\Sigma_1,\dots,\Sigma_M}
w(\Sigma_1,\dots,\Sigma_M)
\tag 3.25
$$
with the weights
$$
w(\Sigma_1,\dots,\Sigma_M)=
\Tr_{\Cal H_\Lambda}
\prod_{t=1}^M K(\Sigma_t).
\tag 3.26
$$
For a given collection of configurations $\Sigma_1,\dots,\Sigma_M$,
$\Sigma_t=(B^{(t)},s_{\Lambda\setminus B^{(t)}}^{(t)})$,
on ``time slices'' $t=1, \dots, M$,
we now assign a variable
$\sigma_{(x,t)}\in\bar S=\{0,1,\dots,q\}$ to each point in ``spacetime''
lattice
$$
\Bbb L=\Bbb Z^d\times\{1,\dots,M\}
\tag 3.27
$$
by defining
$$
\sigma_{(x,t)}=
\cases
g^{(q)}_x & $if $x\notin\Lambda,\\
s_x^{(t)}& $if $x\in\Lambda\setminus B^{(t)},\\
0 & $if $x\in B^{(t)}.\\
\endcases
\tag 3.28
$$
Considering elementary cubes, i.e. the closed unit cubes $C(x,t)$
with center $(x,t)$ in
$$
\Bbb L_\Lambda=\Lambda\times\{1,\dots,M\},
\tag 3.29
$$
we say that a cube $C(x,t)$ is {\it in the ground state $m$},
if the configuration $\sigma_{(y,t)}$ coincides with the configuration
$g^{(m)}$ on all points $y\in U(x)$.
Otherwise, the cube $C(x,t)$ is called an {\it excited cube}.
Note that a cube $C(x,t)$ may be excited for two reasons (possibly both):
either the $R_0$neighborhood $U(x)$ of $x$ contains a point $y\in
B^{(t)}$, corresponding to the insertion of some operator $V_A$ with $y\in
A$, i.e. due to a {\it quantum excitation}, or it contains a point $y$ for
which the classical variable $s^{(t)}_y$ differs from the ground state value
$g^{(m)}_y$, which corresponds to a {\it classical excitation}.
Note also that a configuration where two succesive cubes $C(x,t)$ and
$C(x,t+1)$ are in a classical state, $\sigma_{(x,t)}=s_x^{(t)}$
and $\sigma_{(x,t+1)}=s_x^{(t+1)}$, has weight zero unless
$\sigma_{(x,t)}=\sigma_{(x,t+1)}$; indeed, otherwise one has
$\bra{s_x^{(t)} } K(\Sigma_t)K(\Sigma_{t+1})\ket{s_x^{(t+1)}} =0$.
This is true also for $t=M$ once we identify $t=M+1$ with $t=1$ (periodic
boundary conditions on $\Bbb L$).
Recalling the relation (3.23), we now extract a factor
$e^{\tilde\beta \Phi_x(g^{(m)})}\equiv e^{\tilde\beta e_m}$
for each cube $C$ in the ground state $m$, leading to an overall factor of
$$
e^{\tilde\beta \sum_m e_mW_m},
\tag 3.30
$$
where $W_m$ is the number of cubes in the ground state $m$.
Considering, on the other hand, the union $D$ of all excited cubes, we assign
a label $\alpha_D(F)$ to all elementary faces $F$
in the boundary of $D$,
by defining $\alpha_D(F)=m$ if $F$ is a common face
for a cube $C_D$ in $D$ and a cube $C_m$ outside $D$ in the ground state $m$,
$F=C_D\cap C_m$. Defining the reduced weight
$\omega(\sigma_{\Bbb L})$ by
$$
w(\Sigma_1,\dots,\Sigma_M)
=e^{\tilde\beta \sum_m e_mW_m}
\omega(\sigma_{\Bbb L}),
\tag 3.31
$$
we observe that $\omega(\sigma_{\Bbb L})$
depends only on the configuration $\sigma_D$
and the label $\alpha_D$,
$$
\omega(\sigma_{\Bbb L})=
\omega(\sigma_D,\alpha_D).
\tag 3.32
$$
(The configuration outside $D$ is entirely determined by the labels
$\alpha_D$.) The weight $\omega(\sigma_D,\alpha_D)$
inherits from (3.24) the factorization property
$$
\omega(\sigma_D,\alpha_D)=
\prod_{i=1}^n
\omega(\sigma_{D_i},\alpha_{D_i}).
\tag3.33
$$
Here $D_1,\dots,D_n$ are the connectivity
components of $D$.
At this point, the rest is standard.
One considers the sets
$$
\Bbb T_\Lambda=\bigcup_{(x,t)\in\Bbb L_\Lambda}C(x,t)
\tag 3.34
$$
and
$$
\Bbb T =\bigcup_{(x,t)\in \Bbb L_\infty}C(x,t),
\tag 3.35
$$
imposing periodic bundary conditions in the ``time direction'',
and defines a (labeled) contour $Y$ as a pair $(\supp Y,\alpha)$,
where
$\supp Y\subset \Bbb T_\Lambda$
is a connected union of closed unit cubes
with centers in $\Bbb L_\Lambda$ (considered as a subset of
$\Bbb T$), while
$\alpha$ is an asignment of a label
$\alpha(F)$ to faces of $\partial\supp Y$ which is constant on the boundary
of all connected components of
$\Bbb T\setminus\supp Y$.
The contours $Y_1,\dots,Y_n$ corresponding to a configuration
$\sigma_{\Bbb L_\Lambda}$ are then defined by taking the connected
components of the set $D$ of excited cubes in
$\Bbb T_\Lambda$ for their
supports $\supp Y_1,\dots,\supp Y_n$ and by
taking the labels $m$ of the ground
states for the cubes $C$ in
$\Bbb T_\Lambda\setminus \supp Y_i$ that touch the
face $F$, see above, for the corresponding labels $\alpha_i(F)$.
Resumming over all configurations in (3.25) that lead to the same set of
contours, and taking into account the factorization property (3.33),
it is an easy exercise to show that the resulting weight factors into a
product of contour activities $\rho(Y_i)$ and ground state terms
$e^{\tilde\beta e_m W_m}$. This yields the representation
$$
Z_{q,\Lambda}=\sum_{\{Y_1,\dots,Y_n\}}
\prod_i \rho(Y_i)
\prod_m e^{\tilde\beta e_m W_m},
\tag 3.36
$$
which is exactly of the same form as the contour representation of a
classical spin system. We can therefore apply standard PirogovSinai theory
to analyze the quantum spin system considered here,
provided we can verify its basic assumption  the Peierls condition.
Namely, we should prove a bound of the form
$$
\rho(Y)\leq e^{\gammaY}e^{\tilde\beta e_0Y}
\tag 3.37
$$
where $\gamma$ is a sufficiently large constant,
$e_0=\min_m e_m$, and $Y$ is the number of elementary cubes in $\supp Y$.
This will be done in the next section.
\remark{Remark}
The weights $\rho(Y)$ are in general complex, even if
the coupling constant
$\lambda$ is real. Observing that the operators
$K(\Sigma_t)$ in (3.26) are selfadjoint for real $\lambda$,
we have, however, that
$$
w(\Sigma_1,\dots,\Sigma_M)^\ast
=w(\Sigma_M,\dots,\Sigma_1)
\,.
\tag 3.38
$$
Considering two contours
$Y$ and $Y^\ast$ which can be obtained from each other
by a reflection at a constant time plane, we therefore
get
$$
\rho(Y)^\ast=\rho(Y^\ast)
\tag 3.39
$$
provided $\lambda\in\Bbb R$.
\endremark
\head{4. Exponential decay for contour activities}
\endhead
We first give an explicit expression for the weight $\rho(Y)$. Combining
(3.25), (3.31), (3.32), and (3.33) we have
$$
Z_{q,\Lambda} = \sum_{\Sigma_1,\dots,\Sigma_M}
e^{\tilde\beta \sum_m e_m W_m}
\prod_{i=1}^n \omega(\sigma_{D_i},\alpha_{D_i}),
\tag 4.1
$$
where $D_1,\,\dots,\,D_n$ are the connectivity components
of the union $D$ of all excited cubes corresponding to
$\Sigma_1,\dots,\Sigma_M$.
This expression is equivalent to
(3.36) once we take for a contour
$Y \equiv (D,\alpha_{D})$ the weight
$$
\rho(Y) = \sum_{\sigma_{D} \to Y} \omega(\sigma_{D},\alpha_{D}).
\tag 4.2
$$
Here the sum is over all configurations $\sigma_{D}$ consistent with the
contour $Y$, i.e. over all configurations $\sigma_{D}$ on $D=\supp Y$ that,
if extended outside $\supp Y$ by appropriate ground states determined by the
labels $\alpha_D$, yield the contour $Y$.
\proclaim{Proposition 4.1} Let $\lambda\in\Bbb R$, $\tilde\beta>0$,
and $\gamma_Q\geq 1$ be such that, for all $x\in\Bbb Z^d$,
$$
(e1)\tilde\beta\lambda
\sum_{A\in\Cal A_0:\atop x\in A}
\V_A\e^{\gamma_QA}
\leq 1.
\tag 4.3
$$
Then
$$
\rho(Y)
\leq
e^{(\tilde\beta e_0 + \gamma)Y}
\tag 4.4
$$
where
$$
\gamma=\min\{\tilde\beta\gamma_0,
R_0^{d}(\gamma_Q1)\} 
\log (2S).
\tag 4.5
$$
\endproclaim
The proof of the proposition relies on the following
lemma.
\proclaim{Lemma 4.2}
Let $B\subset\Lambda$,
and let $s_B$, $\tilde s_B$ and $s_{\overline{\partial B}}$
be arbitrary classical configurations on $B$ and
$\overline{\partial B}=\{x\notin B\mid \dist(x,B)\leq 2 R_0\}$,
respectively.
Let $\gamma_Q$, $\tilde\beta$ and $\lambda$
be as in Proposition 4.1.
Then
$$
\left
\bra{s_B}
T_B(s_{\overline{\partial B}})
\ket{\tilde s_B}
\right
\leq
e^{\tilde\beta e_0\bar B}
e^{(\gamma_Q1)B}.
\tag 4.6
$$
\endproclaim
\demo{Proof of Lemma 4.2}
Let
$s_\Lambda$ be an arbitrary extension of the configuration
$s_{B\cup\overline{\partial B}}$ to the full set
$\Lambda$, and let $\tilde s_\Lambda$ be the configuration
which agrees with $s_\Lambda$ on $\Lambda\setminus B$,
and with $\tilde s_B$ on $B$.
Then
$$
\bigl
\bra{s_{B}}
T_B(s_{\overline{\partial B}})
\ket{\tilde s_B}
\bigr
=
\bigl
\bra{s_\Lambda}
T_{\bar B} (B)
\ket{\tilde s_\Lambda}
\bigr
\leq
\bigl\
T_{\bar B} (B)
\bigr\
,
\tag 4.7
$$
where, in agreement with (3.8) and (3.9),
the operator $T_{\bar B} (B)$ is defined by
$$
T_{\bar B} (B) = \sum_{\scriptstyle {\cal A} =
\{A_1,\dots,A_k\} \atop A_i\in\Cal A_0,\, \scriptstyle \cup_j A_j = B}
\tilde T_{\bar B} ({\cal A})
\tag 4.8
$$
and
$$
\tilde T_{\bar B}(\Cal A)=
\sum _{\boldkey n: \supp \boldkey n= \Cal A}
\Bigl[
\prod_{A\in{\cal A}}
\frac{(\lambda)^{n_A}}{n_A!}\int_0^{\tilde\beta}d\tau_A^1\dots
d\tau_A^{n_A}
\Bigr]
T_{\bar B}(\boldsymbol \tau,\boldkey n).
\tag 4.9
$$
The timeordered operator $T_{\bar B}({\boldsymbol \tau}, \bold n)$ is
defined as in (3.6), with the Hamiltonian
$\sum_{x \in \bar B} H_{q,x}^{(0)}$ replacing $H_{q,\Lambda}^{(0)}$.
Observing that for all $s>0$,
$$
\e^{s\sum_{x \in \bar B}
H^{(0)}_{q,x}}\=
e^{s\bar B e_0},
\tag 4.10
$$
we now bound
$$
\bigl\
T_{\bar B}(\boldsymbol \tau,\boldkey n)
\bigr\
\leq
e^{\tilde\beta \bar B e_0}
\prod_{A \in {\cal A}}
\ V_A \^{n_A}.
\tag 4.11
$$
Combining (4.7)  (4.9) with the bound (4.11)
and the fact that
the assumption (4.3) implies that
$\tilde\beta\lambda \ V_A \\leq 1$
for all $A\in\Cal A_0$,
we obtain
$$
\align
\bigl
\bra{s_{B}}
T_{B}(s_{\overline{\partial B}})
\ket{\tilde s_{B}}
\bigr
&\leq
e^{\tilde\beta e_0  \bar B }
\sum_{\scriptstyle {\cal A} = \{A_1,\dots,A_k\}
\atop\scriptstyle A_i\in\Cal A_0,\,\cup_j A_j = B}
\prod_{A \in {\cal A}}
\Bigl(
\sum_{n_A = 1}^\infty
{(\tilde\beta\lambda)^{n_A} \over n_A !}
\ V_A \^{n_A}
\Bigr)
\\
\leq
e^{\tilde\beta e_0  \bar B }
&\sum_{\scriptstyle {\cal A} = \{A_1,\dots,A_k\}
\atop\scriptstyle A_i\in\Cal A_0,\,\cup_j A_j = B}
\prod_{A \in {\cal A}}
\Bigl(
(e1)\tilde\beta\lambda
\ V_A \
\Bigr)
\\
\leq
e^{\tilde\beta e_0  \bar B }
&e^{\gamma_Q B}
\sum_{\scriptstyle {\cal A} = \{A_1,\dots,A_k\}
\atop\scriptstyle A_i\in\Cal A_0,\,\cup_j A_j = B}
\prod_{A \in {\cal A}}
\Bigl(
(e1)\tilde\beta\lambda
\ V_A \
e^{\gamma_QA}
\Bigr).
\tag 4.12
\endalign
$$
We proceed with the bound
$$
\align
\sum_{\scriptstyle {\cal A} = \{A_1,\dots,A_k\}
\atop\scriptstyle A_i\in\Cal A_0,\,\cup_j A_j = B}
&\prod_{A \in {\cal A}}
\Bigl(
(e1)\tilde\beta\lambda \ V_A \
e^{\gamma_QA}
\Bigr)
\leq
\\
&\leq
\sum_{k=1}^\infty
\frac 1{k!}
\prod_{i=1}^k
\Bigl(
\sum_{A_i\in\Cal A_0:\atop A_i\cap B\neq\emptyset}
(e1)\tilde\beta\lambda
\bigl\ V_{A_i} \bigr\ e^{\gamma_QA_i}
\Bigr)
\\
&\leq
\sum_{k=1}^\infty
\frac 1{k!}
\prod_{i=1}^k
\Bigl(
\sum_{x\in B}
\sum_{A_i\in\Cal A_0:\atop x\in A_i}
(e1)\tilde\beta\lambda
\bigl\ V_{A_i} \bigr\ e^{\gamma_QA_i}
\Bigr)
\\
&\leq
\sum_{k=1}^\infty
\frac 1{k!}B^k
\leq e^{B},
\tag 4.13
\endalign
$$
where we have used the assumption (4.3) in the second to
last step. Combining
(4.12) and (4.13),
we obtain the lemma.
\hfill\hfill\qed
\enddemo
\demo{Proof of Proposition 4.1}
In order to bound the sum in
(4.2), we first derive a more explicit representation for
the activity $\rho(Y)$.
We decompose the torus $\Bbb T$ and the
set of excited cubes $D$ into timeslices,
$\Bbb T = \cup_{t=1}^M \Bbb T^{(t)}$ and
$D = \cup_{t=1}^M D^{(t)}$, and recall that
$\sigma_{(x,t)} =0$ iff
the cube $C(x,t)$ belongs to
$B^{(t)}$, the set of sites where transitions between times $t$ and
$(t+1)$ may occur.
Summing over all configurations $\sigma_{D}$ consistent with the
contour $Y$
then corresponds to the following three restrictions:
\roster
\item
The configuration $\sigma_{D}$
is
such that the $R_0$neighbourhood of
each $(x,t)$ with $\sigma_{(x,t)} = 0$ is included in $D$; i.e.
$\overline{B}^{(t)}\subset D^{(t)}$
for each $t=1,\dots,M$.
\item
All cubes $C(x,t)\subset D$ are excited; i.e. either $\sigma_{(x,t)} =0$
or $s^{(t)}_{U(x)} \neq g^{(m)}_{U(x)}$
for all $m=1,\dots,r$.
\item
Let $F$ be a vertical face in the boundary of $D$,
let $m=\alpha_D(F)$ be the label of $F$,
and let $\Bbb T^{(t)}$ be the timeslice
containing $F$. Then $\sigma(x,t)=g^{(m)}_x$
for all $x\in D^{(t)}$
whose distance from $F$ is less then $R_0$.
\endroster
Let us observe that since
$\ket{s_{\Lambda\setminus B_1}}\otimes\ket{s_{B_1}}\equiv
\ket{s_{\Lambda\setminus B_2}}\otimes\ket{s_{B_2}} $ for any $B_1, B_2
\subset \Lambda$, we may write the expansion of unity
$\boldkey 1=\sum_{s_\Lambda}\ket{s_\Lambda}\bra{s_\Lambda} $ on $\Cal
H_\Lambda$ in the form
$$
\boldkey 1=\sum_{s_\Lambda}\ket{s_{\Lambda\setminus B_1}}\otimes\ket{s_{B_1}}
\bra{s_{\Lambda\setminus B_2}}\otimes\bra{s_{B_2}}.
\tag 4.14
$$
Inserting now (3.26) with (3.31) and (3.32) into (4.2), we may use the
above observation to get the expression
$$
\multline
\!\!\!\!\!\rho(Y) =
\sum_{\scriptstyle B^{(1)},\dots,B^{(M)} \atop\scriptstyle \bar B^{(t)}
\subset D^{(t)} } \sum_{s_{D^{(1)} \setminus B^{(1)}}^{(1)},
\dots, s_{D^{(M)} \setminus B^{(M)}}^{(M)} }
\sum_{s_{I^{(1)}}^{(1)}, \dots ,s_{I^{(M)}}^{(M)} }
\prod_{t=1}^M e^{\tilde\beta \sum_{x \in D^{(t)} \setminus \bar B^{(t)} }
\Phi_x(s_{U(x)}^{(t)}) }\times\\ \times
\bra{s_{D^{(t)} \setminus B^{(t)}}^{(t)}} s_{D^{(t)}
\setminus B^{(t)}}^{(t+1)}\rangle
\bra{s_{B^{(t)}}^{(t)}} T_{B^{(t)}}(s_{\overline{\partial B}^{(t)}}^{(t)} )
\ket{s_{B^{(t)}}^{(t+1)}}.
\endmultline
\tag 4.15
$$
Here we defined $I^{(t)} = B^{(t1)} \cap B^{(t)}$.
The three succesive summations are equivalent to the sum in (4.2).
Namely, the sum over $B^{(1)},\dots,B^{(M)}$ in (4.15) obeys automatically
the first restriction above; the second sum must respect the two others (and
there is no restriction on the third sum).
Observe that the choice of a contour
$Y$
(or of several contours),
together with the choice of partial configurations $s_{D^{(t)} \setminus
B^{(t)}}^{(t)}$,
$s_{I^{(t)}}^{(t)}$, at each time $t$ defines
completely the configurations between each time slice.
In order to get a bound on $\rho(Y)$, we estimate the absolute values
of the factors of the terms on the right hand side of (4.15). Taking into
account the condition (2) above and the assumption (2.6) on the classical
part of the Hamiltonian, we have
$$
e^{\tilde\beta \sum_{x \in D^{(t)} \setminus \bar B^{(t)} }
\Phi_x(s_{U(x)}^{(t)}) } \leq e^{\tilde\beta (e_0 + \gamma_0)  D^{(t)}
\setminus \bar B^{(t)}  }.
\tag 4.16
$$
The scalar product between the two base vectors is equal to 0 or 1.
In fact, we could use it to reduce the number of terms appearing in the
second sum in (4.15); however, we just bound it by one.
Combined with Lemma 4.2, we finally get
$$
\rho(Y)
\leq
\sum_{\scriptstyle B^{(1)},\dots,B^{(M)}
\atop\scriptstyle\bar B^{(t)}\subset D^{(t)} }
\prod_{t=1}^M
\sum_{s_{D^{(t)} \setminus B^{(t)}}^{(t)} }
\sum_{s_{I^{(t)}}^{(t)} }
e^{\tilde\beta (e_0 + \gamma_0) D^{(t)}\setminus\bar B^{(t)} }
e^{\tilde\beta e_0 \bar B^{(t)}}
e^{(\gamma_Q1)B^{(t)}}.
\tag 4.17
$$
The summands do not depend any more on the partial configurations
${s_{D^{(t)} \setminus B^{(t)}}^{(t)} }$ and
${s_{I^{(t)}}^{(t)} }$. Their
number is bounded by
$$
\prod_{t=1}^M
r^{D^{(t)} \setminus \bar B^{(t)} + I^{(t)} }
\leq S^{Y}.
\tag 4.18
$$
To estimate the exponential in (4.17) we use the equation
$D^{(t)} \setminus \bar B^{(t)} = D^{(t)}  \bar B^{(t)}$ and
the bound $\bar B^{(t)} \leq R_0^d B^{(t)}$. Thus
$e^{(\gamma_Q1) B^{(t)}}\leq e^{(\gamma_Q1) R_0^{d} \bar B^{(t)}}$,
and
$$
\rho(Y) \leq
\bigl[
r\, e^{\tilde\beta e_0}
\,
\max\{
e^{\tilde\beta\gamma_0}
\,,\,
e^{(\gamma_Q1) R_0^{d}}
\}
\bigr]^{Y}
\prod_{t=1}^M
\sum_{\scriptstyle B^{(t)} \atop\scriptstyle \bar B^{(t)}
\subset D^{(t)} }
1.
\tag 4.19
$$
The last sum can be easily bounded,
yielding
$$
\sum_{\scriptstyle B^{(t)} \atop\scriptstyle \bar B^{(t)}
\subset D^{(t)} }
1
\leq
\sum_{\scriptstyle B^{(t)}\subset D^{(t)} }
1
=2^{D^{(t)}}.
\tag 4.20
$$
Combined with (4.19), this completes the proof of Proposition 4.1.
\hfill\hfill\qed
\enddemo
We close this section with a proposition providing the necessary
bounds on derivatives:
\proclaim{Proposition 4.3}
Let $\lambda\in\Bbb R$, $\tilde\beta>0$,
and $\gamma_Q\geq 1$ be such that, in addition to (4.3)
we have
$$
e\tilde\beta\lambda
\sum_{A\in\Cal A_0:\atop x\in A}
\Bigl\
\frac {\partial}{\partial\mu_i} V_A
\Bigr\
e^{\gamma_QA}
\leq 1.
\tag 4.21
$$
Then
$$
\Bigl
\frac {\partial}{\partial\mu_i}\rho(Y)
\Bigr
\leq
\bigl(\tilde\beta C_0+1\bigr) Y
e^{(\tilde\beta e_0 + \gamma)Y}.
\tag 4.22
$$
Here $C_0$ is the constant from
{\rm (2.7)} and $\gamma$ is the constant
defined in {\rm (4.5)}.
\endproclaim
\demo{Proof}
We first derive
an analogue of the bound (4.6) in Lemma 4.2.
To this end, we have to bound the norm of
$\partial T_{\bar B}(\boldsymbol \tau,\boldkey n)/\partial\mu_i$.
Using the representation (3.6) in conjunction with
the assumption (2.7) and the bound (4.10), we get
$$
\multline
\Bigl\
\frac {\partial}{\partial\mu_i}
T_{\bar B}(\boldsymbol \tau,\boldkey n)
\Bigr\
\leq
e^{\tilde\beta \bar B e_0}
\Bigl\{
\tilde\beta C_0 \bar B\prod_{A \in {\cal A}}
\ V_A \^{n_A}+
\\
+
\sum_{j=1}^k n_{A_j}
\Bigl\
\frac{\partial V_{A_j}}{\partial\mu_i}
\Bigr\
\ V_{A_j} \^{n_{A_j}1}
\prod_{A \in {\cal A}\setminus \{A_j\}}
\ V_A \^{n_A}
\Bigr\}.
\endmultline
\tag 4.23
$$
Proceeding as before, we obtain
$$
\align
\bigl
\bra{s_{B}}
\frac {\partial}{\partial\mu_i}
T_{B}(s_{\overline{\partial B}})
\ket{\tilde s_{B}}
\bigr
&\leq\\
\leq e^{\tilde\beta e_0  \bar B } \!\!\!\!\!\!\!\!
\sum_{\scriptstyle {\cal A} = \{A_1,\dots,A_k\}
\atop\scriptstyle A_i\in\Cal A_0,\,\cup_j A_j = B} \!\!
&\Bigl\{
\tilde\beta C_0 \bar B\prod_{A \in {\cal A}}
\bigl(
(e1)\tilde\beta\lambda
\ V_A \
\bigr)+
\\
&\qquad\qquad+
\sum_{j=1}^k
e\tilde\beta\lambda\,
\Bigl\
\frac{\partial V_{A_j}}{\partial\mu_i}
\Bigr\
\prod_{A \in {\cal A}}
\bigl(
(e1)\tilde\beta\lambda
\ V_A \
\bigr)
\Bigr\}\leq
\\
&\leq
e^{\tilde\beta e_0  \bar B }
e^{\gamma_QB}
\sum_{k=1}^\infty
\frac 1{k!}B^k
\bigl\{
\tilde\beta C_0 \bar B +k
\bigr\}
\\
&\leq
e^{\tilde\beta e_0  \bar B }
e^{\gamma_QB}
\bigl\{
\tilde\beta C_0 \bar B + B
\bigr\}
e^{B}
\\
&\leq
\bigl(\tilde\beta C_0+1\bigr) \bar B
e^{\tilde\beta e_0  \bar B }
e^{(\gamma_Q1)B}.
\tag 4.24
\endalign
$$
Inserted into (4.15), and continuing in the same way as before,
we obtain the bound (4.21).\hfill\hfill\qed
\enddemo
\vfill
\newpage
\head{5. Truncated free energies and the stable phases of the model}
\endhead
While the next section will be devoted to a detailed discussion of the
mean values of general local variables, here we will
anticipate the fact that the state
$\langle \boldsymbol \cdot \rangle_q$ can be linked with the
``probability''
 in the ensemble (3.36) of labeled contours

that a given
site $x$ is in
$W_q$,
the area outside contours and in the ground state
$g^{(q)}$. To discuss the stability of phases in dependence on the
``tuning'' parameter
$\mu$ (cf\. Section 2.1), we can thus use the standard PirogovSinai
theory. There are only two features that are not entirely standard
 the fact that the weights $\rho(Y)$ are in general {\it complex numbers}
and the fact that our model is actually considered {\it on a slab} of
thickness $M$ (linked with the temperature $\beta$) with periodic boundary
conditions in the ``time'' direction.
The former was taken into account in \cite{BI89} (also in
\cite{GKK88}, but here we will base our discussion on \cite{BI89}
and later works based
on it, in particular \cite{BK90} and \cite{BK94}).
The latter is a novel feature of quantum models.
Even though it leads only to small modifications, it is important to
realize that the metastable free energies used to determine the phase
diagram will have contributions comming from contours wrapped around
$\Bbb T_\Lambda$
in the ``time'' direction.
This leads to certain modifications in the definition of
truncatd free energies which will be described in this section.
We start with some notation. As usual (see also Section 3)
a {\it contour} is a pair $(\supp Y,\alpha)$,
where
$\supp Y\subset \Bbb T_\Lambda$
is a connected union of finitely many closed unit cubes
with centers in $\Bbb L_\infty$
(we call those cubes {\it elementary cubes} in the sequel), while
$\alpha$ is an asignment of a label
$\alpha(F)$ to the faces $F$ in $\partial\supp Y$ which is constant
on the boundary
of all connected components of
$\Bbb T\setminus\supp Y$.
Its interior $\operatorname{Int} Y$
is the union of all finite
components of $\Bbb T \setminus\operatorname{supp} Y$
and $\operatorname{Int}_m Y$ the union of all
components of $\operatorname{Int} Y$ whose boundary is labeled by $m$.
Recalling that we assumed $d\geq 2$, we
note that the set
$\Bbb T\setminus (\supp Y \cup \Int Y)$
is a connected set, implying that the
functions $\alpha(\cdot)$ is constant on
the boundary of the set $V(Y)=\supp Y\cup\Int Y$.
We say that $Y$ is a $q$contour, if $\alpha_Y = q$ on this boundary.
Two contours $Y$ and $Y^\prime$ are
called {\it compatible} or not touching if
$\supp Y\cap\supp Y^\prime=\emptyset$, and {\it mutually external}
if
$
V(Y)\cap V(Y^\prime)=\emptyset
$.
Given a finite set of mutually compatible contours
$Y_1,\dots,Y_n$ we say that $Y_i$, $i=1,\dots,n$,
is an {\it external contour} in $\{Y_1,\dots,Y_n\}$,
if $\supp Y_i\cap V(Y_j)=\emptyset$
for all $j\neq i$.
Consider now a set of contours
$\{Y_1,\dots,Y_n\}$
contributing to (3.36). The contours
in $\{Y_1,\dots,Y_n\}$ are then mutually compatible,
and all external contours are $q$contours.
In addition, the labels of these contours
are {\it matching} in the sense that the boundary of each
connected component of
$\Bbb T_\Lambda\setminus (\supp Y_1\cup\cdots\cup\supp Y_n)$
has constant boundary conditions.
Given this setup, it is now standard to derive
a second representation for $Z_{q,\Lambda}$
which does not involve such a matching condition.
To this end, we first
introduce partition functions
$Z_{q}(V)$ for all volumes
$V\subset\Bbb T_\Lambda$
for which
$V^c=\Bbb T_\Lambda\setminus V$
is a (possibly empty)
union of closed elementary cubes in $\Bbb T_\Lambda$.
We say that $Y$ is a contour in $V$,
if $V(Y)\subset V$, and call a set
$\{Y_1,\dots,Y_n\}$ of mutually compatible contours
in $V$ a set of {\it matching contours in $V$}
if the boundary of each
connected component of
$V\setminus (\supp Y_1\cup\cdots\cup\supp Y_n)$
has constant boundary conditions.
Denoting the union of those components which
have boundary condition $q$ by $W_q$,
we define
$$
Z_{q}(V)=\sum_{\{Y_1,\dots,Y_n\} }
\prod_i \rho(Y_i)
\prod_m e^{\tilde\beta e_m W_m}
\,,
\tag5.1
$$
where the sum runs over all sets of mutually
compatible, matching contours in $V$
for which all external contours are $q$contours.
Recalling the expression (3.36) for
$Z_{q, \Lambda}$, we note that the partition function
(5.1) is actually equal to $Z_{q, \Lambda}$
if $V=\Bbb T_\Lambda$,
$$
Z_{q}(\Bbb T_\Lambda)=Z_{q, \Lambda}
\,.
\tag 5.2
$$
We now rewrite the partition function $Z_q(V)$, in a standard way,
as a sum over contours without any matching condition,
see e\.g\. \cite{Zah84} or \cite{BI89}.
Introducing the weights
$$
K_q(Y):=\rho(Y)e^{\tilde\beta e_q Y}
\prod_{m=1}^r \frac{Z_m(\Int_m Y)}
{Z_q(\Int_m Y)},
\tag 5.3
$$
one gets
$$
Z_q(V)=e^{\tilde\beta e_q V}
\sum_{\{Y_1,\dots, Y_n\}\subset V^0}
\prod^n_{k=1}K_q(Y_k).
\tag 5.4
$$
where the sum runs over all sets of mutually disjoint $q$contours
in $V$.
The weights $K_q(Y)$ do not, necessarily, satisfy the bound
$$
K_q(Y)\leq \epsilon^{Y}\,\,
\tag5.5
$$
with a sufficiently small constant $\epsilon>0$. While it
turns out that such a bound can be proven for stable
phases $q$, it is false for unstable phases.
In order to circumvent this problem, we follow the
standard strategy and
construct truncated contour
activities $K_q^\prime(Y)$, truncated partition functions
$$
Z_q^\prime(V)=e^{\tilde\beta e_q V}
\sum_{\{Y_1,\dots, Y_n\}\subset V^0}
\prod^n_{k=1}K_q^\prime(Y_k),
\tag 5.6
$$
and the corresponding free energies
$f_q$ in such a way that the weights $K_q^\prime(Y)$
satisfy the bound (5.5) and, in the same time do not differ from
$K_q(Y)$ whenever the phase $q$ is stable,
i\.e\. whenever the real part of the free energy
$f_q$ of the truncated model is minimal,
$\Re f_q=\min_m\Re f_m$.
Our definition of the truncated weights $K_q^\prime$ follows closely the
treatment from \cite{BK94}. The main difference is that since the contours
can have only a limited extension in the time direction, we take their
``horizontal diameter'' as the parameter to use in inductive definitions
(and proofs). Namely, for a contour $Y$, we take the diameter $\delta(Y)$
defined as the diameter of the $d$dimensional projection
$$
\{y\in \Bbb R^d \text { such that } (y,t)\in C(x,t), \text {for some }
C(x,t) \subset\supp Y \}
\tag5.7
$$
We define $\delta(V)$ in the same way.
Notice that $\delta(\tilde Y) < \delta(Y)$ whenever
$\supp\tilde Y\subset
\Int(Y)$.
To introduce the weight $K_q^\prime(Y)$ in an inductive manner, assume
that it has already been defined for all $q$ and all
contours $Y$ with $\delta(Y) < n$,
$n\in\Bbb N$, and that
it obeys a bound
of the form (5.5). Introduce $f_q^{(n1)}$
as the free energy of a contour model
with activities
$$
K^{(n1)} (Y^q) = \cases K'(Y^q) \ \ & \text{\rm if} \ \delta(Y^q) \leq
n1
\\ 0 & \text{\rm otherwise}.\endcases
\tag5.8
$$
Consider now a
contour $Y$ with $\delta(Y)=n$.
Since $\delta(\tilde Y) < n$ for all
contours $\tilde Y$ in $\Int Y$,
the truncated partition functions
$Z_q^\prime(\Int_m Y)$
are well defined for all $q$ and $m$.
Their logarithm can be controlled by a convergent cluster
expansion, and
$Z_q^\prime(\Int_m Y)\neq 0$ for all $q$ and $m$.
We therefore may define
$K_q^\prime(Y)$ for $q$contours $Y$ with $\delta(Y)=n$ by
$$
K_q^\prime(Y)=\chi_q^\prime(Y)
\rho(Y)e^{\tilde\beta e_q Y}
\prod_m{Z_m(\Int_m Y)\over
Z_q^\prime(\Int_m Y)}
\,\,,
\tag 5.9
$$
with
$$
\chi_q^\prime(Y)=\prod_{m\neq q}\chi
\left(
\alpha
 \tilde\beta(\Re f^{(n1)}_q\Re f^{(n1)}_m) \delta(Y)
\right).
\tag 5.10
$$
Here $\alpha$ is a constant that
will be chosen later
and
$\chi$ is a smoothed characteristic
function.
We assume that $\chi$ is a $C_1$ function that has been defined in such a way
that it obeys the conditions
$$
0\leq\chi (x)\leq 1,\quad
0\leq \frac{d\chi}{d x} \leq 1,
\tag 5.11
$$
and
$$
\chi(x)=0
\quad\text{\it if}\quad x
\leq 1
\quad\text{\it and}\quad\chi(x)=
1\quad\text{\it if}\quad x\geq 1\,\,.
\tag 5.12
$$
As a final element of
the construction of $K_q^\prime$,
we have to establish the bound
(5.5) for contours $Y$ with $\delta(Y)=n$.
The proof of this fact,
together with the proof of the following
Lemma 5.1, follows closely \cite{BK94}. However, since,
on the one hand, the claims of Lemma 5.1 (as well as the definition of
$K_q^\prime(Y)$) slightly differ from similar claims in \cite{BK94} and,
on the other hand, there is certain number of complications in
\cite{BK94} that are not necessary for the present case, we present the
proof in
the appendix.
We use $f_q$ to denote the free
energy corresponding to the partition
function $Z_q^\prime(V)$,
$$
f_q=  \frac1{\tilde\beta}\lim_{V\to \Bbb T}\frac {1}{V}
\text{log}\,
Z_q^\prime(V)\,\,,
\tag 5.13
$$
and
introduce
$f_0$ and $a_q$ as
$$
\align
f_0 &= \min_m \Re f_m\,\,,
\tag 5.14\\
a_q&=\tilde\beta(\Re f_q  f_0)\,\,.
\tag 5.15
\endalign
$$
Finally, we recall that volumes $V$ as well as supports of
contours, $\supp Y$, are unions of elementary cubes and $Y$ or
$\supp Y$ denotes their $(d+1)$dimensional volume. Similarly for the
boundary $\partial V$ of $V$ we use $\partial V$ to denote its
$d$dimensional euclidean area.
\proclaim{Lemma 5.1} Assume that
$\rho(\boldsymbol\cdot)$ obeys the
conditions {\rm (4.4)} and {\rm (4.22)} and let
$$
\epsilon = e^{\gamma+\alpha+2}
\qquad\text{and}\qquad
\bar \alpha = \alpha  2.
\tag 5.16
$$
Then there
exists a
constant $\epsilon_0>0$
(depending only on $d$ and $r$)
such that the following statements
hold provided $\epsilon<\epsilon_0$ and
$\overline\alpha\geq 1$.
\medskip\noindent
\item{i)} The contour activities
$K_q^\prime(Y)$ are well defined
for all $Y$ and obey {\rm (5.5)} and
$$
\Bigl
\frac {\partial}{\partial\mu_i}K^\prime_q(Y)
\Bigr
\leq
(3r\tilde\beta C_0+ 2)V(Y)
\epsilon^{Y}.
\tag 5.17
$$
\item{ii)} {If}
$a_q\delta(Y)\leq\overline\alpha$,
then $\chi_q(Y)=1$ and $K_q(Y)=K_q^\prime(Y)$.
\item{iii)} If
$a_q\delta(V)\leq \overline\alpha$,
then $Z_q(V)=Z_q^\prime(V)$.
\item{iv)} For all volumes $V\subset \Bbb T_\Lambda$
for which $V^c =\Bbb T_\Lambda\setminus V$
is a union of closed elementary cubes, one has
$$
Z_q(V)
\leq
e^{ \tilde\beta f_0V+O(\epsilon)\partial V}
\tag 5.18
$$
and
$$
\Bigl\frac {\partial}{\partial\mu_i}Z_q(V)\Bigr
\leq (2\tilde\beta C_0+ 1)
Ve^{ \tilde\beta f_0V+O(\epsilon)\partial V}.
\tag 5.19
$$
\endproclaim
\remark{Remarks}
i) Here, as in the appendix, $O(\epsilon)$ stands for a
bound $K\epsilon$, where $K<\infty$ is a constant
that depends only on the dimension $d$ and the number of
classical states $r$.
ii) For real $\lambda$,
the free energy $f$
independent of boundary conditions and can be expressed as
$$
f
=\frac{1}{\beta}\lim_{\Lambda\nearrow \Bbb Z^d}
\frac1{\Lambda} \log \Tr_{\Cal H_\Lambda} e^{\beta H_{q,\Lambda}}
=\frac{1}{\beta}\lim_{\Lambda\nearrow \Bbb Z^d}
\frac1{\Lambda}\log Z_{q,\Lambda}.
\tag5.20
$$
Since $H_{q,\Lambda}$ is a selfadjoint operator in this case,
$f$ is real. Rewriting the partition function in terms of
the $(d+1)$dimensional contour model introduced in Section 3,
see equation (5.2), we have
$$
f=\frac{1}{\tilde\beta M}\lim_{\Lambda\nearrow \Bbb
Z^d}\frac1{\Lambda}\log Z_{q}(\Bbb T_\Lambda)\equiv
\frac{1}{\tilde\beta}\lim_{V\nearrow \Bbb T}\frac1{V}
\log Z_q(V).
\tag5.21
$$
iii) By contrast, the metastable free energies $f_q$ defined
in (5.13) do in general depend on $q$, even if $\lambda$ is
real. It is worth noting, however, that the metastable
free energies are realvalued in this case. To see this,
we first observe that by the symmetry property (3.39),
$Z_q(V)$ is real for all $V$, implying that
$K_q(Y)^\ast=K_q(Y^\ast)$. Using the definitions
(5.8) through (5.10), one then establishes by induction
that $Z^\prime_q(V)$ is real while
$K^\prime_q(Y)^\ast=K^\prime_q(Y^\ast)$.
Given this symmetry for $K^\prime_q(Y)$, the reality
of $f_q$ now easily follows from the cluser expansion
of $\log Z^\prime_q(V)$.
\endremark
\head{6. Expectation values of local variables}
\endhead
To distinguish between different phases we have to evaluate
expectation values of
local observables. Whenever we have a local bounded observable $\ob$,
represented by an operator acting on $\Cal H_{\supp \ob}$ with a finite $\supp
\ob\subset
\Bbb Z^d$, we have
$$
\langle \ob\rangle_{q,\Lambda}=
\frac{\Tr_{\Cal H_\Lambda}(\ob e^{\beta H_{q,\Lambda}})}
{\Tr_{\Cal H_\Lambda}(e^{\beta H_{q,\Lambda}})}
=\frac{\Tr_{\Cal H_\Lambda}(\ob T^M)}
{\Tr_{\Cal H_\Lambda}(T^M)}
=\frac{Z_{q,\Lambda}^\ob}{Z_{q,\Lambda}}.
\tag 6.1
$$
Retracing the steps leading to the contour representation (3.36) of
$Z_{q,\Lambda}$, we can get in a straightforward manner a similar
expression for $Z_{q,\Lambda}^\ob$. Namely, introducing
$$
w_\ob(\Sigma_1,\dots,\Sigma_M)=
\Tr_{\Cal H_\Lambda} \bigl(\ob
\prod_{t=1}^M K(\Sigma_t)\bigr)
\tag 6.2
$$
with $K(\Sigma)$ given as before by (3.23), we have
$$
Z_{q,\Lambda}^\ob=\sum_{\Sigma_1,\dots,\Sigma_M}
w_\ob(\Sigma_1,\dots,\Sigma_M).
\tag 6.3
$$
Localizing the observable $\ob$,
{\it by definition}, in the first time slice,
we define the $d+1$ dimensional support
of $\ob$ as
$$
\Cal S(\ob):=\bigcup_{x\in\supp\ob}C(x,1)
\,.
\tag 6.4
$$
Assuming without loss of generality that
$\supp\ob$ and hence $\Cal S(\ob)$
is a connected set, we introduce the contours corresponding
to a configuration $\sigma_{\Bbb L_\Lambda}$
in the same way as before, with the only difference
that the supports of these contours are now the connected
components of $D\cup\Cal S(\ob)$, where, as in Section 3,
$D$ is the set of excited cubes in $\Bbb T_\Lambda$. By this
definition, one of the contours corresponding to
$\sigma_{\Bbb L_\Lambda}$ will contain the set $\Cal S(\ob)$
as part of its support. We denote this contour by $Y_{\ob}$.
Continuing as before, we obtain the representation
$$
Z_{q,\Lambda}^\ob
=\sum_{\scriptstyle \{Y_\ob,Y_1,\dots,Y_k\}
\atop\scriptstyle \supp Y_{\ob}\supset \Cal S(\ob)} \rho_\ob(Y_\ob)
\prod_i \rho(Y_i)
\prod_m e^{\tilde\beta e_m W_m},
\tag 6.5
$$
where
$$
\rho_\ob(Y_\ob)
= \sum_{\sigma_{D} \to Y}
\omega_\ob(\sigma_{D},\alpha_{D})
\,,
\tag 6.6
$$
with the reduced weight $ \omega_\ob$ given by a
formula of the type (3.31).
More generally, we introduce, for a volume $V$ which contains
the $d+1$ dimensional support $\Cal S(\ob)$ of $\ob$,
the diluted partition function
$$
Z_{q}^\ob(V)=\sum_{\scriptstyle \{Y_\ob,Y_1,\dots,Y_k\}
\atop\scriptstyle Y\supset \supp \ob} \rho_\ob(Y_\ob)
\prod_i \rho(Y_i)
\prod_m e^{\tilde\beta e_m W_m},
\tag 6.7
$$
where the sum
runs over all sets of mutually compatible,
matching contours in $V$ for which all external
contours are $q$contours. For $V=\Bbb T_\Lambda$,
$Z_{q}^\ob(V)=Z_{q,\Lambda}^\ob$, which implies that
$$
\langle \ob\rangle_{q,\Lambda}=\frac{Z_q^\ob(\Bbb T_\Lambda)}{Z_q(\Bbb
T_\Lambda)}.
\tag 6.8
$$
For every term contributing to (6.7) we consider the collection
$\Cal Y_\ob$ consisting of the contour $Y_\ob$
as well as all contours $Y$ among $\{ Y_1,\cdots,Y_k\}$ encircling it
($\Int Y \supset \Cal S(\ob)$)
and define
$$
\rho(\Cal Y_\ob) = \rho_\ob(Y_\ob) \prod_{\scriptstyle Y\in \Cal Y_\ob
\atop\scriptstyle Y\neq Y_\ob}\rho (Y).
\tag 6.9
$$
Denoting further $\supp \Cal Y_\ob = \bigcup_{Y\in\Cal Y_\ob} \supp Y$,
$\Int \Cal Y_\ob$ the union of all finite components of
$ \Bbb T\setminus\supp\Cal Y_\ob$,
$\Int_m \Cal Y_\ob$ the union of all components of
$\Int \Cal Y_\ob$ that are labeled by $m$,
and $\Ext \Cal Y_\ob=\bigcap_{Y\in\Cal Y_\ob} \Ext Y$,
we
introduce, in addition to the
weights $K_q(Y)$ defined in the preceeding
section for an arbitrary
$q$contour $Y$, also the weight
$$
K_{q,\ob}(\Cal Y_\ob):=\rho_\ob(\Cal Y_\ob)e^{\tilde\beta e_q \supp \Cal
Y_\ob}
\prod_{m=1}^r \frac{Z_m(\Int_m \Cal Y_\ob)}
{Z_q(\Int_m \Cal Y_\ob)}
\tag 6.10
$$
attributed to the collection $\Cal Y_\ob$.
As a result we get the representation
$$
Z_q^\ob(V)=e^{\tilde\beta e_q V}
\sum_{\Cal Y_\ob, Y_1,\dots, Y_n}
K_{q,\ob}(\Cal Y_\ob)\prod^n_{\ell =1}K_q(Y_\ell).
\tag 6.11
$$
Here the sum goes over set of all collections $\Cal Y_\ob$ and all sets
$ \{Y_1,\dots, Y_n\}
$
of nonoverlapping $q$ contours in $V$,
such that for all contours $Y_i$,
$i=1,\dots, n$, the set $V(Y_i)$ does not intersect the set
$\supp \Cal Y_\ob$.
Assuming for the moment that the weights
$K_{q,\ob}(\Cal Y_\ob)$ and $K_q(Y)$
decay sufficiently fast with the size of
$\Cal Y_\ob$ and $Y$,
respectively, we now
use the standard
Mayer cluster expansion for polymer systems to get
$$
{Z_q^\ob(V)\over Z_q(V)}=
\sum_{\Cal Y_\ob} K_{q,\ob}(\Cal Y_\ob)
\sum_{n=0}^\infty
{1\over n!}\sum_{\{Y_1,\dots, Y_n\}}
\Bigl[ \prod^n_{k=1}K_q(Y_k) \Bigr] \phi_c(\Cal Y_\ob,Y_1,\cdots,Y_n)
\,.
\tag 6.12
$$
Here $\phi_c(\Cal Y_{\ob},Y_1,\cdots,Y_n)$ is a combinatoric factor
defined in terms of the connectivity properties of the
graph $G(\Cal Y_0,Y_1,\cdots,Y_n)$,
introduced
as the graph on the vertex set $\{0,1,\cdots,n\}$ which has an edge
between two vertices $i\geq 1$ and $j\geq 1$, $i\neq j$, whenever
$\supp Y_i\cap\supp Y_j\neq\emptyset$, and
an edge between the vertex $0$ and a
vertex $i\neq 0$ whenever
$V(Y_i)\cap\supp \Cal Y_\ob\neq \emptyset$.
(see for example \cite{Sei82} or \cite{Dob94} for a new simple and very
lucid proof).
The combinatoric factor $\phi_c(\Cal Y_{\ob},Y_1,\cdots,Y_n)$ is
zero if
$G(Y_\ob,Y_1,\cdots,Y_n)$ has more than one component.
To prove the convergence of (6.12) one has to show that the weights
$K_{q,\ob}$ and $K_q$
decay sufficiently fast with the size of
$\Cal Y_\ob$ and $Y$,
respectively.
To this end it is useful to
introduce truncated models.
For a contour $Y$ with $V(Y)\cap\Cal S(\ob)=\emptyset$,
we define $K_q^\prime(Y)$ as before, see
(5.9), while for the collection $\Cal Y_\ob$ we define
$$
K_{q,\ob}^\prime(\Cal Y_\ob)
=\rho_\ob(\Cal Y_\ob)
e^{\tilde\beta e_q \supp \Cal Y_\ob}
\prod_{m=1}^r
{Z_m(\Int_m \Cal Y_\ob)
\over
Z_q^\prime(\Int_m \Cal Y_\ob)}
\prod_{Y\in \Cal Y_\ob}\chi^\prime_q(Y)\,,
\tag 6.13
$$
with $\chi_q^\prime(Y)$ as in (5.10).
Given this definition,
we introduce
$$
{{Z_q^\prime}^\Psi}(V) =
e^{\tilde\beta e_qV}
\sum_{\Cal Y_\ob}
K_{q,\ob}^\prime(\Cal Y_\ob)
\sum_{\{Y_1,\dots, Y_n\}}
\prod^n_{k=1}K_q^\prime(Y_k)
\,
\tag 6.14
$$
and
$$
\langle \ob \rangle_{q,V}^\prime=
{{{Z_q^\prime}^\ob}(V)\over Z_q^\prime(V)}
\,,
\tag 6.15
$$
which can again be expanded as
$$
\langle \ob \rangle_{q,V}^\prime
=
\sum_{\Cal Y_\ob}
K_{q,\ob}^\prime(\Cal Y_\ob)
\sum_{n=0}^\infty
{1\over n!}
\sum_{\{Y_1,\dots, Y_n\}}
\left[
\prod^n_{k=1}K_q^\prime(Y_k)
\right]
\phi_c(\Cal Y_\ob,Y_1,\cdots,Y_n)
\,.
\tag 6.16
$$
The following Lemma
gives
the absolute convergence
of the expansion (6.16), which
in turn will
yield Theorem 2.1 ii), iii) and iv).
\medskip
\proclaim{Lemma 6.1} Let $\epsilon$,
$\epsilon_0$ and $\overline\alpha$
be as in Lemma 5.1, and assume
that $\epsilon<\epsilon_0$ and
$\overline\alpha\geq 1$.
Then:
\medskip\noindent
\item{i)} For every collection $\Cal Y_\ob$ one has
$$
\left
K_{q,\ob}^\prime(\Cal Y_\ob)
\right
\leq
\Vert \ob \Vert
e^{(\alpha+2)\supp \ob}
\epsilon^{\supp\Cal Y_\ob\setminus
\Cal S( \ob)}
\,.
\tag 6.17
$$
\item{ii)} {If}
$$
a_q \max_{Y\in \Cal Y_\ob}
\delta(Y)
\leq\overline\alpha\,,
\tag 6.18
$$
then $K_{q,\ob}^\prime(\Cal Y_\ob)=K_{q,\ob}(\Cal Y_\ob)$.
\item{iii)} The cluster expansion (6.16) is absolutely convergent,
and
$$
\align
\Bigl\langle \ob \rangle_{q,V}^\prime\Bigr
&\leq
\sum_{\Cal Y_\ob}
\BiglK_{q,\ob}^\prime(\Cal Y_\ob)\Bigr
\sum_{n=0}^\infty
{1\over n!}
\sum_{\{Y_1,\dots, Y_n\}}
\left[
\prod^n_{k=1}
\BiglK_q^\prime(Y_k)\Bigr
\right]
\Bigl\phi_c(\Cal Y_\ob,Y_1,\cdots,Y_n)\Bigr
\\
&\leq
\Vert \ob \Vert
e^{(\alpha+2+O(\epsilon))\supp \ob}
\,.
\tag {\rm 6.19}
\endalign
$$
\item{iv)} If
$
a_q\delta(\Bbb T_\Lambda)\leq \bar\alpha
$,
then
$\langle\ob\rangle_{q,\Lambda}=\langle\ob\rangle_{q,\Lambda}^\prime$.
\item{v)}
For an arbitrary volume $V\subset\Bbb T$,
$$
{Z_q^{\ob}(V)
\leq
\Vert \ob \Vert e^{(\gamma +1) \supp\ob}
e^{ \tilde\beta f_0V+O(\epsilon)\partial V}}
\,.
\tag 6.20
$$
\item{vi)}
There is a constant $K=K(d)>0$ such that
for $V=\Bbb T_\Lambda$,
$$
{Z_q^{\ob}(V)
\leq
\Vert \ob \Vert e^{(\gamma +1) \supp\ob}
e^{ \tilde\beta f_0V+O(\epsilon)\partial V}}
\max
\bigl( e^{\frac{a_q}{4}V}, e^{K\gamma \partial V}\bigr)
\,.
\tag 6.21
$$
\endproclaim
\demo{Proof}
Observing that
$
\displaystyle
\delta(W)\leq \max_{Y\in \Cal Y_\ob} \delta(Y)
$ for all components $W$ of $\Int \Cal Y_\ob$,
the statement ii) of Lemma 6.1 immediately follows from
Lemma 5.1.
In order to prove i), we first note
that
$\rho_\ob(Y_\ob)\leq \Vert \ob\Vert \tilde\rho(Y_\ob)$, where
$\tilde\rho(Y_\ob)$
satisfies an analog of the bound (4.4) (however, $Y_\ob$ is not
necessarily excited on $\Cal S(\ob)$), namely
$$
\tilde\rho(Y_\ob)
\leq
e^{\tilde\beta e_0 \supp Y_\ob}
e^{\gamma\supp Y_\ob\setminus\Cal S(\ob)}
(1+e^{\gamma})^{\Cal S(\ob)}.
\tag 6.22
$$
Hence
$$
\rho_\ob( \Cal Y_\ob) e^{\tilde\beta e_q\supp \Cal Y_\ob}
\leq \Vert \ob \Vert e^{\tilde\beta (e_0e_q) \supp \Cal Y_\ob}
e^{\gamma\supp \Cal Y_\ob\setminus\Cal S(\ob)}
(1+e^{\gamma})^{\Cal S(\ob)}.
\tag 6.23
$$
Using
Lemma 5.1 we get
$$
\prod_{m=1}^r
\left
{Z_m(\Int \Cal Y_\ob)
\over
Z_q^\prime(\Int \Cal Y_\ob)}
\right
\leq
e^{a_q\Int \Cal Y_\ob
+O(\epsilon)\partial \Int \Cal Y_\ob}
\,.
\tag 6.24
$$
Combined with the bound $\tilde\beta(e_qe_0)\leq a_q+O(\epsilon)$ we get
$$
\left
K_{q,\ob}^\prime(\Cal Y_\ob)
\right
\leq
\Vert \ob \Vert
e^{a_q\supp\Cal Y_\ob\cup \Int \Cal Y_\ob
+ O(\epsilon) \partial \Int \Cal Y_\ob}
e^{\gamma\supp \Cal Y_\ob\setminus\Cal S(\ob)}
e^{O(\epsilon)\supp \Cal Y_\ob}.
\tag 6.25
$$
Observing that
$\prod_{Y\in \Cal Y_\ob}\chi_q(Y)\neq 0$
implies that
$a_qV(Y)\leq (\alpha+1+O(\epsilon))Y$
for each $Y\in\Cal Y_\ob$, and noting
that $\Cal S(\ob)=\supp \ob$,
we finally get
$$
\left
K_{q,\ob}^\prime(\Cal Y_\ob)
\right
\leq
\Vert \ob \Vert
e^{(\gamma\alpha1O(\epsilon))
\supp \Cal Y_\ob\setminus\Cal S(\ob)}
e^{(\alpha+1+O(\epsilon))\supp
\ob}.
\tag 6.26
$$
Statement iii) follows from i) and Lemma 5.1, i),
while statement iv) follows from ii) and Lemma 5.1,
ii) and iii).
To get the statement v) and vi), we first write $Z_q^{\ob}(V)$
as
$$
Z_q^{\ob}(V) =
\sum_{\Cal Y_\ob}
\rho_\ob( \Cal Y_\ob) Z_q(\Ext \Cal Y_\ob)
\prod_m Z_m(\Int \Cal Y_\ob).
\tag 6.27
$$
Using now
(6.23) and the bound (5.18) in its strengthened
form (A.43), we get
$$
\multline
\bigl Z_q^{\ob}(V) \bigr
\leq
\Vert \ob \Vert e^{\gamma\Cal S(\ob)}
e^{\tilde\beta f_0 V}
e^{O(\epsilon)\partial V}
\sum_{\Cal Y_\ob}
e^{\gamma \supp \Cal Y_\ob}
e^{O(\epsilon)\supp \Cal Y_\ob}
\times \\ \times
\max_{U\subset \Ext \Cal Y_\ob}
\bigl(
e^{\frac{a_q}{4}\Ext \Cal Y_\ob \setminus U}
e^{K(d) \gamma \partial U}
\bigr)
.
\endmultline
\tag 6.28
$$
Extracting a factor
$$
\multline
\max_{\Cal Y_\ob}
e^{\frac{\gamma}2 \supp \Cal Y_\ob}
\max_{U\subset \Ext \Cal Y_\ob}
\bigl(
e^{\frac{a_q}{4}\Ext \Cal Y_\ob \setminus U}
e^{K(d) \gamma \partial U}
\bigr)
\leq\\ \leq
\max_{\Cal Y_\ob}
e^{K(d){\gamma} \partial V(\Cal Y_\ob)}
\max_{U\subset V\setminus V(\Cal Y_\ob)}
\bigl(
e^{\frac{a_q}{4} (V\setminus V(\Cal Y_\ob))\setminus U}
e^{K(d) \gamma \partial U}
\bigr)
\leq\\ \leq
\max_{\tilde U\subset V}
\bigl(
e^{\frac{a_q}{4}V\setminus \tilde U}
e^{K(d) \gamma \partial \tilde U}
\bigr)\,,
\endmultline
\tag 6.29
$$
where $V(\Cal Y_\ob)=V\setminus\Ext \Cal Y_\ob$,
and bounding
$$
\sum_{\Cal Y_\ob}
e^{\frac{\gamma}2 \supp \Cal Y_\ob  }
\leq
(1+ O(e^{\frac {\gamma}2}))^{\supp\ob},
\tag 6.30
$$
we get
$$
\bigl
Z_q^\ob(V)
\bigr
\leq
\\ob\e^{(\gamma+1)\supp\ob}
e^{\tilde\beta f_0 V}
e^{O(\epsilon)\partial V}
\max_{\tilde U\subset V}
\bigl(
e^{\frac{a_q}{4}V\setminus \tilde U}
e^{K(d) \gamma \partial \tilde U}
\bigr)\,.
\tag 6.31
$$
Bounding the last factor by one, we get statement v);
continuing as in the proof of Lemma A.3 in the appendix,
we get statement vi).
\hfill\hfill\qed
\enddemo
\head{7. Proof of Theorems 2.1 and 2.2}
\endhead
\demo{Proof of Theorem 2.1}
In order to prove the theorem,
we will use Lemmas 5.1 and Lemma 6.1, with $\alpha=\gamma/2$.
We therefore need that $\gamma>42\log\epsilon_0$,
which in turn requires $\tilde\beta$ and $\gamma_Q$ are
sufficiently large, see (4.5).
Choosing appropriate $\beta_0$ and
$\gamma_Q$,
we take
$\tilde\beta\in(\frac12\beta_0,\beta_0]$
and choose $\lambda_0$
according to (2.13). Then
(4.3)
and (4.21) are
satisfied once $\lambda \leq \lambda_0$.
For every $\beta\geq \beta_0$ we
now choose $M$ such that
$\tilde\beta=\frac{\beta}{M}\in(\frac12\beta_0,\beta_0]$.
Whenever $a_q=0$ we can then use the
claims ii) from Lemma 5.1 and ii) from
Lemma 6.1 to conclude that $K_q(Y)=K_q^\prime(Y)$ for all $Y$ and
$K_{q,\ob}^\prime(\Cal Y_\ob)=K_{q,\ob}(\Cal Y_\ob)$ for all $\Cal Y_\ob$.
As a result the bounds (5.5), (5.17), and (6.17) are valid for $K_q(Y)$
and $K_{q,\ob}(\Cal Y_\ob)$.
This allows
us
to use cluster expansions for evaluation of $Z_q(V)$ as well as
$Z_q^\ob(V)$.
Hence (2.25) follows directly from
(5.2), (5.13) and Lemma 5.1 iii).
Similarly, we get (2.26)
(with an explicit formula for $\langle \ob\rangle_q$) from (6.12).
Finally, given the absolute convergence of the cluster expansion for
the expectation values of local observables and the exponential
decay of the contour activities, the bound (2.27) is standard.
To evaluate the
expectation value of the projection operator
(2.28),
we apply the expansion (6.10)
for the particular observable
$\ob=P^{(m)}_{U(x)}\equiv \ket{g_{U(x)}^{(m)}}\bra{g_{U(x)}^{(m)}}$.
Using the factor
$\epsilon^{\supp\Cal Y_\ob\setminus\Cal S(\ob)}$
in (6.17), one can show that the sum of all
terms with
$\supp\Cal Y_\ob\neq\Cal S(\ob)
$ is of the order
$\text{const}\,\epsilon$, a term that can be made small by taking
$\epsilon$ small.
We are thus left with
contributions coming from the term
$K_{q,\ob}(\Cal Y_\ob)$ with
$$
\supp\Cal Y_\ob=
\Cal S(\ob)
\equiv\bigcup_{x\in U(X)} C(x,1)
\,.
\tag 7.1
$$
This means that necessarily $\Cal Y_\ob=\{Y_\ob\}$ with
$\supp Y_\ob=
\Cal S(\ob)$
and $\alpha(\partial\Cal S(\ob))=q$.
The only configurations $\sigma$ yielding this $Y_\ob$
are those for which $\sigma=\{\sigma_{y,t}\}$
agrees with $g^{(q)}$, except possibly for the point $(y,t)=(x,1)$.
For the activity
$$
K_{q,\ob}(\Cal Y_\ob)
=\rho_\ob(\{Y_\ob\})
e^{\tilde\beta e_q U(x)}
\,,
\tag7.2
$$
this gives a contribution $O(\epsilon)$ if
$\sigma_{x,1}\neq g_x^{(q)}$, and a contribution
$$
\multline
e^{\tilde\beta e_q \Lambda M }
\Tr_{\Cal H_\Lambda}
\Bigl(
\ob
\ket{g_{\Lambda}^{(q)}}\bra{g_{\Lambda}^{(q)}}
\prod_{t=1}^M
e^{\tilde\beta \sum_{y\in\Lambda}\Phi _y (g^{(q)}_{U(y)})}
\Bigr)
=\\=
\Tr_{\Cal H_\Lambda}
\bigl(
\ob \ket{g_{\Lambda}^{(q)}}\bra{g_{\Lambda}^{(q)}}
\bigr)
=\delta_{m,q}.
\endmultline
\tag7.3
$$
if $\sigma_{x,1}=g_x^{(q)}$.
Putting everything together, we obtain the bound
$$
\langle P^{(m)}_{U(x)}\rangle_q=\delta_{m,q} + O(\epsilon)
\,,
\tag7.4
$$
which proves iv).
Finally, v) is a standard claim in the PirogovSinai theory.
Namely, given the bounds (5.5), (5.17), and the fact that
$f_q$ can be analysed by a convergent cluster
expansion, we get
$f_q  e_q \leq O(\epsilon)$ and a similar bound for the
derivatives of $f_q$. Statement v) then follows from
the corresponding assumptions (2.4) and (2.5) on the functions
$e_q(\mu)$.
\qed
\enddemo
\demo{Proof of Theorem 2.2}
The partition functions
$$
Z_{\per, \Lambda(L)}= \Tr_{\Cal H_\Lambda}e^{\beta H_{\per,\Lambda(L)}}
\tag 7.5
$$
and
$$
Z_{\per, \Lambda(L)}^{\ob}= \Tr_{\Cal H_\Lambda}\ob e^{\beta
H_{\per,\Lambda(L)}}
\tag 7.6
$$
have representations $Z(\Bbb T_{\Lambda_{\per}(L)})$
and $Z^{\ob}(\Bbb T_{\Lambda_{\per}(L)})$ similar to (5.1) and (6.5).
The proof of Theorem 2.2 follows from this representation in a standard manner
\cite{BI 89, BK90}.
One only has to notice that contours contributing to these partition
functions may
be wrapped around the torus
$\Bbb T_{\Lambda_{\per}(L)}$ in time as well as space directions.
Nevertheless, whenever a contour $Y$ satisfies the condition
$\delta(Y)\leq\frac{L}3$, one can define $\Ext Y$ as the largest
component of
$\Bbb T_{\Lambda_{\per}} \setminus \supp Y$.
For every configuration containing
only such
contours, all external contours have clearly the same external label.
Splitting now $Z(\Bbb T_{\Lambda_{\per}(L)})$
(resp\. $Z^{\ob}(\Bbb T_{\Lambda_{\per}(L)})$)
into contributions containing at
least one contour such that
$\delta(Y)>\frac{L}3$ and those where all contributing contours are such that
$\delta(Y)\leq\frac{L}3$, we get
$$
Z(\Bbb T_{\Lambda_{\per}})= Z^{\Bi}(\Bbb T_{\Lambda_{\per}})+
\sum_{m=1}^r Z_{m}(\Bbb T_{\Lambda_{\per}}).
\tag 7.7
$$
Here $Z_{m}(\Bbb T_{\Lambda_{\per}})$ is given as a sum over all
configurations containing only those contours for
which $\delta(Y)\leq\frac{L}3$ and such that the common external label of
external contours is $m$.
Taking into account the
fact that term
$Z^{\Bi}(\Bbb T_{\Lambda_{\per}})$
is exponentially suppressed in $L$ (one can use verbatim
the proof from
\cite{BI89}),
we get
$$
\Bigl Z(\Bbb T_{\Lambda_{\per}})\sum_{m=1}^r Z_m(\Bbb
T_{\Lambda_{\per}})\Bigr
\leq e^{\tilde\beta f_0 M (2L+1)^d}
e^{\text{const}\, \gamma L},
\tag 7.8
$$
and similarly
$$
\Bigl Z^{\ob}(\Bbb T_{\Lambda_{\per}})\sum_{m=1}^r Z^{\ob}_m(\Bbb
T_{\Lambda_{\per}})\Bigr
\leq
\Vert \ob \Vert
e^{(\gamma+1)\supp\ob}
e^{\tilde\beta f_0 M (2L+1)^d}
e^{\text{const}\, \gamma L}.
\tag 7.9
$$
Moreover, whenever $m$ is stable, $m\in Q(\mu)$, we have
$Z_m(\Bbb T_{\Lambda_{\per}})=Z^{\prime}_m(\Bbb T_{\Lambda_{\per}})$
and we can use the cluster expansion of $\log Z^{\prime}_m(\Bbb
T_{\Lambda_{\per}})$ and $f_m$ to show that
$$
\bigl Z_m(\Bbb T_{\Lambda_{\per}}) e^{\tilde\beta f_m M (2L+1)^d} \bigr
\leq
e^{\tilde\beta f_0 M (2L+1)^d} M (2L+1)^d
e^{\text{const}\, \gamma L}.
\tag 7.10
$$
Namely, we just observe that the first terms in which these two cluster
expansions differ are of the order $e^{\text{const}\, \gamma L}$
(clusters wrapped around $\Bbb T_{\Lambda_{\per}}$ in spatial directions).
For $m\notin Q(\mu)$, on the other hand,
we proceed as in the proof of Lemma A.3 to get
$$
\multline
\Bigl Z_m(\Bbb T_{\Lambda_{\per}})\Bigr
\leq
\exp( M (2L+1)^d e^{\text{const}\, \gamma L})
\,e^{\tilde\beta f_0 M (2L+1)^d}
\times\\ \times
\max
\Bigl\{
e^{\frac{a_q}{4}M (2L+1)^d}
\,,\,
e^{K(d) \gamma M(2L+1)^{d1}}
\Bigr\}.
\endmultline
\tag 7.11
$$
For $m\notin Q(\mu)$ and $L$ sufficiently large,
we therefore get
$$
\Bigl Z_m(\Bbb T_{\Lambda_{\per}})\Bigr
\leq
2e^{\tilde\beta f_0 M (2L+1)^d}
e^{K(d) \gamma M(2L+1)^{d1}}
\,.
\tag 7.12
$$
In a similar way, proceeding now as in the proof of
Lemma 6.1, we get
$$
\Bigl Z_m^{\ob}(\Bbb T_{\Lambda_{\per}})\Bigr
\leq
2\\ob\ e^{(\gamma+1)\supp \ob}
e^{\tilde\beta f_0 M (2L+1)^d}
e^{K(d) \gamma M(2L+1)^{d1}}
\,,
\tag 7.13
$$
provided $m\notin Q(\mu)$ and $L$ is sufficiently large.
Combining the bounds (7.8), (7.10) and (7.12),
and using the fact that $f_m=\Re f_m$
if the couling constant $\lambda$ is real,
which in turn implies $f_m=f_0$ if $m\in Q(\mu)$,
we get
$$
\Bigl
Z(\Bbb T_{\Lambda_{\per}})

Q(\mu) e^{\tilde\beta f_0 M (2L+1)^d}
\Bigr
\leq
e^{\tilde\beta f_0 M (2L+1)^d}
O(e^{\text{const}\gamma L})
\tag 7.14
$$
provided $L$ is sufficiently large.
Introducing now the truncated expectation values
$$
\langle \ob\rangle_{q, \Bbb T_{\Lambda_{\per}}}^{\prime}=\frac{
{{Z_q^\prime}^\Psi}(\Bbb T_{\Lambda_{\per}})}{{{Z_q^\prime}}(\Bbb
T_{\Lambda_{\per}})}
\tag 7.15
$$
(cf\. (5.6) and (6.14)),
we get
$$
\multline
\Bigl
Z^{\ob}(\Bbb T_{\Lambda_{\per}})

\sum_{m\in Q(\mu)}
e^{\tilde\beta \Re f_m M (2L+1)^d}
\langle \ob\rangle_{m, \Bbb T_{\Lambda_{\per}}}^{\prime}
\Bigr
\leq \\ \leq
\sum_{m\in Q(\mu)}
\Bigl
\langle \ob\rangle_{m, \Bbb T_{\Lambda_{\per}}}^{\prime}
\bigl(
Z^{\prime}_m(\Bbb T_{\Lambda_{\per}})

e^{\tilde\beta \Re f_m M (2L+1)^d}
\bigr)
\Bigr
+\\+
\sum_{m\not\in Q(\mu)}
\Bigl
Z^{\ob}_m(\Bbb T_{\Lambda_{\per}})
\Bigr
+
\Vert \ob \Vert
e^{(\gamma+1)\supp \ob }
e^{\tilde\beta f_0 M (2L+1)^d}
e^{\text{const}\, \gamma L}.
\endmultline
\tag 7.16
$$
Next, we observe that
$\langle \ob\rangle_{m, \Bbb T_{\Lambda_{\per}}}^{\prime}$
can be bounded in the same way as
$\langle \ob\rangle_{m, \Bbb T_{\Lambda}}^{\prime}$,
namely
$$
\bigl
\langle \ob\rangle_{m, \Bbb T_{\Lambda_{\per}}}^{\prime}
\bigr
\leq
\\ob\ e^{(\alpha+2+O(\epsilon))\supp \ob}
\leq
\\ob\ e^{(\gamma+1)\supp \ob}
\,.
\tag 7.17
$$
Inserting the bounds (7.10), (7.17) and (7.13)
into (7.16), and dividing both sides of the resulting
bound by
$ Q(\mu)e^{\tilde\beta f_0 M (2L+1)^d}$, we get
$$
\Bigl
\langle \ob \rangle_{\per,\Lambda(L)}\sum_{m\in Q(\mu)}
\langle \ob\rangle_{m, \Bbb T_{\Lambda_{\per}}}
\Bigr
\leq
e^{\text{const}\, \gamma L}
\\ob\ e^{(\gamma+1)\supp \ob}
\,,
\tag 7.18
$$
provided $L$ is sufficiently large.
In the limit $L\to\infty$, this yields the claim of Theorem 2.2.
\qed
\enddemo
\vfill
\newpage
\head{Appendix. PirogovSinai theory in thin slabs  proof of Lemma 5.1}
\endhead
In this appendix, we prove Lemma 5.1.
Actually, it is a direct consequence of the Lemma A.1 below. In order to
state the lemma, we recall the definition of $f_q^{(n)}$ as the free
energy of an
auxilliary contour model with activities
$$
K^{(n)} (Y) =
\cases K^\prime (Y) \ \ & \text{\rm if} \ \delta(Y) \leq n, \\
0 & \text{\rm otherwise},\endcases
\tag"{(A.1)}"
$$
and define
$$
\align
f_0^{(n)} & = \min_q \Re f_q^{(n)}, \tag"{(A.2)}" \\
a_q^{(n)} & = \tilde\beta(\Re f_q^{(n)}  f_0^{(n)}).
\tag"{(A.3)}"
\endalign
$$
Observing that $f_0 = \lim_{n\to\infty} f_0^{(n)}$ and
$a_q = \lim_{n\to\infty}a_q^{(n)}$, Lemma 5.1 follows directly from the
following.
\proclaim{Lemma A.1}
Assume that $\rho(\boldsymbol\cdot)$ obeys the conditions
{\rm (4.4)} and {\rm (4.22)}
and
let
$$
\epsilon = e^{\gamma+\alpha+2}
\qquad\text{and}\qquad
\bar \alpha = \alpha  2.
\tag"{(A.4)}"
$$
Then there is a constant $\epsilon_0$,
depending only on $d$ and $r$,
such that the following statements
are true once ($\gamma$ is such that) $\epsilon < \epsilon_0$ and $\alpha
\geq 3$. For all $n \geq 0$ and $Y$ and $V$ such that $\delta(Y) \leq n$,
$\delta(V)
\leq n$, one has:
\medskip
\item{i)}\ \ $K_q^\prime (Y) \leq \epsilon^{Y}$.
\item{ii)} \ \ $\Bigl\displaystyle\frac
{\partial}{\partial\mu_i}K^\prime_q(Y)\Bigr
\leq (3r\tilde\beta C_0 + 2)V(Y)\epsilon^{Y}$.
\item{iii)}\ \ If $a_q^{(n)}\delta(Y)\leq\bar\alpha$
\quad then \quad $\chi_q^\prime (Y) = 1$.
\item{iv)}\ \ If $a_q^{(n)}\delta(Y)\leq\bar\alpha$
\quad then \quad $K_q^\prime (Y) = K_q(Y)$.
\item{v)} \ \ If $a_q^{(n)}\delta(V)\leq\bar\alpha$
\quad then \quad $Z^\prime_q(V) = Z_q(V)$.
\item{vi)}\ \ $Z_q(V)\leq e^{\tilde\beta f_0^{(n)}V}
e^{O(\epsilon)\partial V}$.
\item{vii)}
\ \ $\Bigl\displaystyle\frac {\partial}{\partial\mu_i}Z_q(V)\Bigr
\leq (2\tilde\beta C_0 + 1)V
e^{ \tilde\beta f_0^{(n)}V}e^{O(\epsilon)\partial V}$.
\endproclaim
\bigskip
\demo{Proof} We proceed by induction on $n$.
\enddemo
\noindent{\it I. The case $n=0$.}
\medskip
There are no contours with $\delta(Y) = 0$.
This makes i)  iv) trivial statements and implies that
$f_q^{(0)}=e_q$. On the other hand,
$\delta(V)=0$ implies $V=0$ and
$Z_q(V) = Z_q^\prime (V)=1$, which makes
v)  vii) trivial statements.
\medskip
\noindent{\it II. Induction step $n1\to n$.}
\medskip
\demo{Proof of i) for $\delta(Y) = n$}
Clearly,
$\delta(\Int Y) < n$, and all contours $\tilde Y$ contributing to
$Z^\prime_q(\Int_m Y)$ obey the condition $\delta(\tilde Y) < n$.
This implies that
$K_q^\prime (\tilde Y)
\leq \epsilon^{\tilde Y}$ by the
inductive assumption i).
As a consequence, the logarithm of
$Z^\prime_q(\text{\rm Int}_m Y)$
can be analyzed by a convergent
expansion, and
$$
\left
\log Z_q^\prime (\text{\rm Int}_m Y)
+ \tilde\beta f_q^{(n1)}\text{\rm Int}_m Y
\right
\leq
O(\epsilon) \partial \text{\rm Int}_m Y.
\tag"{(A.5)}"
$$
Combining (A.5) with the induction assumption vi), we get
$$
\align
\prod_m
\left
\frac{Z_m(\text{\rm Int}_m Y)}{Z^\prime_q(\text{\rm Int}_m Y)}
\right
&\leq
e^{a_q^{(n1)} \Int Y}
e^{O(\epsilon)\sum_m \partial \text{\rm Int}_m Y} \\
& \leq e^{a_q^{(n1)} \Int Y} e^{
O(\epsilon)Y}.
\tag"{(A.6)}" \\
\endalign
$$
Observing that
$$
\tilde\beta e_m  \tilde\beta f_m^{(n1)} \leq O(\epsilon),
\tag"{(A.7)}"
$$
which implies
the bound
$$
\tilde\beta (e_q  e_0)  a_q^{(n1)} \leq O(\epsilon),
\tag"{(A.8)}"
$$
we
use the bound (4.4) to evaluate
$$
\rho(Y) e^{\tilde\beta e_qY}
\leq e^{\gammaY}
e^{\tilde\beta (e_q  e_0)Y}
\leq e^{(\gamma  O(\epsilon))Y}
e^{a_q^{(n1)} Y}.
\tag"{(A.9)}"
$$
Applying now the bounds (A.9) and (A.6) to the definition
(5.9)
and using
the equation
$V(Y) =  \text{\rm Int} \, Y + Y$, we
obtain
$$
K_q^\prime (Y) \leq \chi_q^\prime (Y) e^{a_q^{(n1)} V(Y)}
e^{(\gamma  O(\epsilon))Y}.
\tag A.10
$$
Without loss of generality,
we may assume that $\chi^\prime_q(Y) > 0$
(otherwise $K^\prime_q(Y) = 0$
and the statement i) is trivial).
Let us notice that
$$
V(Y)\leq \delta(Y) Y.
\tag A.11
$$
Indeed, considering a disjoint union $T$ of ``rows'' consisting of
elementary cubes $C(x,t)$ with fixed cordinates $x_2, \dots, x_d$ and
$t$, one notices that the set $V(Y)$ intersects at most $\delta(Y)$ of
elementary cubes in each such row and there is at most $Y$ such rows that
have a nonempty intersection with $V(Y)$.
By the definition of
$\chi^\prime_q(Y)$ and (A.11), we get
$$
\tilde\beta f_q^{(n1)}  f_m^{(n1)}\,\, V(Y)
\leq (1 + \alpha) Y
$$
for all $m \neq q$. As a consequence,
$$
a_q^{(n1)} V(Y) \leq (1 + \alpha )
Y,
\tag"{(A.12)}"
$$
provided $\chi_q^\prime (Y) \neq 0$.
Combined with (A.10) and the fact that
$\chi_q^\prime (Y) \leq 1$, this implies that
$$
K_q^\prime (Y) \leq
e^{[\gamma  1  \alpha  O(\epsilon)]Y},
\tag"{(A.13)}"
$$
which
yields the desired bound i) for
$\delta(Y) = n$.
\enddemo % of i)
\demo{Proof of ii) for $\delta(Y) = n$}
Using (4.22), (2.7), (4.4), and (A.8), we get
$$
\multline
\Bigl\frac {\partial}{\partial\mu_i}\bigl(\rho(Y) e^{\tilde\beta
e_qY}\bigr)\Bigr
\leq (2\tilde\beta C_0 +1)Y
e^{(\tilde\beta e_0 +\gamma)Y}
e^{\tilde\beta e_q Y}
\leq\\
\leq (2\tilde\beta C_0 +1)Ye^{(\gamma  O(\epsilon))Y}
e^{a_q^{(n1)} Y}.
\endmultline
\tag A.14
$$
Using inductive assumption i) and ii) for contours contributing to
$Z^\prime_q(\Int_m Y)$, we can apply the cluster expansion to get the bounds
$$
\Bigl\frac {\partial}{\partial\mu_i}\log Z^\prime_q(\Int_m Y)\Bigr
\leq
\bigl[
\tilde\beta C_0 +(1+\tilde\beta C_0)O(\epsilon)
\bigr]
\Int_m Y
\tag A.15
$$
and
$$
\tilde\beta
\Bigl\frac {\partial}{\partial\mu_i}f_m^{(n1)}\Bigr
\leq
\bigl[
\tilde\beta C_0 +(1+\tilde\beta C_0)O(\epsilon)
\bigr]
\,.
\tag A.16
$$
Using further the inductive assumption vi) and vii),
as well as the bounds (A.5) and (A.6), we get
$$
\align
\left\frac {\partial}{\partial\mu_i}\prod_m
\frac{Z_m(\text{\rm Int}_m Y)}{Z^\prime_q(\text{\rm Int}_m Y)}
\right
&\leq
\bigl[(3\tilde\beta C_0 + 1)+
(1+\tilde\beta C_0) O(\epsilon)\bigr]\Int Y
e^{a_q^{(n1)} \Int Y} e^{ O(\epsilon)Y}
\\
&\leq
\bigl[(2r\tilde\beta C_0 + 1)+
(1+\tilde\beta C_0) O(\epsilon)\bigr]\Int Y
e^{a_q^{(n1)} \Int Y} e^{ O(\epsilon)Y}.
\tag A.17
\endalign
$$
With the help of (A.16) and (5.11) we get
$$
\align
\left\frac {\partial}{\partial\mu_i}\chi^\prime_q(Y)
\right
&\leq
2(r1)[\tilde\beta C_0 +(1+\tilde\beta C_0)O(\epsilon)]
\delta (Y)
\\
&\leq
2(r1)[\tilde\beta C_0 +(1+\tilde\beta C_0)O(\epsilon)]
Y.
\tag A.18
\endalign
$$
Combining now (A.14), (A.17), and (A.18) with (A.6), (A.9)
and the observation that $V(Y)=Y + \Int Y $,
we get
$$
%\multline
\Bigl
\frac {\partial}{\partial\mu_i}K^\prime_q(Y)
\Bigr
\leq
\bigl\{
2r\tilde\beta C_0+1+(1+\tilde\beta C_0)O(\epsilon)
\bigr\}
V(Y)
e^{a_q^{(n1)} V(Y)}
e^{(\gamma  O(\epsilon))Y}
\,.
%\endmultline
\tag A.19
$$
Using now again the fact that $K_q^\prime (Y)$ (and its derivatives) vanishes
unless (A.12) is fulfiled, we get the desired bound, provided
$\epsilon$ is sufficiently small.
\enddemo %of ii)
\bigskip
\demo{Proof of iii) for $k=\delta(Y) \leq n$
and $a_q^{(n)} \delta(Y) \leq \bar \alpha$}
We just have proved that i)
is true for all contours $Y$ with
$\delta(Y) \leq n$. As a consequence,
both
$f_m^{(k1)}$
and
$f_m^{(n)}$
may be analyzed by a
convergent cluster expansion.
Using the definition
of $f_m^{(n)}$ and the obvious fact that
$Y\geq \delta(Y)$ (again, $d\geq 2$),
one concludes that all contours $Y$ contributing to the cluster
expansion of the difference
$f_m^{(k1)}  f_m^{(n)}$ obey the bound $Y \geq k.$
As a consequence,
$$
\tilde\beta f_m^{(k1)}  f_m^{(n)}
\leq (K \epsilon)^{k}
\tag A.20
$$
and
$$
\tilde\beta f_m^{(k1)}  f_m^{(n)}
\delta(Y)
\leq (K \epsilon)^{k}\delta(Y)
= k (K \epsilon)^{k}
\leq O(\epsilon).
\tag A.21
$$
where $K$ is a constant depending
only on the dimension $d$ and the number
of phases $r$.
Combining (A.21) with the assumption
$a_q^{(n)} \delta(Y) \leq \bar \alpha$,
we obtain
the lower bound
$$
\alpha  \tilde\beta [f_q^{(k1)}f_m^{(k1)}]\delta(Y)
\geq
\alpha  a_q^{(n)\delta(Y)}O(\epsilon)
\geq
\alpha  \bar\alpha  O(\epsilon)
=2O(\epsilon),
$$
with $\bar \alpha$ defined by (A.4).
Combining this with (5.12) we get
the equality $\chi_q^\prime (Y) = 1$.
\enddemo
\bigskip
\demo{Proof of iv) and v)}
The statement follows from the just proven fact that
$\chi_q^\prime (Y) = 1$ for all contours $Y$
with $\delta(Y) a_q^{(n)}\leq \bar\alpha$,
the definiton (5.9) of $K_q^\prime(Y)$ and
the relations (5.6) and (5.4).
We proceed by a second induction on the
diameters of $Y$ and $V$. For $\delta(Y)=0$
or $\delta(V)=0$ the statement is trivial.
Assume now that $K_q^\prime(Y)=K_q(Y)$ for all
$Y$ with $\delta(Y)\leq k \bar\alpha$. We
then rewrite the partition function
$Z_q(V)$ given in the form (5.1)
by splitting the set
of external contours into
small
and large contours and, for a fixed collection of large external contours
$\{ X_1, \cdots X_{k} \}_{\text{\rm ext}}$, we resum over (mutually
external) small
$q$contours in
$\Ext = V \backslash \overset k \to{\underset i
= 1 \to \cup} V(X_i)$. As a result we get
$$
Z_q(V) = \sum_{\{ X_1,\cdots ,
X_{k}\}_{\text{\rm ext}}}
\
Z_q^{\small} (\Ext)
\prod_{i=1}^{k }
\biggl[
\rho(X_i) \prod_m Z_m (\text{\rm Int}_m X_i)
\biggr]
\tag A.22
$$
with the sum going over sets of mutually
external large contours in $V$.
The partition function $Z_q^{\small}(\Ext)$
is obtained from
$Z_q(\Ext)$ by dropping all
large external $q$contours.
Due to the inductive assumption iv),
$K_q(Y) = K^\prime_q(Y)$ if $Y$ is
small.
Since
$K^\prime_q(Y) \leq \epsilon^{Y}$
by i), $Z_q^{\small}(\Ext)$
can be evaluated
by a convergent cluster expansion, and
$$
\left Z_q^{\small} (\Ext) \right \leq
e^{\tilde\beta \Re f_q^{\small}\Ext}
e^{O(\epsilon)\partial \Ext}.
\tag A.23
$$
Here $f_q^{\small}$
is the free energy of the contour model with
activities
$$
K_q^{\small} (Y) =
\cases K^\prime_q(Y) \ \ & \text{\rm if} \
\delta(Y)
\leq n \ \text{\rm and} \ Y \ \text{\rm is small}, \\
0 & \text{\rm otherwise}.
\endcases
\tag A.24
$$
On the other hand,
$$
\prod_m  Z_m(\text{\rm Int}_m X_i) 
\leq
e^{\tilde\beta f_0^{(n1)} \Int X_i}
e^{O(\epsilon)\partial \Int X_i}
$$
by the induction assumption vi).
Observing that
the smallest contours contributing to the difference of
$f_m^{(n)}$ and $f_m^{(n1)}$ obey
the bound $Y \geq n$,
while
$$
V(X_i)\leq \delta(X_i)\, X_i\leq nX_i
\,,
$$
we may continue as in the proof of
(A.21) to bound
$$
\tilde\beta f_0^{(n1)}  f_0^{(n)}\Int X_i
\leq
\tilde\beta f_0^{(n1)}  f_0^{(n)}V(X_i)
\leq n (K \epsilon)^{n}X_i \leq O(\epsilon)X_i\,.
$$
Thus
$$
\prod_m Z_m(\Int X_i)
\leq
e^{\tilde\beta f_0^{(n)}\Int X_i} e^{O(\epsilon) X_i}\,.
\tag A.25
$$
Combining (A.23) and (A.25) with the bounds
$$
\rho(X_i) \leq
e^{\gammaX_i\tilde\beta e_0X_i}
\leq e^{(\gammaO(\epsilon))X_i}
e^{\tilde\beta f_0^{(n)}X_i}
\tag A.26
$$
and
$$
\partial\Ext
\leq
\partial V + \sum_{i=1}^{k} \partial V(X_i) 
\leq
\partial V + 2 (d+1) \sum_{i=1}^{k} X_i ,
\tag A.27
$$
we conclude that
$$
\!\!\!\!Z_q (V)
\leq e^{O(\epsilon)\partial V}
e^{\tilde\beta f_0^{(n)} V}\!\!\!\!
\sum_{\{X_1, \cdots , X_{k} \}_{\text{\rm ext}}}
\!\!\!\! e^{\tilde\beta [\Re f_q^{\small}  \Re f^{(n)}_q] \Ext}
\prod_{i=1}^{k} e^{(\gammaO(\epsilon))X_i}.
\tag A.28
$$
Next, we bound the difference $f_q^{\small}  f_q^{(n)}$.
First,
for all large contours $X$,
we have
$$
X\geq \delta(X) \geq
\ell_0 := \frac{\bar \alpha}{a_q^{(n)}}.
\tag A.29
$$
Next, we observe that
$$
\tilde\beta f_q^{(n)}  f_q^{\small}
\leq (K\epsilon)^{\ell_0} \leq
\frac{1}{\ell_0 \log (K\epsilon)},
\tag A.30
$$
where $K$ is a constant
depending only on $d$ and $N$.
Recalling the
condition $\bar \alpha \geq 1$, we
get
$$
\tilde\beta f_q^{(n)}  f_q^{\small}
\leq \frac{a^{(n)}_q}{\log (K\epsilon)}
\leq \frac{1}{2} a_q^{(n)},
\tag A.31
$$
provided $\epsilon$ is chosen small enough.
Combining (A.28) with
(A.31), we finally obtain
$$
Z_q (V)
\leq
e^{O(\epsilon)\partial V}
e^{\tilde\beta f_0^{(n)}V}
\sum_{\{X_1, \cdots , X_{k } \}_{\text{\rm ext}}}
e^{\frac{a_q^{(n)}}{2} \Ext}
\prod_{i=1}^{k} e^{\tilde\gamma X_i}
\tag A.32
$$
with
$$
\tilde \gamma = \gamma  1.
\tag A.33
$$
At this point we need the following Lemma A.2,
which is a variant of a
lemma first proven in \cite{Zah84} (see also \cite{BI89} and
\cite{BK94} for the proof of this lemma exactly in the following
formulation).
\bigskip
\proclaim{Lemma A.2}
{Consider an arbitrary contour activity $\tilde
K_q(Y) \geq 0$, and let $\tilde Z_q$ be the partition function}
$$
\tilde Z_q(V) =
\sum_{\{Y_1, \cdots , Y_n \}} \prod_{i=1}^n (\tilde K_q
(Y_i) e^{Y_i}).
\tag A.34
$$
{\it Let $\tilde s_q$ be the corresponding free energy, and assume
that $\tilde
K_q(Y) \leq \tilde \epsilon^{Y}$, where
$\tilde \epsilon$ is small
(depending on $r$ and $d$). Then for any
$\tilde a \geq \tilde s_q$
the following bound is true}
$$
\sum_{\{Y_1, \cdots , Y_k \}_{\text{\rm ext}}}
e^{\tilde a\Ext}
\prod_i \tilde K_q (Y_i)
\leq
e^{O(\tilde \epsilon)\partial V}\,,
\tag A.35
$$
{with the sum running over all sets of mutually external $q$contours in $V$}.
\endproclaim
\bigskip
In order to apply the lemma, we define
$\tilde K_q(Y) = e^{\tilde
\tauY}$ if $Y$ is a large $q$contour,
and $\tilde K_q(Y) = 0$
otherwise. With this choice,
$$
0 \leq \tilde s_q \leq (K\epsilon)^{\ell_0} \leq \frac{1}{\ell_0
\log(K\epsilon)}.
\tag A.36
$$
As a consequence,
$$
\tilde s_q \leq \tilde a := \frac{a_q^{(n)}}{2}
\tag A.37
$$
provided $\epsilon$ is small enough. Applying Lemma A.2 to the
right hand side of (A.32), and observing that $\tilde
\epsilon := e^{\tilde\tau} \leq \epsilon$, we finally obtain
the desired inequality
$$
Z_q(V) \leq e^{O(\epsilon)\partial V}
e^{\tilde\beta f_0^{(n)} V}.
\tag A.38
$$
\enddemo
\bigskip
\subsubhead{Proof of vii) for $\delta(V) =n$}
\endsubsubhead
\smallskip
Beginning from the formula (A.22) above, we first notice that
$$
\Bigl\frac {\partial}{\partial\mu_i}\log Z^{\small}_q(\Ext Y)\Bigr
\leq (\tilde\beta C_0+(1+\tilde\beta C_0) O(\epsilon))\Ext Y
\tag A.39
$$
(cf\. (A.15)) since we can use ii) for small $Y$. Using further the bound
(A.23) we
can conclude that
$$
\left \frac {\partial}{\partial\mu_i}Z_q^{\small} (\Ext) \right \leq
(\tilde\beta C_0
+(1+\tilde\beta C_0) O(\epsilon))\Ext Y
e^{\tilde\beta \Re f_q^{\small}\Ext}
e^{O(\epsilon)\partial \Ext}.
\tag A.40
$$
Combining this with the bounds (4.22) as well as (A.23) for
$Z_q^{\small} (\Ext) $ and the inductive assumptions (vii) and (vi) for
$Z_m (\Int_m X_i)$, we get
$$
\multline
\left \frac {\partial}{\partial\mu_i}Z_q (V)\right
\leq e^{O(\epsilon)\partial V}
e^{\tilde\beta f_0^{(n)} V}\!\!\!\!\\
\sum_{\{X_1, \cdots , X_{k} \}_{\text{\rm ext}}}
\bigl(\tilde\beta C_0 +
(1+\tilde\beta C_0)O(\epsilon))\Ext Y +
(2\tilde\beta C_0 +1)\Int Y +(\tilde\beta C_0 +1)Y \bigr\\
\!\!\!\! e^{\tilde\beta [\Re f_q^{\small}  \Re f^{(n)}_q] \Ext}
\prod_{i=1}^{k} e^{(\gammaO(\epsilon))X_i}.
\endmultline
\tag A.41
$$
Proceding now as in the proof of vi) from (A.28), we finally get vii).
This concludes the inductive proof of Lemma A.1. \hfill \qed
\bigskip
For $V=\Bbb T_\Lambda$,
the bound from Lemma A.1 vi) can be actually strengthened:
\bigskip
\proclaim{Lemma A.3}
Under the assumptions of Lemma A.1 we have
$$
Z_q(\Bbb T_\Lambda)
\leq
e^{\tilde\beta f_0 \Bbb T_\Lambda}
e^{O(\epsilon)\partial \Bbb T_\Lambda}
\max
\bigl\{
e^{\frac{a_q}{4}\Bbb T_\Lambda}
\,,\,
e^{K(d) \gamma \partial \Bbb T_\Lambda}
\bigr\}
\tag A.42
$$
with a constant $K(d)$ that depends only on the dimension $d$.
\endproclaim
\bigskip
\demo{Proof}
Extracting from (A.32) the factor
$\displaystyle{\max_{\{X_1, \cdots , X_{k } \}_{\text{\rm ext}}}
e^{\frac{a_q^{(n)}}{4} \Ext}
\prod_{i=1}^{k} e^{\frac12 \tilde\gamma X_i}}$, we still get the same bound.
This factor can be, in the limit $n\to\infty$, clearly bounded by
$\displaystyle{\max_{U\subset V}
\bigl( e^{\frac{a_q}{4}V \setminus U}
e^{K(d) \gamma \partial U}\bigr)}$.
As a consequence,
$$
Z_q(V)
\leq
e^{\tilde\beta f_0 V} e^{O(\epsilon)\partial V}
\max_{U\subset V}
\bigl(
e^{\frac{a_q}{4}V \setminus U}
e^{K(d) \gamma \partial U}
\bigr)
\,,
\tag A.43
$$
which is still true for arbitrary $V\subset\Bbb T$.
We now decompose $U$ and $V$ into timeslices,
$U=\cup_t U_t$ and $V=\sum_t V_t$,
and observe that $V\setminus U=\sum_t\bigl(V_tU_t\bigr)$,
while $\partial U\geq \sum_t \partial\pi(U_t)$,
where $\pi(\cdot)$ denotes the projection
onto $\Bbb R^d$.
Using the isoperimetric inequality on $\Bbb R^d$,
we now bound
$$
\align
\max_{U\subset V}
\bigl(
e^{\frac{a_q}{4}V \setminus U}
e^{2d K(d) \gamma U^{\frac {d1}d}}
\bigr)
&\leq
\prod_{t=1}^M
\max_{U_t\subset V_t}
\bigl(
e^{\frac{a_q}{4}(V_tU_t)}
e^{2d K(d) \gamma U_t^{\frac {d1}d}}
\bigr)
\\
&\leq
\prod_{t=1}^M
\max
\bigl\{
e^{\frac{a_q}{4}V_t},
e^{2d K(d) \gamma V_t^{\frac {d1}d}}
\bigr\}
\,.
\tag A.44
\endalign
$$
Restricting ourselves to $V=\Bbb T_\Lambda$,
we observe that $V_t=V/M$
is independent of $t$ in this case.
As consequence,
$$
\align
\max_{U\subset V}
\bigl(
e^{\frac{a_q}{4}V \setminus U}
e^{2d K(d) \gamma U^{\frac {d1}d}}
\bigr)
&\leq
\biggl(
\max
\bigl\{
e^{\frac{a_q}{4}V/M},
e^{2d K(d) \gamma (V/M)^{\frac {d1}d}}
\bigr\}
\biggr)^M
\\
=
\max
\bigl\{
e^{\frac{a_q}{4}V},
e^{K(d) \gamma \partial V}
\bigr\}
\,,
\tag A.45
\endalign
$$
where we used that
$\partial V=2dM (V/M)^{\frac {d1}d}$ in the last step.
\hfill\hfill\qed
\enddemo
\vfil\eject
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\vfill
\newpage
\enddocument
\end