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\title{Low-temperature phase diagrams of quantum lattice systems. I.
Stability for quantum perturbations of classical systems with
finitely-many ground states}
\author{Nilanjana Datta\\
Paul Scherrer Institut\\
Badener Str.\ 569, CH-8048 Z\"urich, Switzerland\\
and\\
Institut f\"ur Theoretische Physik, ETH-H\"onggerberg\\
CH-8093 Z\"urich, Switzerland\\
{\tt datta@itp.ethz.ch}
\and
Roberto Fern\'andez\thanks{Present address: Facultad de Matem\'atica,
Astronom\'{\i}a y F\'{\i}sica,
Universidad Nacional de C\'ordoba, Ciudad Universitaria, 5000
C\'ordoba, Argentina. E-mail: {\tt fernande@fis.uncor.edu}}\\
Institut de Physique Th\'eorique, EPFL\\
CH-1015 Ecublens, Lausanne, Switzerland\\
{\tt fernandez@eldp.epfl.ch}
\and
J\"urg Fr\"ohlich\\
Institut f\"ur Theoretische Physik, ETH-H\"onggerberg\\
CH-8093 Z\"urich, Switzerland\\
{\tt froehlich@czheth5a.bitnet}
}
\date{\ }
%\date{\datetitle}
\maketitle
\begin{abstract}
Starting from classical lattice systems in $d\ge 2$ dimensions
with a regular zero-temperature phase diagram, involving a finite
number of periodic ground states, we prove that the addition of a
small quantum perturbation and/or increasing the temperature produce
only smooth deformations of their phase diagrams. The quantum
perturbations can involve bosons or fermions and can be of infinite
range but decaying exponentially fast with the size of the bonds.
For fermions, the interactions must be given by monomials of
even degree in creation and annihilation operators.
Our methods can be applied to some anyonic systems as
well. Our analysis is based on an extension of Pirogov-Sinai theory to
contour expansions in $d+1$ dimensions obtained by iteration of the
Duhamel formula.
\bigskip
\noindent
{\bf Keywords:} Phase diagrams; quantum lattice systems; Pirogov-Sinai
theory; contour expansions; low-temperature expansions.
\end{abstract}
\tableofcontents
\section{Introduction}
The study of phase diagrams of quantum lattice systems is a much less
developed subject than its classical counterpart. There has been extensive numerical work on quantum phase diagrams at zero
temperature, but rigorous studies, which are often unexpectedly difficult
and rich in surprises, have been very few in number. A systematic exploration of the different
types of ground states has been carried out only
for one-dimensional systems
--~ chains ~-- (see for instance \cite{fannacwer92,kentas92,aiznac94} and
references therein).
Our present understanding of phase diagrams at low, but nonzero temperature is
very limited, as well, although many important quantum-spin lattice systems have been rigorously
studied at low temperatures (see for instance
\cite{rob69,gin69,dysliesim78,frolie78,thoyin83,thoyin84,ken85,kenlie86,%
lemmac93,mesmir94,mazsuh94,albdat94,alb94}). These studies provide
useful illustrations of some of the phenomena involved, but they
focus on extracting detailed information for special models,
rather than on developing a general formalism of wider applicability.
In this paper, we take the opposite attitude: We
present some general black-box type results which, although typically
far from optimal for each specific model, allow us to understand the
broad features of some regions in the phase diagrams of
quantum lattice systems dominated by a classical interaction.
Usually, the study of low-temperature phase diagrams involves a
two-step process: First, the zero-temperature phase diagram is drawn,
and, second, one analyzes which of the zero-temperature phases survive
at nonzero temperatures. For classical systems, there is a general
theory to handle the second part of this process, namely
Pirogov-Sinai theory \cite{pirsin75,pirsin76,sin82,zah84,borimb89}.
The bottom line
of this theory can be summarized as follows: If the zero-temperature
phase diagram is a {\em regular}\/ phase diagram (in the sense of
satisfying the Gibbs phase rule) involving a {\em finite}\/ number of {\em
periodic}\/ ground states, and if in addition, the excitations of these
ground states have an energy proportional to the size of their {\em boundaries}\/ (Peierls
condition), then the phase diagrams for low enough temperatures are
only small deformations of the zero-temperature phase diagrams.
In other words, the theory says that for systems with finitely
degenerate ground states and obeying the Peierls condition, the entropy contribution
to the free energy, present at non-zero temperatures, is only a small
correction to the internal energy at low
temperatures.
In this paper we extend this theory to systems with small quantum
perturbations, and we conclude that the
addition of these perturbations only leads to small deformations of
the phase diagram if the temperature is low (in particular equal to zero). It
is somehow surprising that, except for a pioneer announcement
\cite{pir78} which was never followed by full proofs, this
natural extension of Pirogov Sinai theory has never been considered previously\footnote{We
have recently been informed by C.\ Borgs and R.\ Koteck\'y that they have also constructed an extension of
Pirogov-Sinai theory to systems of the type analyzed in this paper.}.
One may argue that this is because the extension (apparently) refers
to the least interesting regions of a quantum phase diagram, namely
those where the quantum part does not
trigger any new effect. This is, however, a poor reason
on two accounts: First, a ``no-go'' result is needed and
useful, because it allows people hunting for quantum effects to rule
out large regions of the phase diagram, saving effort and
misunderstandings. Second, and more importantly, some quantum
effects can, in fact, be studied by using our results. Indeed, we shall
show in a subsequent paper \cite{pap2} how, by combining the present theory with a
perturbation scheme, one can analyze degeneracy-breaking
effects induced by quantum perturbations and the associated phase
transitions.
A noteworthy difference between our approach and many of the previous
ones (eg.\ \cite{rob69,gin69,ken85}), is that, instead of using the Trotter formula, we resort to an imaginary-time Dyson
expansion based on an iteration of Duhamel's formula. The resulting
expansion
roughly corresponds to performing part of the limit involved in the
Trotter formula, so a sum over a large number of small subintervals
is replaced by an integral. Thus, we work in subregions
of $\zed^d\times[0,\beta]$; the last coordinate ---~the
``time-direction''~--- being
a continuous one (with periodic boundary conditions), and
our contours are piecewise-cylindrical surfaces whose
``time''-sections are
ordinary Pirogov-Sinai classical contours. We think that this approach
has several advantages. On one hand, contour considerations are
based on the surfaces naturally associated to the expansion, without
the additional projection introduced in approaches based on the Trotter
formula. This allows for simpler and clearer geometrical and
combinatorial arguments, a fact also exploited, for instance, in
\cite{aiznac94}. On the other hand, the effects of quantum
perturbations have a nice visualization: they change the sections of
contours. If the system were purely classical, the contours would be
straight cylinders of constant section; the quantum terms produce
deformations or the appearance of ``vacuum fluctuations'' in the form of contours that appear
and/or disappear at intermediate values of the ``time'' coordinate.
These observations permit us to make rigorous the usual
heuristics about quantum perturbations having an ``entropy effect''
comparable to temperature. Indeed, all our bounds are in terms of the
maximum of the quantum-coupling parameter, $\lambda$, and
a temperature-parameter of the form $e^{-\beta \tilde J}$, for some coupling $\widetilde J>0$. In particular,
by letting $\beta\to\infty$ our
formalism yields information about ground states: the
``classical-like'' contours extending all the way
along the interval $[0,\beta]$ disappear, and all that remains are vacuum
fluctuations in a ``sea of spins'' configured as in a classical ground state. The fact that these expected features are exhibited in
such a simple and immediate way is, we believe, a nice feature of our approach.
For the convenience of the reader we summarize the
hypotheses and results in the following section, which can be read as
a ``recipe'' section. Readers
interested in the method itself can
continue with the proofs and technicalities of the remaining sections.
%_____________ INSERTION ________________________
\section{Hypotheses and results}
\subsection{A formalism for quantum lattice systems}
\label{ssetup}
We consider particles with a finite number, N, of internal degrees of freedom,
on a d-dimensional lattice $\zed^d$. The Hilbert space associated with each
site of the lattice is isomorphic to $\bbbc^N$. The system is governed by a
Hamiltonian of the form
\begin{equation}
{\bf H} \;=\; {\bf H}^{\rm cl} + {\bf V}
\label{s.1}
\end{equation}
where ${\bf H}^{\rm cl}$ is interpreted as the ``classical part'', and ${\bf
V}$ as the ``quantum perturbation''. The former consists of finite range
interactions, and is assumed to have a
finite number of periodic ground states. The interactions
constituting ${\bf V}$ can be of infinite range, {\em provided}\/ their
strengths decay exponentially with the size of their supports. To make these
assumptions precise, we need some standard definitions.
A quantum lattice model can be interpreted either as a {\em spin system}\/ or as
a {\em lattice gas}\/. In a quantum spin system, there is a particle at each site of the lattice having a finite number of internal degrees of freedom. In
describing such a system there is no need to refer to the statistics of the
particles. In contrast, the particles in a lattice gas are allowed to hop from
site to site. Hence their statistics plays an important role, and it is
necessary to introduce Fock spaces to describe them. The mathematical
framework required to describe these systems has been introduced in [5, 15, 32]. We
summarize the essential features below.
\newline
{{\it Quantum Spin Systems}:
For a quantum spin system the Hilbert space, ${\cal H}_{\Lambda}$, associated
with a finite
subset $\Lambda$ of the lattice is given by the tensor product
\begin{equation}
{\cal H}_\Lambda \;\bydef\; \bigotimes_{x\in\Lambda} {\cal H}_x
\end{equation}
where each ${\cal H}_x$ is isomorphic to $\bbbc^N$ [An infinite tensor
product of Hilbert spaces is intentionally avoided, since it is not uniquely
defined and is complicated to deal with. Infinite volume limits are considered
only at the level of observable algebras and states].
Bases of ${\cal H}_{\Lambda}$ can be put in correspondence with configurations
on $\Lambda$ in the following way: We choose an orthonormal basis
\begin{equation}
\{e^x_\sigma\}_{\sigma \in I} \,\,\,\,\,\,{\rm with}\,\,\,\, I:=\{1, \ldots ,
N\} \label{exo}
\end{equation}
in ${\cal H}_x$. Let $\Omega_{\Lambda}$ be the set of configurations $\{\omega_{\Lambda}\}$ in $\Lambda$, defined
as the set of all assignments $\{{\sigma_x}\}_{x\in \Lambda}$ of an element
${\sigma_x} \in I$ to each $x$. If $X \subset \Lambda$, then $\omega_X$
denotes the restriction of the configuration $\omega_{\Lambda}$ to the subset $X$. For each
configuration $\omega_\Lambda = \{{\sigma_x}\}_{x\in \Lambda} \in \Omega_\Lambda$, let $e_{\omega_\Lambda}$ be the vector defined as
\begin{equation}
e_{\omega_\Lambda}:= \otimes_{x \in \Lambda} \, e^x_{{\sigma_x}} \label{emo}
\end{equation}
The set of vectors $\{e_{\omega_\Lambda}\}_{\omega_\Lambda \in \Omega_\Lambda}$ is an orthonormal basis of ${\cal H}_{\Lambda}$.
A {\em state}\/ of a quantum lattice system is defined as a positive linear
functional on a suitable $C^*$-algebra. To construct the latter, we start with the
algebra ${\cal A}_{\Lambda}$ of all bounded operators (matrices), acting on ${\cal H}_{\Lambda}$,
with the usual operator norm and with hermitian conjugation as the
$*$-involution. The algebras ${\cal A}_{\Lambda}$ can be considered to be partially nested, i.e.,
\begin{equation}
{\cal A}_{\Lambda_1} \subset {\cal A}_{\Lambda_2} \,\,\,{\rm if}
\,\,\,{\Lambda_1}\subset {\Lambda_2},
\end{equation}
by identifying
each operator $A_1 \in {\cal A}_{\Lambda_1}$ with the operator $A_1 \otimes
{\bf 1}_{\Lambda_2 \setminus \Lambda_1} \in {\cal A}_{\Lambda_2}$, where {\bf 1} denotes the identity
operator. Moreover, the algebras ${\cal A}_{\Lambda}$ are {\em local}\/ , i.e. if
$A_1 \in {\cal A}_{\Lambda_1}$ and $A_2 \in {\cal A}_{\Lambda_2}$, and
${\Lambda_1} \cap \Lambda_2 = \emptyset$, then
\begin{equation}
A_1\,A_2 = A_2\,A_1.
\end{equation}
The norm closure of $\bigcup_{\Lambda
\nearrow \zed^{d}} {\cal A}_{\Lambda}$ defines an algebra which we denote by ${\cal A}$. It
is the quasilocal $C^*$-algebra of observables associated to the infinite
lattice $\zed^{d}$. All local algebras ${\cal A}_\Lambda$ are subalgebras of ${\cal A}$.
The group ${{\bf{T}}_{\zed^{d}}}$ of space translations acts as a
$*$-automorphism group $\{\tau_a : a \in \zed^{d}\}$ on ${\cal A}$,
with
\begin{equation}
{\cal A}_{\Lambda + a} = \tau_a \,{\cal A}_{\Lambda},
\end{equation}
for $\Lambda \subset \zed^{d}$. (The definition of $\tau_a$ is obvious.)
An {\em interaction}\/ of a quantum spin system is a function $\Phi$ from
finite, nonempty subsets $B$ of $\zed^{d}$ to selfadjoint observables $\Phi_B \in {\cal A}_B$. The set of
interactions, $\{\Phi_B \}$, constitutes the Hamiltonian of the system. We shall assume that
the interactions are translation invariant, i.e,
\begin{equation}
\tau_a\,\Phi_B = \Phi_{B+a}, \,\,\,\,{\rm for }\,\,\,{\rm each} \,\,a \in
\zed^{d}, \,\,B \subset \zed^{d}.
\end{equation}
An interaction will be called {\em classical}\/ if we can choose a basis
$\{e^x_\sigma\}_{\sigma\in I}$ in
${\cal H}_x$ such that, for every finite $\Lambda \subset \zed^{d}$, the matrices $\Phi_B$, $B \subset \Lambda$, are diagonal in the
basis $\{e_{\omega_\Lambda}\}_{\{\omega_\Lambda\}\in {\Omega_\Lambda}}$, defined
through \reff{emo}. In this case the interactions are uniquely
defined by the numbers
\begin{eqnarray}
\phi_B(\omega) &\bydef& \langle e_{\omega_B}| \Phi_B|
e_{\omega_B}\rangle \nonumber\\
&=& \langle e_{\omega_\Lambda}| \Phi_B|
e_{\omega_\Lambda}\rangle \quad,\quad \forall \Lambda\supset B\;.
\end{eqnarray}
\newline
{{\it Quantum lattice gases}}:
%\label{ssetup}
In order to describe the itinerant particles of a quantum lattice gas, one
starts with the one-particle Hilbert space
\begin{equation}
{\cal H}^{(1)} \bydef \ell^2(\zed^d) \otimes \bbbc^N
\label{fs.1}
\end{equation}
It represents a single particle which has $N$ internal degrees of freedom and
is confined to a lattice $\zed^d$. A basis of this space can be obtained from
the bases $\{e^x_{{\sigma_x}}\}_{{\sigma_x}\in I}$ of ${\cal H}_x$,
introduced above, (eqn.\reff{exo}). We shall also
use $e^x_{\sigma_x}$ to denote the vector in ${\cal H}^{(1)}$ which has all other summands equal to zero.
%i.e. for which $e^{x'}_{\sigma_{x'}} = 0$ if $x' \ne x$.
To incorporate the statistics of the particles we construct Fock spaces
\begin{equation}
{\cal F}_P({\cal H}^{(1)}) \;=\; P\bigoplus_{n\ge 0} \left({\cal
H}^{(1)}\right)^n \label{foc}
\end{equation}
where
\begin{equation}
\left({\cal H}^{(1)}\right)^n = {\cal H}^{(1)} \otimes {\cal H}^{(1)} \otimes \cdots
\otimes {\cal H}^{(1)}
\end{equation}
denotes the $n$-fold tensor product of ${\cal H}^{(1)}$ with itself, $\bigl({\cal H}^{(1)}\bigr)^0\bydef\bbbc$, and $P$
is the orthogonal projection onto the subspace with the right symmetry properties. We shall consider bosons and fermions.
\newline
(i) For {\em bosons}\/, $P$ is the
projection onto the symmetric subspace, defined on each $\left({\cal
H}^{(1)}\right)^n$ by
\begin{equation}
P_{\rm Bose} (e^{x_1}_{\sigma_1}\otimes\cdots\otimes
e^{x_n}_{\sigma_n}) \;=\;
{1\over n!}\, \sum_{\pi}\,
e^{x_{\pi_1}}_{\sigma_{\pi_1}}\otimes\cdots\otimes
e^{x_{\pi_n}}_{\sigma_{\pi_n}},
\label{fs.4}
\end{equation}
where $e^{x_i}_{\sigma_i} \in {\cal H}^{(1)}$, for all $i$, and $\pi$ ranges over all permutations of the indices
$(1,\ldots n)$.
\newline
(ii) For {\em fermions}\/, $P$ is the projection onto the
antisymmetric subspace
\begin{equation}
P_{\rm Fermi} (e^{x_1}_{\sigma_1}\otimes\cdots\otimes
e^{x_n}_{\sigma_n}) \;=\;
{1\over n!}\, \sum_{\pi}\, \sign(\pi)\,
e^{x_{\pi_1}}_{\sigma_{\pi_1}}\otimes\cdots\otimes
e^{x_{\pi_n}}_{\sigma_{\pi_n}},
\label{fs.5}
\end{equation}
where $\sign(\pi)$ is $+1$ if the permutation $\pi$ is even and $-1$ if it is
odd. The RHS of \reff{fs.5} vanishes if any vector $e^{x_i}_{\sigma_i}$ appears more
than once in the tensor product. This implies that it is impossible to create
two fermions in the same state, in accordance with the {\em Pauli exclusion
principle}\/.
Identical constructions can be made for finite volumes, i.e. when ${\cal H}^{(1)}$ is
replaced by
\begin{eqnarray}
{\cal H}_\Lambda^{(1)} &\bydef& \ell^2(\Lambda) \otimes \bbbc^N\nonumber\\
&\simeq& \bigoplus_{x\in\Lambda} {\cal H}_x\;,
\label{fs.6}
\end{eqnarray}
for finite $\Lambda\subset\zed^d$. The fermion Fock space for a finite volume
$\Lambda$ is given by
\begin{equation}
{\cal F}_{P_{\rm Fermi}}({\cal H}^{(1)}_{\Lambda}) \;=\; P_{{\rm Fermi}}\bigoplus_{n\ge 0} \left({\cal
H}^{(1)}_{\Lambda}\right)^n \label{focc}
\end{equation}
It follows from the Pauli principle that
the direct sum in \reff{focc} terminates at
\newline
$n = N|\Lambda|$.
The formulae \reff{fs.4} and \reff{fs.5} can be used to define some bases in the Fock
spaces. One must take into account the fact that different vectors of $\left({\cal
H}^{(1)}\right)^n$, namely those which differ only in a
permutation of the factors $e^{x_i}_{\sigma_i}$, are mapped by the projection operator $P$
onto the {\em same}\/ vector of the Fock space (up to a sign).
To avoid ambiguities, we choose a total ordering of the sites in $\zed^d$. For future convenience, we choose the {\em spiral
order}\/, depicted in Figure
\ref{order} for $d=2$.
\begin{figure}
%\vspace{4cm}
\begin{center}
\begin{picture}(100,75)(-50,-25)
\put(0,0){\circle*{5}}
\put(0,8){\makebox(0,0){$1$}}%{$x_1\!\!=\!\!0$}}%\overline x$}}
\put(0,-8){\makebox(0,0){$(0\,,0)$}}
\put(25,0){\circle*{5}}
\put(25,8){\makebox(0,0){$2$}}
\put(25,25){\circle*{5}}
\put(25,33){\makebox(0,0){$3$}}
\put(0,25){\circle*{5}}
\put(0,33){\makebox(0,0){$4$}}
\put(-25,25){\circle*{5}}
\put(-25,33){\makebox(0,0){$5$}}
\put(-25,0){\circle*{5}}
\put(-25,8){\makebox(0,0){$6$}}
\put(-25,-25){\circle*{5}}
\put(-25,-17){\makebox(0,0){$7$}}
\thicklines
\put(0,-25){\vector(1,0){50}}
\put(50,-25){\vector(0,1){75}}
\put(50,50){\vector(-1,0){100}}
\put(-50,50){\vector(0,-1){37.5}}
\end{picture}
\end{center}
\caption{Spiral order in $\zed^2$}
\label{order}
\end{figure}
We shall say that $(x_1,\sigma_1)$ is {\em earlier}\/ than
$(x_2,\sigma_2)$ ---~and write $(x_1,\sigma_1) \preceq
(x_2,\sigma_2)$~--- if $x_1\preceq x_2$ and, for $x_1=x_2$,
$\sigma_1\le\sigma_2$. The spiral order has the
convenient property that
\begin{equation}
\begin{minipage}{305pt}
{\em the set of sites earlier than those
in a given, finite set $B$ is also a {\em finite}\/ set $B^{(\preceq)}$}\/.
\end{minipage}
\label{spiral}
\end{equation}
This property will be useful in defining states corresponding to ``classical''
boundary conditions. [See discussion following \reff{fs.45}].
An orthonormal basis of ${\cal F}_P({\cal H}^{(1)})$ is
given by the vectors
\begin{equation}
|n_{x_1\sigma_1}\cdots n_{x_k\sigma_k}\rangle \;\bydef\;
\biggl( \underbrace{e^{x_1}_{\sigma_1}\otimes\cdots\otimes
e^{x_1}_{\sigma_1}}_{n_{x_1\sigma_1}\;{\rm times}} \otimes\ldots
\otimes\underbrace{e^{x_k}_{\sigma_k}\otimes\cdots\otimes
e^{x_k}_{\sigma_k}}_{n_{x_k\sigma_k}\;{\rm times}} \biggr)_\preceq
\label{fs.7}
\end{equation}
where ``$(\,\cdots\,)_\preceq$'' indicates that the braced
factors must be ordered such that each subscript $(x_i\sigma_i)$ is
earlier than the ones to its right. The vectors
\reff{fs.7} involve infinitely many occupation numbers $n_{x\sigma}$,
but only finitely many, namely the indicated ones, are nonzero. By restricting the sites $x_i$ to those in a finite region $\Lambda\subset\zed^d$, one obtains a
basis of ${\cal F}_P({\cal
H}_\Lambda^{(1)})$ in a similar manner. For fermions, each $n_{x\sigma}$ is
either $0$ or $1$.
Having introduced the Fock spaces appropriate for the description of
bosons and fermions, we proceed to define suitable $C^*$-{\em algebras}\/ of
observables. The $C^*$-algebras are generated by the
creation and annihilation operators on Fock space obeying
the canonical anticommutation relations (CAR), for fermions, and the Weyl form
of the canonical commutation relations (CCR), for bosons.
The annihilation and creation operators on ${\cal F}_P({\cal
H}^{(1)})$ are defined as
\begin{equation}
{\bf c}_{x\sigma} \bydef P\, {\bf a}_{x\sigma} P \label{seex}
\end{equation}
and
\begin{equation}
{\bf c}^*_{x\sigma} \bydef P\, {\bf a}^*_{x\sigma} P, \label{seexx}
\end{equation}
with
\begin{equation}
{\bf a}_{x\sigma} (e^{x_1}_{\sigma_1}\otimes\cdots\otimes
e^{x_n}_{\sigma_n}) \bydef \sqrt n\, (e^{x}_{\sigma},
e^{x_1}_{\sigma_1})\,\, e^{x_2}_{\sigma_2}\otimes\cdots\otimes
e^{x_n}_{\sigma_n},
\end{equation}
\begin{equation}
{\bf a}_{x\sigma}^{*} (e^{x_1}_{\sigma_1}\otimes\cdots\otimes
e^{x_n}_{\sigma_n}) \bydef \sqrt{n+1} \,
\, e^{x}_{\sigma}\otimes e^{x_1}_{\sigma_1}\otimes\cdots\otimes
e^{x_n}_{\sigma_n}
\end{equation}
where $(e^{x}_{\sigma},
e^{x_1}_{\sigma_1})$ denotes the scalar product of the vectors $e^{x}_{\sigma}$ and $e^{x_1}_{\sigma_1}$. Furthermore,
\begin{equation}
{\bf a}_{x\sigma} |0> \bydef 0
\end{equation}
and
\begin{equation}
{\bf a}^*_{x\sigma} |0> \bydef e^{x}_{\sigma}
\end{equation}
where
\begin{equation}
|0\rangle := (1,0,0,\cdots)\in
\bigoplus_{n\ge 0}({\cal H}^{(1)})^n
\end{equation}
denotes the {\em vacuum}\/, i.e., the zero-particle state.
For bosons, the operators defined through eqs.\reff{seex} and \reff{seexx} satisfy the
canonical commutation relations (CCR):
\begin{eqnarray}
[{\bf c}_{x_1\sigma_1},{\bf c}_{x_2\sigma_2}] &= &
[{\bf c}^*_{x_1\sigma_1},{\bf c}^*_{x_2\sigma_2}] \,=\, 0\nonumber\\[0pt]
[{\bf c}_{x_1\sigma_1},{\bf c}^*_{x_2\sigma_2}] &= &
(e^{x_1}_{\sigma_1}, e^{x_2}_{\sigma_2}){\bf{1}},
\label{fs.20}
\end{eqnarray}
where ${\bf{1}}$ is the identity operator.
For fermions, the corresponding operators satisfy the canonical
anticommutation relations (CAR):
\begin{eqnarray}
\{{\bf c}_{x_1\sigma_1},{\bf c}_{x_2\sigma_2}\} &= &
\{{\bf c}^*_{x_1\sigma_1},{\bf c}^*_{x_2\sigma_2}\} \,=\, 0\nonumber\\
\{{\bf c}_{x_1\sigma_1},{\bf c}^*_{x_2\sigma_2}\} &= &
(e^{x_1}_{\sigma_1}, e^{x_2}_{\sigma_2}){\bf{1}}
\label{fs.21}
\end{eqnarray}
It follows from the CAR that $\|{\bf c}_{x\sigma}\| = \|{\bf c}^*_{x\sigma}\|=
1$.
The basis vectors, $|n_{x_1\sigma_1}\cdots n_{x_k\sigma_k}\rangle$, defined by eqn.\reff{fs.7}, can be alternatively expressed
in terms of the action of the creation operators on the vacuum.
\begin{equation}
|n_{x_1\sigma_1}\cdots n_{x_k\sigma_k}\rangle\;=\;
{1 \over \sqrt{\prod_{i=1}^k n_{x_i\sigma_i}!}}\,
\Bigl( ({\bf c}^*_{x_1\sigma_1})^{n_{x_1\sigma_1}} \cdots
({\bf c}^*_{x_k\sigma_k})^{n_{x_k\sigma_k}} \Bigr)_\preceq\,|0\rangle\;.
\label{fs.25}
\end{equation}
The labelling is consistent with the fact that these vectors are
simultaneous eigenvectors of the {\em number operators}\/
\begin{equation}
{\bf n}_{x\sigma} \bydef {\bf c}^*_{x\sigma}{\bf c}_{x\sigma}
\label{fs.26}
\end{equation}
with the eigenvalues $n_{x\sigma}$ taking values 0 or 1 for fermions, and 0 or any
natural number for bosons. More generally
\begin{equation}
{\bf c}^*_{x_i\sigma_i} |n_{x_1\sigma_1}\cdots n_{x_k\sigma_k}\rangle
\;=\; e^{i\alpha_\preceq}\,\sqrt{n_{x_i\sigma_i}+1}\,\,
|n_{x_1\sigma_1}\cdots n_{x_i\sigma_i}+1\cdots n_{x_k\sigma_k}\rangle
\label{fs.30}
\end{equation}
and similarly
\begin{equation}
{\bf c}_{x_i\sigma_i} |n_{x_1\sigma_1}\cdots n_{x_k\sigma_k}\rangle
\;=\; e^{i\alpha_\preceq}\,\sqrt{n_{x_i\sigma_i}}\,\,
|n_{x_1\sigma_1}\cdots n_{x_i\sigma_i}-1\cdots n_{x_k\sigma_k}\rangle
\label{fs.31}
\end{equation}
where $\alpha_\preceq$ is a phase which depends on the $n_{x_j\sigma_j}$ with $(x_j,\sigma_j)$ strictly earlier than
$(x_i,\sigma_i)$:
\begin{eqnarray}
\alpha_\preceq &=& 0 \,\,\,\,\,{\rm for}\,\,\,{\rm bosons}, \\
\alpha_\preceq &=& 0, \pi \,\,\,{\rm for}\,\,\,{\rm fermions}.
\end{eqnarray}
%--- change of page 9 to 12 from here -----
Let ${\cal B}_\Lambda$ be the $*$-algebra generated by the identity and the
fermionic annihilation operators, ${\bf c}_{x\sigma}$, with $x\in\Lambda$.
It is
referred to as the field algebra and is larger than the algebra of
observables. The algebra ${\cal A}_\Lambda$ of local observables is the
subalgebra of ${\cal B}_\Lambda$ consisting of all those operators which can
be expressed as sums of monomials of even degree in the
creation and annihilation operators associated with the lattice sites $x \in
\Lambda$.\footnote{It is often reasonable to demand that observables are
gauge invariant. A gauge
transformation ${\cal G}_\phi$ is defined by its action on the annihilation
operators
$$
{\cal G}_\phi: {\bf c}_{x\sigma} \longmapsto e^{i\phi} {\bf
c}_{x\sigma};\,\,\,\phi\in {\bbbr}
$$
The algebra ${\cal A}_\Lambda$ of local observables could also be defined as the
subalgebra of ${\cal B}_\Lambda$ consisting of all those elements which are
invariant under the above transformation, i.e.,
$$A\in {\cal A}_\Lambda \,\,\,{\rm if} \,\,\, A \in {\cal
B}_\Lambda \,\,\,\,\,{\rm and}\,\,\,{\cal G}_\phi(A) = A.
$$
Consequently a gauge invariant
observable is given by a sum of terms each having an equal number of creation
and annihilation operators.} For $A_1\in {\cal A}_{\Lambda_1}$ and $A_2\in {\cal A}_{\Lambda_2}$
\begin{equation}
[A_1, A_2] = 0 \,\,\,{\rm if} \,\,\, \Lambda_1 \cap \Lambda_2 = \emptyset
\end{equation}
and hence the algebras ${\cal A}_{\Lambda}$ are {\it local}.
%\end{document}
For bosons the creation and annihilation operators are not bounded. This is
because there is no bound on the number of particles in the same one-particle
state. The technical
difficulties posed by this unboundedness can be avoided by considering bounded
functions of these operators. One such choice yields the Weyl
operators which are defined as
\begin{equation}
W_{x\sigma}(a,b) = {\rm exp}\,(ia\Phi_{x\sigma} + ib\Pi_{x\sigma}),\,\,\,\,\,a,b \in \bbbr
\end{equation}
where
\begin{equation}
\Phi_{x\sigma} = \frac{{\bf c}_{x\sigma} + {\bf c}^{*}_{x\sigma}}{\sqrt 2}
\end{equation}
and
\begin{equation}
\Pi_{x\sigma} = \frac{{\bf c}_{x\sigma} - {\bf c}^{*}_{x\sigma}}{{\sqrt 2}i}
\end{equation}
The operators ${\bf c}_{x\sigma}$ and ${\bf c}^{*}_{x\sigma}$ are the bosonic
annihilation and creation operators satisfying the CCR, \reff{fs.20}. The Weyl
operators satisfy the commutation relations
\begin{equation}
W_{x\sigma}(a,b)\,W_{x'\sigma'}(a',b') = {\rm exp}\Bigl(i(ab'-a'b)\delta_{xx'}\,\delta_{\sigma\sigma'}\Bigr)W_{x'\sigma'}(a',b')\,W_{x\sigma}(a,b)
\end{equation}
which are called the
Weyl form of the CCR. A quasilocal $C^*$-algebra suitable for the description
of bosons can be generated from these Weyl operators.
In both cases, the fermionic and the bosonic case, the {\em local}\/ algebras are
nested w.r.t. inclusions of the localization regions, i.e.,
\begin{equation}
{\cal A}_{\Lambda_1} \subseteq {\cal A}_{\Lambda_2} \,\,\,{\rm if}\,\, {\Lambda_1} \subseteq {\Lambda_2}
\end{equation}
and the quasilocal algebra of local observables is the norm closure of $\bigcup_{\Lambda \nearrow
Z^{\nu}} {\cal A}_{\Lambda}$, in complete analogy with quantum spin systems. Furthermore, let ${\cal B}$ be the quasilocal
$C^*$-algebra defined as
\begin{equation}
{\cal B} = {\overline{\bigcup_{\Lambda \nearrow
Z^{\nu}} {\cal B}_{\Lambda}}}^{\,\rm {norm}},
\end{equation}
where ${\cal B}_{\Lambda}$ is the $*$-algebra generated by the identity and
annihilation operators for fermions, and by the Weyl operators for bosons.
As in a quantum spin system, an interaction in a quantum lattice gas is given
by selfadjoint operators $\Phi_B \in {\cal A}_B$, for {\em finite}\/ subsets $B$ of the
lattice. It is to be noted that it is implicitly assumed here that the $\Phi_B$ are bounded
operators. This imposes severe restrictions on the allowed interactions in a
bosonic lattice gas. For fermions, the interactions $\Phi_B$ are given by sums
of monomials of even degree in creation and annihilation operators. We write
this symbolically as
\begin{equation}
\Phi_B = \sum_{\llB} \Phi_{\llB}
\end{equation}
where each $\Phi_{\llB}$ is an even monomial and $\sum_{\llB}$ denotes the
sum over all such monomials (with support in $B$) comprising the interaction $\Phi_B$. In our
formulation of the low temperature expansion, it is necessary to express the
fermionic interactions in terms of their constituent monomials, in order to
arrive at a precise
definition of quantum contours [see Sect. \ref{qcont}]. In order to have a
unified treatment for bosons and fermions, we shall denote all quantum
interactions in the sequel by $\sphiq{}$, with the understanding that, for
fermions, $\sphiq{}$ is an even monomial (as mentioned above), whereas for
bosons $\sphiq{}=\phiq{B}$. Moreover, the notations
\begin{equation}
{\underline{B}} \ni 0 \quad ; {\underline{B}} \cap \Lambda \ne 0
\label{conds}
\end{equation}
will be used to denote that the set $B$, corresponding to the monomials
$\sphiq{}$ (for fermions), and to the interaction $\phiq{B}$ (for bosons), satisfies the above conditions, \reff{conds}. For simplicity we shall assume that the interactions are translation
invariant. However, our formalism can be easily extended to periodic
interactions.
If an interaction, $\Phi_B$, is diagonal in the basis formed by the vectors defined in
\reff{fs.7}, then it is called {\em classical}\/ and is denoted by
$\Phi^{\rm cl}_{B}$.
A {\em bond}\/ is defined as a set $B \subset \zed^d$, for which
$\Phi_{B} \ne 0$. In particular, a {\it quantum bond} is defined as follows:
\begin{definition}
A {\em quantum bond} is a set $B$ for which $\phiq{\llB}\neq 0$. In the
sequel, the symbol $\llB$ will often be used to denote
the support of $\Phi^q_{\llB}$ and will also be referred to
as a {\em quantum bond}.
\end{definition}
The {\em range}\/ of the interactions is defined as the maximum of the diameters of
the bonds (defined with any convenient translation invariant notion of distance on $\zed^d$). The classical part of the
interactions will be assumed to be of finite range. The quantum perturbation
can be of infinite range, but is assumed to satisfy a summability
condition of the form
\begin{equation}
\sum_{\llB\ni 0} \|\Phi^q_{\llB}\|
e^{\kappa \diam B} \;<\; \infty
\label{pt.1}
\end{equation}
for $\kappa>0$ large enough.
If the occupancy of {\it every} site of the lattice
$\zed^d$ is chosen to be one then the {\it lattice gas} reduces to a {\it
spin system}.
We now introduce the notion of {\em boundary conditions}\/ which is of
crucial importance in the determination of phase diagrams. As mentioned
before, we restrict our attention to finite subsets $\Lambda$ of the lattice
$\zed^d$. We choose some periodic configuration $s$ on
$\zed^d$, which is defined by the occupation numbers $n_{x\sigma}=s_{x\sigma}$
for $x\in\zed^d, 1\le\sigma\le N$. The exterior configuration, $s_{\Lambda^c}$,
with ${\Lambda^c} := \zed^d \setminus \Lambda$, is
called a {\em boundary condition}\/.
It is evident that our formalism must be generalized, because the configuration $s$ does not
correspond to a vector in the Fock space ${\cal F}_P({\cal H}^{(1)})$.
The vectors in Fock space are square-summable superpositions of vectors
with {\em finite total occupation number}\/, whereas, for the vector $|s
\rangle$, corresponding to a configuration $s$, this number is {\em
infinite}\/, (unless $s_{x\sigma}= 0$, for all $x, \sigma$).
Thus, instead of considering the Fock
space ${\cal F}_P({\cal H}^{(1)})$, we consider a Hilbert space of states corresponding to {\it
local alterations} of the configuration $s$. Technically, this means that we
construct a space ${\cal F}^s_P({\cal H}^{(1)})$ which has
$|s\rangle$ as a {\em cyclic vector}. This construction is possible if we can
prove that $s$ defines a {\em state}\/, i.e.,
a positive, normalized linear functional on the quasilocal $C^*$-algebra ${\cal B}$. Since
${\cal B}$ is generated by the identity and the annihilation operators, this
amounts to showing that we can define expectation values of the form
\begin{equation}
\langle s | {\bf b}_{x_1\sigma_1}\cdots {\bf b}_{x_n\sigma_n}
| s \rangle\;,
\label{fs.40}
\end{equation}
where ${\bf b}_{x_i\sigma_i} = {\bf c}_{x_i\sigma_i}$, or ${\bf
b}_{x_i\sigma_i} = {\bf c}^{*}_{x_i\sigma_i}$, for $1\le i \le n$. We do this through the following limiting procedure: We consider the vectors
$|s_\Lambda\rangle\in {\cal F}_P({\cal H}^{(1)})$ defined by the occupation
numbers
\begin{equation}
n_{x\sigma} \;=\; \left\{\begin{array}{ll}
s_{x\sigma} & \hbox{if } x\in\Lambda\\
0 & \hbox{else}\;,
\end{array}\right.
\end{equation}
and we set
\begin{equation}
\langle s | {\bf b}_{x_1\sigma_1}\cdots {\bf b}_{x_n\sigma_n}
| s \rangle \;\bydef\; \lim_{\Lambda\nearrow\zed^d}\,
\langle s_\Lambda | {\bf b}_{x_1\sigma_1}\cdots {\bf b}_{x_n\sigma_n}
| s_\Lambda \rangle\;.
\label{fs.45}
\end{equation}
This limit exists due to property \reff{spiral} which implies that the
phases of the matrix elements on the RHS of \reff{fs.45} stabilize once $\Lambda\supset
\{x_1,\cdots,x_n\}_{(\preceq)}$. Our procedure defines the Hilbert spaces
${\cal F}^s_P({\cal H}^{(1)})$ via the standard GNS construction \cite[Section
2.3.3]{brarob87}. For $\Lambda \subset \zed^d$ and boundary condition $s$, we
define the finite-volume {\it partial trace} for ${\bf A}\in {\cal
B}_\Lambda$ as
\begin{equation}
\tr^s_\Lambda {\bf A} \;\bydef\;
\sum_{v_\Lambda}
\langle v_\Lambda \otimes s_{\Lambda^c}| {\bf A} |
v_\Lambda \otimes s_{\Lambda^c}\rangle\;,
\label{fs.50}
\end{equation}
where $\{|{v_\Lambda}\rangle\}$ is an orthonormal basis of ${\cal F}_P({\cal
H}_\Lambda^{(1)})$ of the form \reff{fs.25}. (Note that $|v_\Lambda \otimes s_{\Lambda^c}\rangle \in {\cal F}^{s}_P({\cal
H}^{(1)})$.)
\bigskip
We define the Hamiltonian, ${\bf H}_\Lambda$, associated with a finite subset
$\Lambda$ of the lattice as follows:
\begin{equation}
{\bf H}_\Lambda \;\bydef\; \sum_{\llB\cap\Lambda\neq\emptyset}
{\bf P}^s_\Lambda \Phi_{\llB}{\bf P}^s_\Lambda \;,
\label{fs.65}
\end{equation}
where $\Phi_{\llB}\in {\cal A}_B$, and ${\bf
P}^s_\Lambda$ is the orthogonal projection operator onto the subspace
$$
\{{\bf A}|s\rangle : {\bf A} \in
{\cal B}_\Lambda\}
$$
of ${\cal F}^s_P({\cal H}^{(1)})$. That is, we eliminate those matrix elements
of the operators $\Phi_{\llB}$, with $B$ intersecting both $\Lambda$ and
its complement, that would lead to a change in the configuration outside $\Lambda$.
The {\em finite-volume free energy density}\/
for a set of interactions $\{\Phi_{\llB}\}$, boundary condition $s$, and inverse temperature $\beta$ is given by the expression
\begin{equation}
f_s(\Lambda) \;\bydef\; {-1\over \beta|\Lambda|}
\ln \tr^s_\Lambda e^{-\beta {\bf H}_\Lambda} \;.
\label{f.free1}
\end{equation}
Its infinite-volume limit
\begin{equation}
f\;\bydef\; \lim_{\Lambda\nearrow\zed^d}\, f_s(\Lambda)\;,
\label{f.free2}
\end{equation}
is the {\em free-energy density}\/. The limit may be taken in the sense of van
Hove.
The {\em finite-volume Gibbs state}\/ for a set of interactions
$\{\Phi_{\llB}\}$, boundary condition $s$ and inverse temperature $\beta$ is the
linear functional on ${\cal B}_{\Lambda}$ defined by
\begin{equation}
{\cal B}_{\Lambda} \ni {\bf A} \;\mapsto\;
{\tr^s_\Lambda {\bf A}\, e^{-\beta {\bf H}_\Lambda} \over
\tr^s_\Lambda e^{-\beta {\bf H}_\Lambda} }\;.
\label{fs.60}
\end{equation}
%\end{document}
%----------- end insertion ----------------------
\subsection{Assumptions on the interactions}
\label{shyp}
We consider interactions of the form
$\phit{B}=\phicl{ B} + \phiq{\llB}$
which give rise to a class of Hamiltonians of the form
\begin{eqnarray}
{\bf H}_\Lambda &=& {\bf H}_{\underline\mu\,\Lambda}^{\rm cl} +
{\bf V}_\Lambda\nonumber\\
&=& \biggl[\sum_{B\cap \Lambda\neq\emptyset} \phicl{B}\biggr]
+ \biggl[\sum_{\llB\cap \Lambda\neq\emptyset}
\phiq{\llB}\biggr]\;.
\label{fsh.1}
\end{eqnarray}
{\it Note}: We remark that the {\em self-adjointness of the interactions}
plays no essential role in this work. Hence we do not assume it. In particular, this
means that the eigenvalues of the classical interactions, $\phicl{B}$, are
allowed to be complex.
\bigskip
\noindent
We requires the following hypotheses:
\bigskip
\noindent
{\bf (H1)} $\{\phicl{ B}\}$ is a set of {\em classical,
finite-range}\/ interactions parametrized by $\underline \mu
:=(\mu_1,\ldots,\mu_{P-1})$. The ``coordinate
axes'' of the phase diagram are labelled by $\mu_i,\,1\le i\le P-1$. The range of the interactions is assumed to be independent
of $\underline\mu$. We shall assume translation invariance, but
analogous results can be obtained for periodic interactions as well.
The ``classical'' Hamiltonian ${\bf H}_{\underline\mu\,\Lambda}^{\rm cl}$ is
assumed to satisfy the standard hypotheses of Pirogov-Sinai theory, namely:
\newline
There is a non-empty open set ${\cal O}\subset\IR^{P-1}$ such that the
following properties are satisfied:
\medskip
%\noindent
{\bf (H1.1)} {\em Existence of a common eigenbasis}\/.
There is a basis of the form given in \reff{fs.7} in which all operators $\phicl{ B}$ are simultaneously diagonal, for all $\underline\mu\in{\cal O}$.
\medskip
%\noindent
{\bf (H1.2)} {\em Smoothness properties}\/. The functions
${\cal O}\ni\underline\mu\mapsto \phicl{ B}$
are {\em differentiable}\/ in operator norm. The functions, as well as their
derivatives, are uniformly bounded in norm. Typically, ${\cal O}$ is a bounded region. If the functions $\phicl{ B}$ have a {\em linear}\/ dependence on
$\underline\mu$, the parameters
$\mu_i$ correspond to fields or chemical potentials.
\medskip
%\noindent
{\bf (H1.3)} {\em Finite degeneracy}\/.
The set formed by {\it all} {\it periodic} ground states of $\{\phicl{
B}\}$, for all $\underline\mu\in{\cal O}$, is a finite family
\begin{equation}
\kk\;=\;\{s_1,\ldots,s_P\}\;.
\end{equation}
In the present situation, a
periodic configuration $s$ is a ground state for $\{\phicl{B}\}$ if
\begin{equation}
{\rm Re}\, e_{\underline\mu}(s) \;=\; \min_{\tilde s\;{\rm periodic}}
{\rm Re}\, e_{\underline\mu}(\widetilde s)\;,
\label{fx.1}
\end{equation}
where
\begin{equation}
e_{\underline\mu}(s) \;\bydef\; \lim_{\Lambda\nearrow\zed^d}\,
{1\over |\Lambda|} \sum_{B\cap \Lambda\neq\emptyset}
\phicl{ B}(s)\;.
\label{fii.1}
\end{equation}
The symbol $|\Lambda|$ denotes the cardinality of the set $\Lambda$, and the limit is taken, for instance, via sequences of growing
parallelepipeds. [{\it {Note}}: The
definition of classical ground states is more complicated in the presence
of infinite degeneracy or non-periodicity. See for example
\cite[Appendix B]{vEFS_JSP} and references therein.]
For {\it periodic} configurations the limit \reff{fii.1} exists, and the specific
energy is equivalently given by the average energy contribution of each
fundamental cell of the configuration, i.e.,
\begin{equation}
e_{\underline\mu}(s) \;=\; {1 \over |W|} \sum_{x\in W}
e_{\underline\mu\,x}(s)
\label{nil.1}
\end{equation}
where $W\subset\zed^d$ is a choice of a fundamental cell of $s$, [i.e., a
parallelepiped in which the length of each side is a multiple of the
corresponding period of $s$], and
\begin{equation}
e_{\underline\mu\,x}(s) \;\bydef\; \sum_{B\ni x} \frac{\phicl{ B}(s)}{\mid B \mid}
\label{nil.2}
\end{equation}
can be interpreted as the contribution of the site $x$ to the energy. It will
be referred to as the specific energy ``at $x$'' of the configuration $s$.
\medskip
%\noindent
{\bf (H1.4)} {\em Peierls condition}\/.
For all $\underline\mu\in{\cal O}$ the Peierls condition is satisfied, for some $\underline\mu${\em -independent}\/ Peierls constant $J>0$.
Roughly speaking, this means that the insertion of an excitation corresponding to a
ground state configuration that is different from the one on the rest of the
lattice costs an energy proportional to the surface
area of the inserted droplet. The constant of proportionality is the
Peierls constant. For a precise statement of this condition, see
Definition \ref{d.peierls} below.
\medskip
%\noindent
{\bf (H1.5)} {\em Regularity of the phase diagram}\/. The
zero-temperature phase diagram for $\underline\mu\in{\cal O}$ is
regular. We shall explain this notion below.
At zero temperature, the phase diagram is drawn using the set of ground states
\begin{equation}
{\cal Q}^{(\infty,0)}(\underline\mu) \;\bydef\;
\Bigl\{ s_p\in\kk \,:\, \re e_{\underline\mu}(s_p) =
\min_{s_u\in\kk} \, \re e_{\underline\mu}(s_u)\Bigr\}
\label{fx.dd.3}
\end{equation}
for each value of $\underline\mu$. The superscripts $(\infty,0)$ correspond
to the values of $\beta$ (proportional to the inverse temperature) and the quantum perturbation
parameter $\lambda$. The classical zero-temperature phase diagram is the family of manifolds
\begin{equation}
{\cal S}^{(\infty,0)}_{\{s_{p_1},\ldots,s_{p_k}\}} \;\bydef\;
\Bigl\{ \underline\mu \,:\, {\cal Q}^{(\infty,0)}(\underline\mu)=
\{s_{p_1},\ldots,s_{p_k}\}\Bigr\}
\label{fx.dd.4}
\end{equation}
for $1\le k\le P$, $s_{p_1},\ldots,s_{p_k}\in\kk$. These manifolds
are called the {\em strata}\/ of the phase diagram.
The phase diagram defined by these strata is {\em regular}\/ if the map
\begin{equation}
\underline \mu \; \longmapsto \;
\Bigl(\re e_{\underline \mu}(s_1)-\min_{1\le i\le P}
\re e_{\underline \mu}(s_i)\,,\, \ldots\,,\,
\re e_{\underline \mu}(s_P)-
\min_{1\le i\le P} \re e_{\underline \mu}(s_i)\Bigr)
\label{fx.dd.5}
\end{equation}
is a homeomorphism of ${\cal O}$ into the boundary of the positive octant in the space $\IR^{P}$.
This means that the stratum of maximum coexistence, ${\cal
S}_\kk$, is a single
point ($\cong$ the origin of $\IR^{P}$), the strata with $P-1$
ground states are curves emanating from it ($\cong$ the coordinate
semiaxes), and so on. The strata with $P-k$ ground states are
$k$-dimensional manifolds bounded by the strata with $P-k+1$ ground
states. (This geometry is also known as the {\em Gibbs phase rule}\/.)
\bigskip
\noindent
{\bf (H2)} The quantum perturbation $\{\phiq{B}\}$ is
a translation invariant interaction satisfying {\em exponential decay}\/.
The precise expression of this decay is based on a choice of
{\em sampling plaquettes}\/
$W_a(x)= \{y\in\zed^d \colon |x_i-y_i|\le a \hbox{ for } 1\le i\le d
\}$.
[The constant $a$ is chosen so as to have a one-to-one correspondence between
configurations and classical contours, (see Section \ref{ccon} below).]
For a finite $B\subset\zed^d$, let
\begin{equation}
\begin{minipage}{305pt}
$g(B) \;\bydef\; $
\parbox[t]{200pt}{
{\em minimal number of plaquettes needed to cover
$B$ with a connected set}\/.}
\end{minipage}
\label{fx.10}
\end{equation}
Then the decay condition is given by
\begin{equation}
\| \phiq{\llB}\| \;\le\; c\,\lambda^{g(B)}
\label{f.n.0}
\end{equation}
for some constant $c$ and some $0<\lambda<1$.
\medskip
\noindent
%We emphasize that the operators $\phiq{B}$ may or may not
%commute with the operators $\phicl{B}$, that is, we may
%absorb ``classical'' terms (eg.\ some long-range part)
%in ${\bf V}$ as well. However, we shall
\subsection{Examples}
\label{sexamples}
In this section we give some simple examples of
models to which our theory can be applied. For simplicity we consider models
in $\zed^2$, but analogous results hold for models on $\zed^d$, $d>2$.
\bigskip
\noindent
{\bf Example 1:} {\em Fisher antiferromagnet}\/. This is an example
of a quantum-spin system. For simplicity we choose the spin at each site to
be 1/2. Hence ${\cal H}_x\simeq \bbbc^2$. The system has a Hamiltonian given by
\begin{eqnarray}
{\bf H}_\Lambda &=& \biggl[ \sum_{\,\touch\Lambda}
{\bf \sigma}^{(3)}_x {\bf \sigma}^{(3)}_y -
K \sum_{\ll x,y\gg\,\touch\Lambda}
{\bf \sigma}^{(3)}_x {\bf \sigma}^{(3)}_y
- h \sum_{x\in\Lambda} {\bf \sigma}^{(3)}_x
- h^{\rm stagg} \sum_{x\in\Lambda} (-1)^{|x|} {\bf \sigma}^{(3)}_x
\biggr]\nonumber\\
&& + \biggl[ t \sum_{\,\touch\Lambda}
\Bigl( {\bf \sigma}^{(1)}_x {\bf \sigma}^{(1)}_y
+ {\bf \sigma}^{(2)}_x {\bf \sigma}^{(2)}_y \Bigr)
+ {\rm h.c.}\biggr]\;.
\label{fi.1}
\end{eqnarray}
${\bf \sigma}^{(i)}_x$, $i=1,2,3$, are the spin operators (Pauli matrices);
$<\!\!x,y\!\!>$ and $\ll\!\! x,y \!\!\gg$
denote nearest neighbour
and next-nearest neighbour pairs, respectively, and the notation
$B\touch\Lambda$ is used to refer to the set:
\begin{equation}
\{B\subset \zed^d\, : B\cap\Lambda\neq\emptyset\}. \label{touch}
\end{equation}
Finally, $t$ is an exchange coupling constant.
\newline
{\it Note}: In this and the following examples we use square
brackets to separate the classical and quantum parts of the Hamiltonian (as in
\reff{fsh.1}). Moreover, any perturbation satisfying hypothesis
(H2) can be added to the quantum parts.
\begin{figure}[htbp]
% \begin{center}
% \mbox{\epsfig{file= ../figure/fisher.eps,height=17 cm}}
\vspace{17cm}
% \end{center}
\caption{Zero temperature phase diagram of the Fisher antiferromagnet
(Example 1) for (a) $K>0$ (b) $K=0$ and $|h|<2$. }
\label{fisher}
\end{figure}
This model gives rise to phase diagrams of different degrees of complexity
depending on which of the couplings are varied. Let us first consider
$\underline\mu=(h,h^{\rm stagg})$. The parameter $h^{\rm stagg}$
modulates a {\em staggered field}\/ whose sign changes as
the parity of $|x|\bydef |x_1|+\cdots+|x_d|$ changes. The
ferromagnetic coupling $K$ is assumed to have a {\em
fixed}\/ non-negative value. We use the symbols ``$+$'' and ``$-$'' to denote
spin up and spin down respectively.
For $K>0$,
the set of ground states of the classical part is
\begin{equation}
\kk \;=\; \{s_+, s_-, s_{+-}, s_{-+}\}\;,
\end{equation}
where $s_+$ is the all-``$+$'' configuration, $s_-$ the all-``$-$'',
and $s_{+-}$ and $s_{-+}$ are the two N\'eel configurations with ``$+$''
spins in one sublattice and ``$-$'' spins in the other one.
The fundamental cell of these periodic configurations can be chosen to
be a $2\times2$ square. Hence we write symbolically:
\begin{equation}
s_+ =\conf{+}{+}{+}{+} \;,\;
s_- =\conf{-}{-}{-}{-} \;,\;
s_{+-} =\conf{+}{-}{-}{+} \;,\;
s_{-+} =\conf{-}{+}{+}{-} \;.
\end{equation}
The corresponding zero-temperature phase diagram is depicted in Figure
\ref{fisher}(a). The oblique lines for $h>0$ are given by the equation
\begin{equation}
h= 2 + |h^{\rm stagg}|.
\end{equation}
The corresponding lines for $h<0$ are given by the equation
\begin{equation}
h= - 2 - |h^{\rm stagg}|.
\end{equation}
To construct the ground state phase diagram, it is convenient to rewrite
the classical part of the Hamiltonian as a sum over terms corresponding to
$2\times 2$ blocks, M, i.e., $\sum_M \Phi_M^{cl}$ with
\begin{eqnarray}
\Phi_M^{cl}=\frac{1}{2}\sum_{\,\subset M}
{\bf \sigma}^{(3)}_x {\bf \sigma}^{(3)}_y &-&
K \sum_{\ll x,y\gg\,\subset M}
{\bf \sigma}^{(3)}_x {\bf \sigma}^{(3)}_y
- \frac{h}{4} \sum_{x\in M} {\bf \sigma}^{(3)}_x \nonumber\\
&-& \frac{h^{\rm stagg}}{4} \sum_{x\in M} (-1)^{|x|} {\bf \sigma}^{(3)}_x
\end{eqnarray}
and find the minimal energy configurations over any such block $M$. This is
because the operators $\Phi_M^{cl}$ constitute an $m$-potential, \cite{holsla78}.
This diagram is regular in the
vicinity of the maximal-coexistence points $P$ and $Q$. At zero temperature,
this model exhibits a transition between ferromagnetic and
antiferromagnetic order when any one of the oblique coexistence lines in the phase diagram is crossed. Our theory will show that this transition
survives at nonzero temperatures and/or in the presence of small
quantum perturbations, like the spin-flipping term added in the second
line of \reff{fi.1}. An alternative proof of this fact is presented
in \cite{alb94}.
If we set $K=0$, the oblique coexistence lines emanating from $P$ and
$Q$ acquire infinitely many periodic ground states. For the upper
lines these ground states result from periodic arrangements of the
configurations:
\begin{equation}
\conf{+}{-}{+}{+}
\,\,\,\,\conf{+}{+}{-}{+}\,\,\,\,\conf{+}{+}{+}{-}\,\,\,\,\conf{-}{+}{+}{+}
\end{equation}
For the lower lines the
configurations which contribute are the ones obtained from the above set by
a spin-flip. These sectors of the phase diagram,
therefore, lie outside the scope of our theory [violation of (H1.3)].
Nevertheless, we can still analyze the regions around the open
vertical segment joining $P$ with $Q$. This corresponds to fixing
the parameter $h$ at some value, such that $|h|<2$, and considering the model
to be parametrized
by $h^{\rm stagg}$ alone. This yields the phase diagram of Figure \ref{fisher}
(b).
The fact that in each case the relevant classical part of the Hamiltonian satisfies the Peierls condition follows from
a general theorem of Holsztynski and Slawny \cite{holsla78} and the
stability of the Peierls condition under perturbation (Proposition
\ref{p.stab} below).
%
%\begin{figure}[p]
%\vspace{15cm}
%
%\caption{Zero-temperature phase diagrams of: (a) Fisher
%antiferromagnet (Example 1), (a1) for $K>0$ and (a2) for $K=0$;
%(b) simple fermionic model of Example 2, (b1) for $K>0$ and (b2) for
%$K=0$; (c) model of fermionic spins of Example 3.}
%\label{fisher}
%\end{figure}
%
\medskip
\noindent
{\bf Example 2:} {\em Simple fermionic model}\/. We consider
spinless fermions with interaction
\begin{eqnarray}
{\bf H}_\Lambda &=& \biggl[ \sum_{\,\touch\Lambda}
{\bf n}_x {\bf n}_y - K \sum_{\ll x,y\gg\,\touch\Lambda}
{\bf n}_x {\bf n}_y - \mu \sum_{x\in\Lambda} {\bf n}_x
- \mu^{\rm stagg} \sum_{x\in\Lambda} (-1)^{|x|} {\bf n}_x
\biggr]\nonumber\\
&& + \biggl[ t \sum_{\,\touch\Lambda} {\bf c}^*_x {\bf c}_y
+ {\rm h.c.}\biggr]\;.
\label{fi.2}
\end{eqnarray}
A lattice site can either be empty, or occupied by a single fermion. Hence
${\cal H}_x\simeq \bbbc^2$. This model can be obtained from the one of the previous example by a
transformation of spin variables to lattice gas variables.
%ferromagnetism translates into nearest-neighbour repulsion, the
%stabilizing antiferromagnetic term becomes a
%next-nearest-neighbour attraction and the spin-flipping quantum
%perturbation turns into a kinetic (particle-hopping) term.
By suitably transcribing the results of Example 1, we obtain, for $K >0$,
the zero-temperature phase diagram shown in Figure \ref{example} (a).
\begin{figure}[htbp]
% \begin{center}
% \mbox{\epsfig{file= ../figure/example.eps,height= 15cm}}
\vspace{15cm}
% \end{center}
\caption{Zero-temperature phase diagram of the simple fermionic model of
Example 2 (a) for $K>0$ and (b) for
$K=0$ and $0\le \mu < 4 -2K$.}
\label{example}
\end{figure}
%
\medskip
\noindent
The latter involves the ground states
\begin{equation}
\kk \;=\; \{s_{\textstyle\bullet},
s_{\textstyle\bullet\circ}, s_{\textstyle\circ\bullet}\}\;,
\end{equation}
where $s_{\textstyle\bullet}$ is the configuration with exactly one
fermion at each site, while
$s_{\textstyle\bullet\circ}$ and $s_{\textstyle\circ\bullet}$ are the
half-filled configurations having one fermion at each site of one of
the sublattices and no particle in the other sublattice.
Diagrammatically,
\begin{equation}
s_{\textstyle\bullet} =\conf{{\displaystyle\bullet}}
{{\displaystyle\bullet}}{{\displaystyle\bullet}}
{{\displaystyle\bullet}} \;,\;
s_{\textstyle\bullet\circ} =\conf{\displaystyle\bullet}
{\displaystyle\circ}{\displaystyle\circ}{\displaystyle\bullet} \;,\;
s_{\textstyle\circ\bullet} =\conf{\displaystyle\circ}
{\displaystyle\bullet}{\displaystyle\bullet}{\displaystyle\circ} \;. \label{diagg}
\end{equation}
The oblique lines in Figure \ref{example} (a) are given by the equation
\begin{equation}
\mu = 4 - 2K + |\mu^{\rm stagg}|.
\end{equation}.
This diagram is regular in the vicinity of the maximal-coexistence
point $P$. For $K=0$ the oblique lines become lines of infinite
degeneracy, with ground states having, in addition to the above configurations, \reff{diagg},
the following one:
$$
\conf{\displaystyle\bullet}{\displaystyle\bullet}
{\displaystyle\bullet}{\displaystyle\circ},
$$
and the three others obtained from it by rotations. Hence our
theory can only be applied in the region around the vertical coexistence
line up to, but {\em excluding}, the point $P$, i.e., to phase diagrams as in
Figure \ref{example} (b). The stability of these phase diagrams at low, but
nonzero temperatures, and under quantum perturbations, has been studied in
\cite{lemmac93}, in which the $K=0$ model was introduced,
and in \cite{albdat94}, where the
analogous region of the $K>0$ model was analysed.
\bigskip
\noindent
{\bf Example 3:} {\em Simple model of fermions with spin}\/.
By combining elements from the previous two examples, we can easily
generate simple models involving itinerant particles with spin, which satisfy the hypotheses of Section \ref{shyp}. For instance,
consider spin-1/2 fermions with Hamiltonians
\begin{eqnarray}
{\bf H}_\Lambda &=& \biggl[ \sum_{\,\touch\Lambda}
\biggr({1+ {\bf \sigma}^{(3)}_x {\bf \sigma}^{(3)}_y \over 2}\biggr)
{\bf n}_x {\bf n}_y -
K \sum_{\ll x,y\gg\,\touch\Lambda}
\biggr({1+ {\bf \sigma}^{(3)}_x {\bf \sigma}^{(3)}_y \over 2}\biggr)
{\bf n}_x {\bf n}_y \nonumber\\
& & \ {}- \mu \sum_{x\in\Lambda} {\bf n}_x
- \mu^{\rm stagg} \sum_{x\in\Lambda} (-1)^{|x|} {\bf n}_x
- h \sum_{x\in\Lambda} {\bf \sigma}^{(3)}_x{\bf n}_x
- h^{\rm stagg} \sum_{x\in\Lambda} (-1)^{|x|} {\bf \sigma}^{(3)}_x{\bf n}_x
\biggr]\nonumber\\
&& {}+ {\bf V}_\Lambda\;.
\label{fi.10}
\end{eqnarray}
We have defined ${\bf n}_x\bydef\sum_{\sigma=-1,1} {\bf n}_{x\sigma}$.
In the most general case, namely when we choose some fixed $K>0$ and
consider the other four constants as parameters, the phase diagram is
regular around the maximal-coexistence point
\begin{equation}
P\;=\;\Bigl(\mu=2-2K\,,\,\mu^{\rm stagg}=0\,,\, h=2\,,\, h^{\rm
stagg}=0\Bigr)\;,
\label{fi.max}
\end{equation}
where there are five degenerate ground states:
\begin{eqnarray}
s_{++} =\conf{+}{+}{+}{+} \;,\;
s_{+-} &\hspace{-1em}=&\hspace{-1em} \conf{+}{-}{-}{+} \;,\;
s_{-+} =\conf{-}{+}{+}{-} \;,\nonumber\\
s_{+{\textstyle\circ}}=\conf{+}{\displaystyle\circ}
{\displaystyle\circ}{+} &,&
s_{{\textstyle\circ}+}=\conf{\displaystyle\circ}{+}{+}
{\displaystyle\circ}\;.
\end{eqnarray}
In Figure \ref{eg3} we present the cross section of the $h>0$
region of the phase diagram through the plane $h^{\rm stagg}=\mu^{\rm
stagg}=0$.
\begin{figure}[p]
\vspace{12cm}
\caption{Zero-temperature phase diagram of the model of fermionic spins of Example 3.}
\label{eg3}
\end{figure}
%
\medskip
\noindent
The validity of the Peierls condition is again a
consequence of the results of Holsztynski and Slawny \cite{holsla78}
and Proposition \ref{p.stab} given in Section 3.
In subsequent papers \cite{pap2,luc} we consider a broader class of
Hamiltonians whose classical part need not have a finite number of ground
states (and hence may violate the Peierls condition). In \cite{pap2} we develop a
perturbation technique which, together with the contour expansion methods of
this paper, permits us to study the degeneracy-breaking effects of a quantum
perturbation on the classical part and to analyse the phase diagram of the
Hamiltonian at low temperatures.
\subsection{The main theorem}
\label{smain}
Our results show that, under the hypotheses listed in
Section \ref{shyp}, the phase diagrams obtained at low temperatures, and
for small quantum perturbations, are only small deformations of
the zero-temperature diagram corresponding to the classical part
$\{\phicl{B}\}$.
The precise statement of this result requires a notion of stability of
phases.
\begin{definition}
We shall say that a configuration $s_p\in\kk$ defines a stable phase
(or that the $s_p$-phase is stable),
for the interactions $\{\phit{\llB}\}$, if there exists a
neighbourhood, ${\cal D}$, of the origin of $\IR^2$, such that, for each pair
$(e^{-\beta},\lambda) \in {\cal D}$ and any local operator ${\bf A}$, the infinite-volume limit
\begin{equation}
\lim_{\Lambda\nearrow\zed^d}
{\tr^{s_p}_\Lambda \,{\bf A}
\, e^{-\beta {\bf H}_\Lambda} \over
\tr^{s_p}_\Lambda \, e^{-\beta {\bf H}_\Lambda} }
\;\bydef\; \langle {\bf A} \rangle^{s_p}_{\beta\,\lambda}
\label{ffx.1}
\end{equation}
exists and satisfies
\begin{equation}
\lim_{\scriptstyle \beta\to\infty \atop \scriptstyle\lambda\to 0}
\langle {\bf A} \rangle^{s_p}_{\beta\,\lambda} \;=\;
\langle s_p | {\bf A} | s_p \rangle \;.
\label{ffx.2}
\end{equation}
\end{definition}
In analogy to \reff{fx.dd.3}, we introduce the sets
\begin{equation}
{\cal Q}^{(\beta,\lambda)}(\underline\mu) \;\bydef\;
\Bigl\{ s_p \,:\, \hbox{the }s_p\hbox{-phase is stable for }
\{\phit{B}\}\Bigr\}
\label{fi.20}
\end{equation}
to define the strata
\begin{equation}
{\cal S}^{(\beta,\lambda)}_{\{s_{p_1},\ldots,s_{p_k}\}} \;\bydef\;
\Bigl\{ \underline\mu \,:\, {\cal Q}^{(\beta,\lambda)}(\underline\mu)=
\{s_{p_1},\ldots,s_{p_k}\}\Bigr\}\;.
\label{fi.21}
\end{equation}
\newline
Our paper presents a proof of the following theorem:
\begin{theorem}\label{t.bout}
Under the hypotheses of Section \ref{shyp}, there are constants
$\widetilde J>0$ and $\varepsilon_0>0$ such that, for each $\beta$ and
$\lambda$ in the region
\begin{equation}
%\varepsilon_0 \;\bydef\;
\max \bigl(e^{-\beta\tilde J}\,,\, \lambda\bigr) \;<\;
\varepsilon_0,
\label{sh.1}
\end{equation}
there exists a non-empty open set
${\cal O}_{\beta\lambda}\in \IR^{P-1}$
such that:
\begin{itemize}
\item[(i)] The phase diagram defined by the strata
${\cal O}_{\beta\lambda}\cap
{\cal S}^{(\beta,\lambda)}_{\{s_{p_1},\ldots,s_{p_k}\}}$
is regular [in the sense described below \reff{fx.dd.5}] and these
strata are differentiable manifolds.
%Higher levels of smoothness can
%be obtained by strengthening the smoothness requirements (H1.2) in
%Section \ref{shyp}.
\item[(ii)] As $\varepsilon_0 \to 0$, the strata
${\cal O}_{\beta\lambda}\cap
{\cal S}^{(\beta,\lambda)}_{\{s_{p_1},\ldots,s_{p_k}\}}$
tend to the zero-temperature classical
strata
${\cal O}\cap
{\cal S}^{(\infty,0)}_{\{s_{p_1},\ldots,s_{p_k}\}}$, pointwise in
$\underline\mu$. In particular, the distance
between the maximal-coexistence manifolds
${\cal S}^{(\beta,\lambda)}_\kk$ and
${\cal S}^{(\infty,0)}_\kk$ is $\OO(\varepsilon_0)$.
\end{itemize}
\end{theorem}
We shall say that a zero-temperature, classical phase diagram is {\em
stable}\/, under temperature and quantum perturbations, if the
conclusions (i) and (ii) of the theorem hold.
Using our contour expansion methods, we can further prove that, for a fixed
value of $\underline\mu$ corresponding to a single--phase region of the phase
diagram, the free energy
density and the quantum expectations, defined in \reff{ffx.1}, are analytic functions of $\beta$ and $\lambda$, provided $({\rm
Re}\,\beta)^{-1}$ and $|\lambda|$ are small enough.
As an illustration, let us describe the consequences of this theorem for the
examples of Section \ref{sexamples}. For the Fisher antiferromagnet,
we conclude that, for $K>0$ and $\varepsilon_0$ small
enough, the phase diagram around maximal-coexistence points looks like
a smooth deformation of the diagram of Figure \ref{fisher}(a) in the
vicinity of the points $P$ and $Q$. Symmetry considerations imply
that the coexistence line between N\'eel phases remains at $h^{\rm
stagg}=0$. These results have also been obtained in \cite{alb94}, using model-tailored dressing transformations. Likewise,
our theory implies that for $K=0$ and $|h|<2$ the phase diagram of
Figure \ref{fisher}(b) remains valid for small
$\varepsilon_0$. In fact, the diagram remains {\em
unchanged}\/ because, by symmetry, the coexistence point stays at
$h^{\rm stagg}=0$.
Similar conclusions apply to the spinless fermion system of Example
2. In particular, we conclude that, for $K=0$, the phase diagram
of Figure \ref{example} (b) remains unchanged when small kinetic terms (i.e.,
quantum perturbations) are added
and the temperature is increased, as already proven in
\cite{lemmac93,albdat94}. Besides, we derive the stability of the
phase diagram for $K>0$ [Figure \ref{example}(a)] around the point $P$.
For the spin-1/2 fermion model of Example 3, we obtain the stability
of the phase diagram around the maximal-coexistence point
\reff{fi.max}. By symmetry, the coexistence between N\'eel phases
(defined by boundary conditions $s_{+-}$ and $s_{-+}$)
remains at $h^{\rm stagg}=0$, and that of the half-filled phases
(boundary conditions $s_{+{\textstyle\circ}}$ and
$s_{{\textstyle\circ}+}$) at $\mu^{\rm stagg}=0$. Hence we also
obtain the stability of the (non-regular) phase diagram of Figure
\ref{eg3} around the point $P$.
The implications of Theorem \ref{t.bout} for more interesting,
$t$-$J$-type, models will be the subject of a forthcoming paper \cite{luc}.
%***************
%\end{document}
\section{Low-temperature expansion for quantum perturbations}
\label{sbasic}
The first step in the proof of our main result, Theorem \ref{t.bout}, consists in
constructing a suitable low-temperature expansion. This is the content of the present section.
Our expansion is a type of polymer expansion in which the
polymers are called quantum contours (and the consistency rules are
more complicated than plain non-intersection). They are a generalization of the well-known {\it classical} contours of
Pirogov-Sinai theory (see eg.\ \cite[Chapter II]{sin82}). We first recall the definition of
these classical contours and of the associated Peierls condition.
In this and the following section we work with a {\em fixed}\/
value of the parameters $\underline \mu\in {\cal O}$.
Consequently, the parameters play no role and are hence not displayed. The
definition of the contours depends only on the reference
configurations $\kk=\{s_1,\ldots, s_P\}$ and on the range $r$ of the
interactions.
%The only point where the parameters should be
%considered is when checking the Peierls condition; we shall add an
%appropriate comment at the end of Section \ref{ccon}.
The parameters will be reintroduced in Section \ref{sps}, where we will
study the effect of varying them.
\subsection{The classical contours. The Peierls condition}
\label{ccon}
To define these contours we start with a set of periodic {\em
reference configurations}\/ $\kk=\{s_1,\ldots, s_P\}$ and a number
$r>0$ which is an apriori bound on the range of the classical interactions to
be considered. We fix {\em sampling plaquettes}\/
$W_a(x)= \{y\in\zed^d \colon |x_i-y_i|\le a \hbox{ for } 1\le i\le d
\}$. The size $a$ must be strictly larger than (i) the
periods of the reference configurations $s_1,\ldots, s_P$, and (ii)
the range $r$.
\newline
Condition (i) implies
the following {\em extension property}\/:
\begin{equation}
\begin{minipage}{305pt}
{\em If $\omega$ coincides
with the configuration $s_p$ on a plaquette $W_a(x)$ and with $s_q$ on
$W_a(y)$ with $\dist(x,y)\le 1$, then $s_p=s_q$.}
\end{minipage}
\label{f.extension}
\end{equation}
%The condition on the range will be of use when considering energies
%[discussion after formula \reff{s.4.-1}] and in reference to the
%Peierls condition (comment at the end of this subsection).
%For instance, for all the examples of Section \ref{sexamples} we can
%take $a\ge 3$.
To simplify the notation, we shall henceforth measure the cardinality of subsets
$A$ of $\zed^d$ in units of sampling plaquettes:
\begin{equation}
|A|\;\bydef\; {\card A\over a^d}\;.
\label{f.n.1}
\end{equation}
Two sets, $A$ and $B$, in
$\zed^d$ are said to be connected if $\dist(A,B)\le 1$ in lattice units.
A subset $M$ of a set $A\subset\zed^d$ is called a {\it component} of $A$ if $M$ is a
maximal connected subset of $A$, i.e., $M$ is connected and $M\subset
M'\subset A$, $M \ne M'$ imply that $M'$ cannot be connected.
The classical contours are constructed out of ``incorrect'' plaquettes.
A site $x$ is said to be
$p$-correct, for a configuration $\omega$, if the latter coincides with
$s_p$ on {\em every}\/ sampling plaquette that contains $x$. The set of
sites that are not $p$-correct for any $p$, $1\le p\le P$, are referred to as
``incorrect''. The set of plaquettes for which at least one site
is ``incorrect'' form the
{\em defect set}\/, $\partial\omega$, of the configuration $\omega$.
Note that
\begin{equation}
\partial\omega \;=\; \bigcup_{x\in\zed^d} \left\{W_a(x) \,:\,
\omega_{W_a(x)} \neq (s_p)_{W_a(x)} \hbox{ for all } 1\le p\le
P\right\} \;.
\label{s.2}
\end{equation}
Typically we will consider configurations $\omega$ equal to some reference
configuration $s \in {\cal K}$ almost everywhere, i.e., $\omega$ differs from
$s$ only on a finite set of lattice sites. In this situation $\partial\omega$
is a finite set. We shall refer to the plaquettes belonging to the defect set
as {\it excited} plaquettes and the components of the defect set as {\it excitations}.
A {\em (classical) contour}\/ of a configuration $\omega$ is a pair
$\gamma=(M,\omega_M)$ where $M$ is a component of
the defect set $\partial\omega$. The set $M$ is the {\em support}\/
of $\gamma$, to be denoted by $\supp\gamma$. We shall often refer to the
support of a contour $\gamma$ again using the symbol $\gamma$ and use the abbreviation
\begin{equation}
|\gamma| \;\bydef\; |\supp\gamma|\;.
\label{s.2.1}
\end{equation}
According to our definition, the smallest contour is the one obtained when
only a single site is ``incorrect''; e.g. for a quantum spin system this
results when one spin is misaligned, the corresponding contour being formed by all the plaquettes
containing this spin.
Hence the minimal nonzero value of $|\gamma|$ is given by
\begin{equation}
(2a-1)^d/a^d \;\ge\; 1\;.
\label{s.2.2}
\end{equation}
%**** MISSING BIT ****
Each configuration defines a unique family of contours from which it
can be reconstructed, but {\it not} all families of contours
correspond to admissible configurations. The additional restrictions are
that contours must not intersect and that configurations in the interiors
and exteriors of nested contours must match. A family of contours which
corresponds to an admissible configuration will be called {\em compatible}\/.
Henceforth, we shall only consider {\em finite}\/ contours (i.e.,
$|\supp\gamma|<\infty$).
For each such contour, $\gamma$, the space
$\zed^d\setminus \supp\gamma$ is divided into a finite number of components. Moreover, by the extension property
\reff{f.extension}, we can extend the configuration on a single plaquette in
a component to a unique configuration of
$\kk$ in that component. In this way we can label each connected component of
$\zed^d\setminus\supp\gamma$ by a particular reference configuration. Thus, we
obtain the unique configuration
$\omega^\gamma$ that has $\gamma$ as its {\em only}\/ contour.
We shall refer to such a configuration as a
one-contour configuration. The only infinite component of $\zed^d\setminus\gamma$ is called the
{\em exterior}\/ of the contour, $\exter(\gamma)$, and the union of
the other components constitute the {\em interior}\/, $\inter(\gamma)$.
The union of components of $\inter(\gamma)$ labelled by a
reference configuration $s_q$ is called the $q$-interior,
$\inter_q(\gamma)$. The contour is called a {\em $p$-contour}\/ if
its exterior is labelled by the configuration $s_p\in\kk$.
A contour $\gamma$
of a configuration $\omega$ is called an {\it exterior contour} of $\omega$ if
its support is not contained in the interior of any other contour of $\omega$,
i.e., if $\gamma \subset {\rm Ext}({\gamma}')$ holds, for any other contour
${\gamma}'$ of $\omega$.
Let $\{\phi^{\rm cl}_{ B}\}$ be a set of classical interactions of range not
exceeding $r$.
The one-contour configurations can be used to compute energies of any allowed
configuration, as we now explain.
Let $\omega^\gamma$ be a one-contour configuration which has the $p$-contour
$\gamma$ as its only contour. The {\em energy cost}\/ of $\gamma$,
relative to its exterior configuration $s_p$, is given by
\begin{equation}
H^{\rm cl}_{\Lambda}(\omega^\gamma|s_p) \;=\; \sum_{ B\touch [\inter(\gamma)
\cup \supp\gamma]}[\Phi^{\rm
cl}_{ B}(\omega^\gamma) - \Phi^{\rm cl}_{ B}(s_p)]\;,
\label{s.3}
\end{equation}
where we have used the notation of \reff{touch}.
It is convenient to use the decomposition (see \cite{zah84})
\begin{eqnarray}
\sum_{B\touch [\inter(\gamma)
\cup \supp\gamma]}\Phi_B (\omega^\gamma) &=&
\sum_B {|B\cap \supp\gamma|\over |B|} \,\Phi_B(\omega^\gamma)
\,+\, \sum_B {|B\cap \inter\gamma|\over |B|} \,\Phi_B(\omega^\gamma)\nonumber\\
&&\qquad {}\,+\, \sum_{B\touch [\inter(\gamma)
\cup \supp\gamma]} {|B\cap \exter\gamma|\over |B|}
\,\Phi_B(\omega^\gamma)\;,
\label{nil.5}
\end{eqnarray}
to write \reff{s.3} in the form
\begin{equation}
H^{\rm cl}(\omega^\gamma|s_p) \;=\; E(\gamma) + \sum_{u=1}^P\,
\sum_{x\in\inter_u(\gamma)} [e_x(s_u) - e_x(s_p)]
\label{s.4}
\end{equation}
where $e_x(s_u)$ is the specific energy ``at $x$'', (see \reff{nil.2}), of the
configuration $s_u$, and
\begin{equation}
E(\gamma) \;=\; \sum_{ B} {|B\cap \supp\gamma|\over |B|}
\,\Bigl[\Phi^{\rm cl}_B(\omega^\gamma) - \Phi^{\rm cl}_B(s_p)
\Bigr]\label{s.4.-1}
\end{equation}
is the {\em contour energy}\/ of $\gamma$
relative to the energy of its exterior configuration.
\newline
In obtaining \reff{s.4.-1} we have profited from having chosen the plaquette
size $a$ {\it larger} than the range $r$, so that
\begin{equation}
(\omega^\gamma)_B=(s_u)_B, \,\,\, {\rm if}\,\, B \cap \inter_u(\gamma) \ne \emptyset,
\end{equation}
for any $B$ with $\Phi^{\rm cl}_B \ne 0$, and, since $\gamma$ is a $p$-contour,
\begin{equation}
(\omega^\gamma)_B=(s_p)_B \,\,\, {\rm if}\,\, B \cap \exter (\gamma) \ne
\emptyset. \label{elf}
\end{equation}
Hence the latter bonds do not contribute to the contour energy $E(\gamma)$.
In situations of maximal coexistence, all the reference configurations $s_p
\in \kk$ are groundstate configurations and have the same specific energy. In
this case, it follows from \reff{s.4} that the energy cost of a contour $\gamma
$ is simply given by $E(\gamma)$.
Consider a region $\Lambda$ such that
\begin{equation}
(\omega^\gamma)_{\zed^d \setminus \Lambda} = (s_p)_{\zed^d \setminus \Lambda}
\end{equation}
The total energy of the configuration $\omega^\gamma$ [needed for the partial
trace of ${H}_{\Lambda}$ in \reff{fs.50}] is given by
\begin{equation}
H^{\rm cl}_{\Lambda\,s_p}(\omega^\gamma) = H^{\rm cl}_{\Lambda}(\omega^\gamma|s_p) +
\sum_{B\touch\Lambda} \Phi^{\rm cl}_B(s_p) \label{fixx.0}.
\end{equation}
We notice that a decomposition analogous to \reff{nil.5} yields
\begin{equation}
\sum_{B\touch\Lambda} \Phi^{\rm cl}_B(s_p)\;=\;
\sum_{x\in\Lambda} e_x(s_p) +
\sum_{B\touch\Lambda} {|B\cap \Lambda^\comp|\over |B|}\,
\Phi^{\rm cl}_B(s_p)
\label{fix.1}
\end{equation}
However, the last term of \reff{fix.1} is a {\it boundary} term which does not contribute to the free energy density or to the
expectation values of observables, \reff{fs.60}, and is independent of the configuration $\omega$. Hence we shall
neglect it.
The energies of configurations with a {\it finite} number of contours (which
are the only ones relevant in the sequel) can be reconstructed from energies of its contours.
Let $\omega=\omega^\Gamma$ be a configuration corresponding to a compatible
family of contours, $\Gamma=\{\gamma_1,\ldots,\gamma_k\}$, and coinciding at infinity with some reference configuration $s_p$. This implies that
the exterior contours of the configuration $\omega$ are $p$-contours. Let
$\omega^{\gamma_1},\ldots,\omega^{\gamma_k}$ be the corresponding one-contour
configurations. The energy cost
\begin{equation}
H^{\rm cl}(\omega^\Gamma|s_p) \;=\; \sum_{B\touch [\inter(\gamma_j)
\cup \supp\gamma_j]}[\Phi^{cl}_B (\omega^\Gamma) - \Phi^{cl}_B (s_p)]
\end{equation}
of $\omega^{\Gamma}$, relative to the exterior
configuration $s_p$, is simply the sum of one-contour energy costs:
\begin{equation}
H^{\rm cl}(\omega^\Gamma|s_p) \;=\; \sum_{j=1}^k
H^{\rm cl}(\omega^{\gamma_j}|s(\gamma_j))\;,
\label{s.6}
\end{equation}
where $s(\gamma_j)$ is the reference configuration in the exterior
of $\gamma_j$. It follows from eqs.\reff{fixx.0},\reff{fix.1} and \reff{s.6} that the total energy of $\omega^{\Gamma}$ is given by
\begin{equation}
H^{\rm cl}_{\Lambda}(\omega) \;=\; \sum_{x\in\Lambda} e_x(s_p) +
E(\Gamma) + \sum_{u=1}^P\, \sum_{x\in \ell_u} \Bigl[e_x(s_u)-e_x(s_p)\Bigr] \;,
\label{s.6.1}
\end{equation}
where
\begin{equation}
E(\Gamma) \;=\; \sum^{k}_{j} E(\gamma_j),
\label{s.8}
\end{equation}
and $\ell_u$ is
the set of sites in $\Lambda$ that are either
$u$-correct or belong to a $u$-contour. This expression is to be used for the
partial trace ${\rm Tr}^{s_p}_{\Lambda}$ in \reff{fs.50}. Hence the
configuration $\omega^\Gamma$ must be such that
\begin{equation}
(\omega^\Gamma)_{\zed^d \setminus \Lambda}= (s_p)_{\zed^d \setminus \Lambda}
\end{equation}
\newline
{\it Remark}: The contours $\gamma_j$ may extend outside $\Lambda$.
This happens if $\omega$ has some incorrect site {\it on} the
boundary, $\partial \Lambda$. In this case all those
plaquettes which contain this site, but extend outside $\Lambda$, also belong
to a contour. Hence, in general, the contours are contained in the larger
set formed by the plaquettes that touch $\Lambda$:
\begin{equation}
\widehat\Lambda \;\bydef\; \bigcup\, \{W_a(x) \,:\,
W_a(x)\cap\Lambda\neq\emptyset\} \;.
\label{fix.10}
\end{equation}
This means that in $E(\gamma_j)$ one may be counting bonds
$B\subset\Lambda^\comp$ that are not counted in
$H^{\rm cl}_{\Lambda\,s_p}$. However, the identity \reff{s.6.1} remains valid,
because these bonds do not contribute to the energy of a contour, [see
sentence following \reff{elf}].
\medskip
\noindent
{{\it Note}}: We use the letter $\gamma$ to denote
individual contours and $\Gamma$ to denote {\em families}\/ of contours.
\bigskip
\noindent
The Peierls condition can now be stated in terms of the contour energies
defined in \reff{s.4.-1}.
\begin{definition}\label{d.peierls}
An interaction $\Phi^{\rm cl}$ satisfies the Peierls condition with
Peierls constant $J$ if
\begin{equation}
\re E(\gamma) \;\ge\; J |\gamma|\;,
\label{s.5}
\end{equation}
where $E(\gamma)$ is the contour energy defined through \reff{s.4.-1}, and
$|\gamma|$ is as in \reff{s.2.1}.
\end{definition}
In general it is not simple to prove that the Peierls condition
is satisfied for a particular model. One way of doing so is to show that the
excess energy of each excited plaquette of the configuration is nonzero, {\it irrespective} of the
particular configuration on the plaquettes surrounding it. However, this is
true only in severely constrained systems or for highly
symmetric situations (as in the Ising model \cite{gri64.f,dob65.f}). In most
systems, it is often energetically favourable for a plaquette to have an
``incorrect'' configuration if the surrounding
plaquettes are already excited. Hence, calculating the excess energy of a
single excited plaquette is not sufficient for verifying the Peierls
condition. Instead, one may need to compute the energy balance of a possibly
complex arrangement of plaquettes. However, one can avoid the
complicated calculations that this involves by resorting to a theorem due to Holsztynski and Slawny \cite{holsla78}, which states that the Peierls condition is satisfied if the interaction can be written as an
{\em $m$-potential}\/, i.e., a potential admitting a finite number of
ground states that minimize the contribution of each bond
simultaneously. The only drawback of this important result is
that its proof does not provide any estimate of the Peierls
constant, a fact that in turn prevents one from explicitly estimating the range of
temperatures for which our results concerning the phase diagram, are valid.
In this paper we have the additional complication
of having to verify the Peierls condition simultaneously for a whole
family of interactions, parametrized by $\underline\mu$
[Hypothesis (H1.4)]. However, the condition imposed on the size of the
sampling plaquettes, namely $a>r$, simplifies the situation, since it allows us to make
use of
some perturbative results (discussed for instance
in \cite[pages 1126--1127]{vEFS_JSP}) which can be summarized in the
following statement:
\begin{proposition}\label{p.stab}
Consider a family of interactions $\{\phicl{B}\}$ differentiable in
$\underline\mu$. Assume that, for some value
$\underline\mu_0$ of the parameters, the interaction
$\{\Phi^{\rm cl}_{\underline\mu_0 \,B}\}$ has a finite number of periodic
ground states $\kk=\{s_1,\ldots, s_P\}$ and that it satisfies the Peierls
condition with Peierls constant $J_0$. Then, for $\delta>0$ small
enough, there exist open neighbourhoods ${\cal
O}_\delta\ni\underline\mu_0$ such that all the interactions
$\{\phicl{B}\}$ with $\underline\mu\in{\cal O}_\delta$ satisfy the
Peierls condition with Peierls constant $J_0-\delta$ and for the same set of
reference configurations $\kk$.
\end{proposition}
For instance, to verify the uniform-Peierls condition hypothesis
[(H1.4) in Section \ref{shyp}] for the examples of Section
\ref{sexamples}, it is enough to check it at the points of maximal
coexistence, which is an easy application of
Holsztynski-Slawny theory.
%_____________ insertion_______________[low-temp.tex]
\subsection{The Duhamel expansion}
\label{sduh}
We start by establishing a low-temperature expansion for the partition
functions
\begin{equation}
\Xi_s(\Lambda) \;=\; \tr^s_\Lambda \, e^{-\beta {\bf H}_\Lambda}
\label{fi.50}
\end{equation}
for finite regions $\Lambda\subset\zed^d$ with boundary condition
$s=s_p\in\kk$. To compute this trace we use the basis of
${\cal F}^s_P({\cal H}^{(1)})$ spanned by the vectors $|v_\Lambda \otimes
s_{\Lambda^c}\rangle $ corresponding to configurations $v_\Lambda s_p$ which coincide with $s_p$
outside $\Lambda$. This basis is chosen because $H^{\rm cl}_\Lambda$ is
diagonal in it [Hypothesis (H1.1)]. In turn, each of the configurations
$v_\Lambda s_p$ is uniquely determined by a compatible family of
contours $\Gamma^p=\Gamma^p(v_\Lambda s_p)$.
The superscript, $p$, indicates that the {\em
exterior contours}, i.e., the contours of $\Gamma^p$
whose supports are not contained in the interior of any other contour of
$\Gamma^p$, are $p$-contours. We can, therefore, relabel the basis
in terms of these contours and write
\begin{equation}
\Xi_p(\Lambda) \;=\; \sum_{\Gamma^p}
\langle \Gamma^p | e^{-\beta{\bf H}_\Lambda} |
\Gamma^p \rangle,
\label{fi.51}
\end{equation}
(henceforth we shall denote $\Xi_{s_p}\equiv\Xi_p$).
The presence of $\Lambda$ in the above formula implies that all the contours
involved must have supports contained in the larger set $\widehat\Lambda$ defined in \reff{fix.10}.
\bigskip
Our starting point for the expansion is the (formal) series
\begin{eqnarray}
e^{-\beta {\bf H}_\Lambda} &=& e^{-\beta {\bf H}^{\rm cl}_\Lambda}
\,+\, \sum_{n\ge 1} \, \int_0^\beta d\tau_1
\,\cdots\,\int_0^\beta d\tau_n \, \Theta\Bigl( \beta -
{\textstyle \sum_{i=1}^n} \tau_i\Bigr) \nonumber\\
&& \quad {}\times e^{-(\beta-\sum\tau_i) {\bf H}^{\rm cl}_\Lambda}
\,(-{\bf V}_\Lambda) \, e^{-\tau_1 {\bf H}^{\rm cl}_\Lambda}
\,\cdots\, (-{\bf V}_\Lambda) \, e^{-\tau_n {\bf H}^{\rm cl}_\Lambda} \;,
\label{fi.52}
\end{eqnarray}
which is obtained by iterating Duhamel's formula
\begin{equation}
e^{-\beta [{\bf H}^{\rm cl}_\Lambda+{\bf V}_\Lambda]} \;=\;
e^{-\beta {\bf H}^{\rm cl}_\Lambda} \,-\, \int_0^\beta d\tau\,
e^{-(\beta-\tau) {\bf H}^{\rm cl}_\Lambda} \,{\bf V}_\Lambda\,
e^{-\tau [{\bf H}^{\rm cl}_\Lambda+{\bf V}_\Lambda]} \;.
\end{equation}
We perform the following manipulations:
\begin{itemize}
%
\item[(a)] Take $\tr^s_\Lambda$ of \reff{fi.52} as in \reff{fi.51}.
%
\item[(b)] Insert
${\bf 1}_{{\cal F}^{s_p}_P({\cal H}_\Lambda^{(1)})}
= \sum_{\Gamma^p} \left|\Gamma^p \rangle \langle \Gamma^p \right|$
around each operator ${\bf V}_\Lambda$ in
\reff{fi.52}.
%
\item[(c)] Use formula \reff{s.6.1} to compute the (diagonal) matrix
elements
$\langle \Gamma^p| e^{-\tau {\bf H}_\Lambda^{\rm cl}} |
\Gamma^p \rangle$.
%
\item[(d)] Expand each ${\bf V}_\Lambda$ as a sum of
$\phiq{\llB}$'s. In this way, at each time step, we obtain matrix elements
involving only {\it one} quantum bond, or, for fermions, {\it one} creation-annihilation
monomial.
%
\end{itemize}
%\newpage
The result is
\begin{eqnarray}
\lefteqn{
\Xi_p(\Lambda) \;=\; \exp\Bigl[-\beta\sum_{x\in\Lambda} e_x(s_p)\Bigr]
\,\times} \nonumber\\
&&\Biggl\{1\,+\, \sum_{n\ge 1}
\sum_{\scriptstyle \Gamma^p_0, \Gamma^p_1, \ldots, \Gamma^p_n \atop
{\scriptstyle \Gamma^p_i\;{\rm compatible\;family} \atop
\scriptstyle \Gamma^p_0 = \Gamma^p_n\;{\rm supported\;on}\;\Lambda}}
\sum_{B_1,\ldots,B_n} \, \int_0^\beta d\tau_1
\,\cdots\,\int_0^\beta d\tau_n \,
\Theta\Bigl( \beta - {\textstyle \sum_{i=1}^n} \tau_i\Bigr)\,
\nonumber\\[20pt]
&& \qquad \qquad{}\times
\langle \Gamma^p_n | -\phiq{\llB_n} | \Gamma^p_{n-1} \rangle
\cdots \langle \Gamma^p_{2} | -\phiq{\llB_2} |
\Gamma^p_1\rangle \langle \Gamma^p_{1} | -\phiq{\llB_1} |
\Gamma^p_0\rangle \nonumber\\
& & \qquad \qquad\ {} \times \exp\biggl\{-\Bigl(\beta-{\textstyle \sum_{i=1}^n}
\tau_i\bigr) E(\Gamma^p_0)
-\tau_1 E(\Gamma^p_1) \,\cdots \, -\tau_n E(\Gamma^p_n)\biggr\}
\nonumber\\[20pt]
&& \qquad{} \times \prod_{u=1}^P \exp\biggl\{-\Bigl[
\Bigl(\beta-{\textstyle \sum_{i=1}^n}\tau_i\Bigr)
\sum_{x\in\ell_u(\Gamma^p_0)} \,+\,
\tau_1\sum_{x\in\ell_u(\Gamma^p_1)} \,+\, \cdots \, +\,
\tau_n\sum_{x\in\ell_u(\Gamma^p_n)}\Bigr] \nonumber\\
&& \qquad \qquad\qquad\qquad \Bigl[e_{x}(s_u)-e_{x}(s_p)\Bigr]
\biggr\}\;,
\nonumber\\
&&\quad \label{fi.55}
\end{eqnarray}
where $\Theta$ is the step function [i.e., $\Theta(t)=1$ if $t>0$ and $0$
otherwise], and $\ell_u(\Gamma^p_i)$ refers to the set of sites $\{x\}$ in the subset
$$
\bigcup_{\gamma \in \Gamma^p_i} \Bigl[\supp\gamma \cup {\rm {Int}}\gamma \Bigr] \label{supg}
$$
of the lattice, which are either $u$-correct or belong to a $u$-contour.
Expression \reff{fi.55} can be interpreted as a ``sum'' of terms each of which is labelled by a
sequence of the form
\begin{equation}
\Upsilon^p \;=\; \left(\Gamma_0^p, \llB_1,\Gamma_1^p,\tau_1,
\ldots, \llB_n,\Gamma_0^p,\tau_n\right),
\label{fi.60}
\end{equation}
where $n$ is zero or a natural number. Each $\Gamma_i^p$ is a compatible
family of classical contours in $\widehat\Lambda$ having $s_p \in {\cal K}$ as
its exterior configuration. The $\tau_i$ are real numbers in the interval $[0,\beta]$ with
\begin{equation}
\sum_{i=1}^{n} \tau_i \le \beta,
\end{equation}
and each $\llB_i$ is a quantum bond.
The sequence \reff{fi.60} can be visualized as
a piecewise cylindrical surface in $d+1$ dimensions formed by
cylindrical pieces of sections $\Gamma^p_i$ and ``flat'' bridges
corresponding to the quantum bonds $\llB_i$, defined in Sect. \ref{ssetup}.
We shall refer to $[0,\beta]$ as the
``time'' axis, and to $\zed^d \simeq \zed^d\times \{0\}$ as the
``spatial'' coordinates. In our construction, the boundary condition
in the spatial direction is defined by one of the ground states. We always
impose periodic boundary conditions in the ``time'' direction, i.e.,
throughout our analysis, the interval
$[0,\beta]$ is endowed with the structure of a circle. This corresponds to
taking the trace of the Boltzmann factor $e^{-\beta {\bf H}_\Lambda}$ as in
\reff{fi.51}.
Let $V$ be a piecewise-cylindrical region in $d+1$ dimensions of the form
\begin{equation}
V=\Lambda\times[\tau_1,\tau_2], \quad
\Lambda\subset\zed^d,\quad\tau_1,\tau_2\in [0,\beta].
\end{equation}
In the following we shall use the symbol $\int_V$ to denote a summation over sites $x$ in
$\Lambda$ (divided by
$a^d$ in accordance with our choice of sampling-plaquette units) and
integration over the continuous variable $\tau$, i.e.,
\begin{equation}
\int_{V} := \int_{\tau_1}^{\tau_2} d\tau \frac{1}{a^d} \sum_{x \in
\Lambda}.
\label{spt}
\end{equation}
The surface $\Upsilon^p$ can be considered
to be constructed in the following manner: $\Upsilon^p$ has a section
$\Gamma_0^p$ at ``time'' zero, which grows cylindrically during a ``time''
interval of length $\bigl(\beta-\sum_{i=1}^n \tau_i\bigr)$ at the end of which
$\llB_1$ is placed transversely, and the
section changes suddenly to $\Gamma^p_1$. This results from the action of
$\phiq{\llB_1}$. The section $\Gamma^p_1$ then grows cylindrically
during a ``time'' interval $\tau_1$ and so on.
The action of the last quantum interaction
$\phiq{\llB_n}$ restores the section to
$\Gamma^p_0$. This section propagates unchanged over a final ``time'' interval of length $\tau_n$.
This space-time picture motivates us to rewrite\reff{fi.55} in the
following abbreviated
form
\begin{eqnarray}
\Xi_p(\Lambda) &=& \exp\biggl[-\int_{\Lambda\times[0,\beta]}\;
e_{x}(s_p)\biggr]\nonumber\\[10pt]
&& {} \times ``\sum_{\Upsilon^p}\hbox{''}\, w(\Upsilon^p)
\,\prod_{u=1}^P \exp\biggl[-\int_{L_u}\,
\left[e_{x}(s_u)-e_{x}(s_p)\right]\biggr]\;,\nonumber\\
\ \label{fi.56}
\end{eqnarray}
%where, in accordance with \reff{spt}, we have defined
%\begin{equation}
%\int_{\Lambda\times[0,\beta]} := \int_{0}^{\beta} d\tau \frac{1}{a^d} \sum_{x %\in
%\Lambda}.
%\end{equation}
The space-time region $L_u$ is the union of cylinders of bases
$\ell_u(\Gamma^p_i)$ and heights $\tau_i$. Its volume is given by
\begin{equation}
|L_u| = \sum_{i=0}^{n}\tau_i |l_u(\Gamma_i^p)|
\label{fax.2}
\end{equation}
where we have denoted $\tau_0\bydef \beta - \sum_{i=1}^n \tau_i$. Hence, using
the notation of \reff{spt}, we have that
\begin{equation}
\int_{L_u} = \sum_{i=0}^{n} \int_{0}^{\tau_i} d\tau \frac{1}{a^d} \sum_{x \in l_u(\Gamma_i^p)}
\end{equation}
The definition
of the weights $w(\Upsilon^p)$ can be readily inferred
from \reff{fi.55}.
By convention, the case $n=0$
corresponds to $\Upsilon^p=\emptyset$, and we define
$w(\emptyset)=1$ and $l_u(\emptyset)=\emptyset$.
%\end{document}
Following the analogy with classical contours, it would be natural to
refer to the maximally connected components of
the surface $\Upsilon^p$ as {\em quantum contours}\/. This is meaningful {\em only}\/ if the ``sum'' \reff{fi.56} can be
written as a ``sum'' over compatible families of such putative
contours. This is possible if the integrals over $\tau_i$'s factorize, and if the weights $w(\Upsilon^p)$ can be written as a product of weights
corresponding to individual, disjoint contours. In Section \ref{sfcon} we
shall prove that these factorization properties are indeed satisfied.
\subsection{Quantum Contours}
\label{qcont}
In this section we give a precise definition of quantum contours, discuss
their properties and
introduce the quantum Peierls condition.
\newline
\begin{definition}
A $p$-{\em quantum contour}\/ for an interaction satisfying the
hypothesis {\bf H2} of Section \ref{shyp} is a sequence of the form
\begin{equation}
\zeta^p \;=\; \left(\Gamma^p_0,\llB_1,\Gamma^p_1, \tau_1,
\ldots, \llB_n,\Gamma^p_0, \tau_n\right)
\label{s.9}
\end{equation}
where $n$ is a natural number (to be referred to as the {\em number of
transitions}\/). Each $\Gamma_i^p$ is a compatible
family of classical contours having $s_p\in\kk$ as exterior
configuration. Each $\tau_i$ is a non-negative real
number such that $\sum_{i=1}^n\tau_i\le\beta$, and each
$\llB_i$ is a
quantum bond. In addition we have the restrictions:
\begin{itemize}
\item[(i)] $\Gamma_i$ arises from $\Gamma_{i-1}$ through the action
of $\phiq{\llB_i}$. This action can change the ``spins'' or the occupation
numbers only in a
subset of $B_i$ (which can even be empty). Therefore
\begin{equation}
0\;\le\;\Bigl||\Gamma_i|-|\Gamma_{i-1}|\Bigr| \;\le\; |B_i|\;.
\label{s.9.1}
\end{equation}
\item[(ii)] The surface resulting from the toroidal boundary
conditions in $[0,\beta]$ is connected [Figure \ref{sf.1}(a)] or linked
[Figures \ref{sfb.1}(b)]. [The condition of linking is relevant only in the
case of anyons.]
\end{itemize}
\end{definition}
\begin{figure}
\vspace{20cm}
\caption{Examples of quantum contours:
(a) Connected contours: (a1) Long contour (a2) Long contour with no connected section;
(a3) Long contour (connectedness results from periodicity in the
``time''-direction) (a4) Short contour.}
\label{sf.1}
\end{figure}
\begin{figure}
\vspace{20cm}
\caption{Examples of quantum contours:
(b1), (b2), (b3) Linked contours (b4) A linked contour whose projection is not connected;
(c) A surface that is not a quantum contour, even though its spatial
projection (= orthogonal projection onto $\zed^d\times\{0\}$) is
connected. }
\label{sfb.1}
\end{figure}
We shall omit the superscript indicating the
exterior configuration whenever it plays no role in our discussion.
Due to the periodic boundary condition, the ``time'' axis has the topology of a
circle. We make a distinction between contours that extend from ``time'' zero
to ``time'' $\beta$ and ones which do not. The former will be referred
to as {\it long contours}, while the latter will be called {\it short
contours}. Some examples of these have been illustrated in Figures \ref{sf.1}
and \ref{sfb.1}.
As mentioned above, a contour $\zeta^p$ represents a surface in $d+1$
dimensions formed by successive cylinders of spatial sections
$\Gamma_i$ and time-height $\tau_i$ and flat pieces ${B}_i$,
$i=1,\ldots,n$, located at each transition.
A quantum contour may have no connected section [Figures
\ref{sf.1}(a2), (a3) and \ref{sfb.1} (b1), (b2)], but the different connected
components cannot be very far away from each other, because they must
become connected or linked through the actions of
$\phiq{\llB_1},\ldots, \phiq{\llB_n}$. As a
consequence
\begin{equation}
\begin{minipage}{305pt}
{\em Sections of a quantum contour are such that
no more than $|B_1|+\cdots+|B_n|$ additional plaquettes are needed to
make them connected}\/.
\end{minipage}
\label{s.9.3}
\end{equation}
(This statement is false in $d=1$.)
Because of this, we shall think of the quantum bonds $B$ as ``glue'',
and we shall refer to their cardinality $|B|$ as the ``number of glue
plaquettes''. Note that observation \reff{s.9.3} is also true for contours with no connected
projections [Figure \ref{sfb.1}]. This is because if $|B_1|+\cdots+|B_n|$
glue plaquettes are needed for one component of a quantum contour to encircle
another one, then an even smaller number of glue plaquettes is required
to connect the two components.
A quantum contour, $\zeta$, also has a well-defined notion of exterior and
interior, with a unique configuration corresponding to each of its connected
components. In analogy with the classical case, we shall use the
notations $\exter(\zeta)$ and $\inter_q(\zeta)$ to refer to them.
We also define the support, $\supp\zeta$, of a quantum contour $\zeta$ as the union of the
corresponding defect set (in $\zed^d\times[0,\beta]$) {\em and}\/ the
sites occupied by each of the quantum bonds $B_i$. Let
\begin{equation}
|\Gamma_i|:= \sum_{\gamma \in \Gamma_i} |\gamma|.
\end{equation}
Then the area $|\zeta| := |\supp\zeta|$ is computed by adding
\begin{equation}
|\zeta|_\perp \;\bydef\; |\Gamma_0|
\Bigl(\beta-\sum_{i=1}^n\tau_i\Bigr) +
|\Gamma_1| \tau_1 + \cdots + \tau_n |\Gamma_0|\;,
\label{s.12}
\end{equation}
which is the sum of the areas of the cylindrical portions, to the
number of sites only contained in the glue plaquettes. This
last number is bounded above by the total number of glue
plaquettes:
\begin{equation}
|B(\zeta)| \;\bydef\; |B_1|+\cdots+|B_n|\;,
\label{f.bb}
\end{equation}
hence
\begin{equation}
|\zeta| \;\bydef\; |{\rm {supp}}\zeta| \;\le\; |\zeta|_\perp + |B(\zeta)|\;.
\label{s.18}
\end{equation}
A quantum contour $\zeta$ is said to be an {\it exterior} contour of a family
of contours if its support is not contained in the interior of any other
contour of the family.
As in the classical case, we shall say that two contours are {\em
compatible}\/ if their supports do not intersect or form linked
surfaces, {\em and} the labels of the configurations match, i.e., the exterior labels
are the same if the contours are mutually exterior or, if they are
nested, the exterior label of the internal contour coincides with the
label of the component of the interior of the larger contour that
contains it. A family of contours is compatible if its members are
pairwise compatible. Such families can be associated to
configurations on $\zed^d\times[0,\beta]$.
\newline
{\it Note}: The condition of non-linking of surfaces is not relevant for
bosons or fermions. However, in view of applications of our theory to particles with
other statistics, we shall include this condition of {\it
non-linking} in the definition of compatibility.
The {\it weight} of a quantum contour $\zeta$ is given by
\begin{eqnarray}
\lefteqn{w(\zeta) \;= \;
\left[\prod_{i=1}^n \langle\Gamma_i|-\phiq{\llB_i}|
\Gamma_{i-1}\rangle\right]}
\nonumber\\
&& {}\times \exp\{-\left[E(\Gamma_0) (\beta - {\textstyle \sum_{i=1}^n}
\tau_i) + E(\Gamma_1) \tau_1 + \cdots +
E(\Gamma_0) \tau_n \right]\}\;.
\label{s.10}
\end{eqnarray}
The decay law \reff{f.n.0} and the Peierls bound \reff{s.5} [along with the
linearity of $E(\Gamma)$, Eq.~\reff{s.8}] imply the bound
\begin{equation}
|w(\zeta)| \;\le\; \lambda^{|B(\zeta)|}
\, \exp[-J|\zeta|_\perp]\;.
\label{s.11}
\end{equation}
This bound is the {\bf quantum Peierls condition}\/.
\bigskip
% -------- remove factorization -----------------
\section{Factorization properties}
\label{sfcon}
\subsection{Factorization of the $\tau$-integrals}
\label{ssint}
This property follows from the fact that each quantum interaction $\Phi^q_{\llB_i}$ affects a bond $B_i$ in only {\it one} component of
$\Upsilon^p$. Hence, at the end of each ``time'' interval, $\tau_i$, only the
section of {\it one} of the components changes. The other components are not
affected by the action of the quantum interaction at time $\tau_i$, and hence their sections, and the
corresponding exponential weights, remain unchanged. We shall explain this statement through a
simple example:
%\end{document}
%^^^^^^^^^^^^^^^^^^^^^^^^^^^^66
Let us assume that we are ``summing'' over a surface $\Upsilon^p$
consisting of two connected components, which we shall label by the symbols
` \raisebox{-0.5ex} {$\widetilde{~}$} ' and ` \raisebox{-0.5ex}{$\widehat{~}$}
'. We also assume that only four bonds are affected by
the quantum interactions, successive ones belonging to different components. This gives rise to sequences of the form:
%\end{document}
\begin{equation}
\Bigl(\widehat\Gamma_0\widetilde\Gamma_0\,,\,
\widehat {\llB}_1\,,\,
\widehat\Gamma_1\widetilde\Gamma_0\,,\,
\tau_1\,,\,\widetilde {\llB}_1\,,\,
\widehat\Gamma_1\widetilde\Gamma_1\,,\,
\tau_2\,,\,\widehat {\llB}_2\,,\,
\widehat\Gamma_2\widetilde\Gamma_1\,,\,
\tau_3\,,\,\widetilde {\llB}_2\,,\,
\widehat\Gamma_2\widetilde\Gamma_2\,,\,
\tau_4 \Bigr)
\end{equation}
[We have omitted the superscript $p$ to simplify the
notation]. We note that we do {\it not} assume that
$\widehat\Gamma_2=\widehat\Gamma_0$.
%\end{document}
We need to consider integrals of the form
\begin{eqnarray}
I &=& \int_0^\beta d\tau_1 \,\cdots\,\int_0^\beta d\tau_4\,
\Theta\Bigl( \beta -
{\textstyle \sum_{i=1}^4} \tau_i\Bigr)
e^{-\left(\beta-\sum\tau_i\right)
[f(\hat\Gamma_0)+f(\tilde\Gamma_0)]}\,
e^{-\tau_1 [f(\hat\Gamma_1)+f(\tilde\Gamma_0)]}\nonumber\\
&&\qquad{}\times
e^{-\tau_2 [f(\hat\Gamma_1)+f(\tilde\Gamma_1)]}\,
e^{-\tau_3 [f(\hat\Gamma_2)+f(\tilde\Gamma_1)]}\,
e^{-\tau_4 [f(\hat\Gamma_2)+f(\tilde\Gamma_2)]}\;,
\end{eqnarray}
where $f$ is some contour energy.
\newline
By regrouping the exponentials and performing the change of variables
\begin{equation}
\widehat\tau_1=\tau_1+\tau_2 \;;\;
\widehat\tau_2=\tau_3+\tau_4 \;;\;
\widetilde\tau_1=\tau_2+\tau_3 \;;\;
\widetilde\tau_2=\tau_4\;,
\end{equation}
we find that $I$ factorizes as follows:
\begin{equation}
I \;=\; \widehat I\,\widetilde I
\end{equation}
with
\begin{equation}
\widehat I \;=\; \int_0^\beta d\tau_1 \,\int_0^\beta d\tau_2\,
\Theta( \beta - \hat\tau_1-\hat\tau_2)\,
e^{-(\beta-\hat\tau_1-\hat\tau_2) f(\hat\Gamma_0)}\,
e^{-\hat\tau_1 f(\hat\Gamma_1)} \,
e^{-\hat\tau_2 f(\hat\Gamma_2)}
\end{equation}
and $\widetilde I$ being given by a similar integral, but with the
` \raisebox{-0.5ex}{$\widehat{~}$} ' replaced by `
\raisebox{-0.5ex}{$\widetilde{~}$} '.
%\end{document}
\subsection{Factorization of the weights}
\label{sfcon.2}
For {\it bosons} it is easy to see that the weights $w(\Upsilon)$ can be
written as a product of weights of pairwise disjoint, connected components.
In this case the phase $\alpha_\preceq$ in \reff{fs.30} and \reff{fs.31} is
zero. Hence we can absorb the matrix elements of the operators $\phiq{B}$ in
the states \reff{fs.25} into
contour weights.
The situation for particles with other statistics is more
complicated because the action of each $\phiq{B_i}$ gives rise to a phase that depends on the other contours present.
However, in the case of {\it fermions}, the weights factorize because the
interactions are assumed to be of the form
\begin{equation}
\phiq{B} = \sum_{\llB} \sphiq{},
\end{equation}
where each $\sphiq{}$ is a monomial of even degree in the fermionic creation
and annihilation operators.
%---------- INSERTION ------------------------
We explain the factorization argument for a periodic space-time surface, $\Upsilon^p$, consisting
of two connected components. An example of such a surface is given in Figure \ref{fact}.
\begin{figure}[htb]
\vspace{12cm}
\caption{A space-time surface, $\Upsilon^p$, with two connected components,
$\zeta_B$ and $\zeta_C$. The sections of $\zeta_B$ and $\zeta_C$ at ``time''
zero are denoted by $\gamma^0_B$ and $\gamma^0_C$ respectively. The
corresponding sections at ``time'' $\beta$ are denoted by $\gamma^{\beta}_B$ and $\gamma^{\beta}_C$.}
\label{fact}
\end{figure}
\medskip
The surface corresponds to the successive actions of a sequence
of operators $\Phi^q_{{\llD}_{1}}\,\cdots\,\Phi^q_{{\llD}_{n+m}}$ belonging to the quasilocal
algebra ${\cal A}$. Let $\zeta_B$ and $\zeta_C$ be the two connected
components of the surface $\Upsilon^p$. They correspond to the two families of
operators ${\cal B}=\{\sphiq{1},\ldots,\sphiq{n}\}$
and ${\cal C}=\{\Phi^q_{\llC_1},\ldots,\Phi^q_{\llC_m}\}$. The sequence $({\llD}_1,\ldots, {\llD}_{n+m})$
is a permutation of the sequence
$({\llB}_1,\ldots, {\llB}_n,
{\llC}_1,\ldots, {\llC}_m)$ and is uniquely determined by the surface $\Upsilon^p$.
We assume that, for any subsequence of ${\llD}_i$'s, the corresponding
subsequences of $\llB_i$'s and $\llC_i$'s maintain an
increasing order in $i$.
%INSERTION OF BACK PAGE HERE - page 34 use \reff{f4.7
The surface $\Upsilon^p$ corresponds to a product,
\begin{equation}
\prod_{i=1}^{n+m} \langle \Gamma_i^p|\Phi^q_{{\llD}_{i}}|\Gamma_{i-1}^p\rangle,
\label{f4.7}
\end{equation}
of matrix elements of the operators $\Phi^q_{{\llD}_{i}}$, with
$\Gamma_{n+m}^p = \Gamma_{0}^p$. In \reff{f4.7}, $\Gamma_{i}^p$ is a compatible
family of contours describing the configuration
%at ``time'' $\tau$
that
corresponds to the section of $\Upsilon^p$ at ``time'' $\tau$, with $\tau_i <
\tau < \tau_{i+1}$. Our definition of the operators $\Phi^q_{{\llD}_{i}}$
guarantees that if $\langle
\Gamma_i^p|\Phi^q_{{\llD}_{i}}|\Gamma_{i-1}^p\rangle \ne 0$ then the family
$\Gamma_i^p$ of compatible contours is uniquely determined by the operator
$\Phi^q_{{\llD}_{i}}$ and by $\Gamma_{i-1}^p$ and hence by
$\Phi^q_{{\llD}_{i}},\cdots,\Phi^q_{{\llD}_{1}}$ and by $\Gamma_0^p$, for
all $i=1,\ldots,n+m$. Moreover, since we are computing traces, $\Gamma_{n+m}^p
= \Gamma_0^p$.
The periodicity and the disjointness of the components $\zeta_B$ and $\zeta_C$
implies properties (P1) and (P2) described below. To state them we introduce the following nomenclature:
We
shall say that an operator ${\bf c}_{x{\sigma}}$ (resp.\
${\bf c}^*_{x{\sigma}}$) {\em occurs}\/ in
$\sphiq{i}$, if the operator is part of the monomial
defining $\sphiq{i}$. We shall also say that the pair
$(x,{\sigma})$ occurs in ${\llB}_i$ if ${\bf c}_{x{\sigma}}$ (resp.\
${\bf c}^*_{x{\sigma}}$) {\em occurs}\/ in
$\sphiq{i}$. The phrase ``occurrence of $(x,{\sigma})$ in ${\cal B}$''
shall mean that the corresponding creation or annihilation operator {\it
occurs} in some factor $\Phi^q_{{\llD}_i}$, with
${\llD}_i={\llB}_k$, for some $1\le k\le n$. A site $x$ is said to belong to a
space-time surface $\zeta$ at a particular instant of time, if it is contained
in the spatial section of $\zeta$ at that time.
The following properties are satisfied:
\begin{itemize}
\item[(P1)] Given the vector $|\Gamma_0^p\rangle$,
there exists some complex number $w_{\cal B}\neq0$, depending on $\Gamma_0^p$ such that
\begin{equation}
\sphiq{n}\,\cdots\,\sphiq{1}
|\Gamma_0^p\rangle \;=\; w_{\cal B} | \Gamma_0^p\rangle\;.
\label{ffix.22}
\end{equation}
This follows from the remarks after \reff{f4.7}, in particular from the
periodicity of the component $\zeta_B$ corresponding to the action of the $\Phi^q_{\llB_i}$'s. We get a
similar relation for the action of the $\Phi^q_{\llC_i}$'s.
\item[(P2)] Between two subsequent occurrences of $(x,{\sigma})$ in
${\cal B}$ there is an {\em even}\/ number of occurrences of
$(x,{\sigma})$ in ${\cal C}$. This corresponds to the non-intersecting
character of the components: If a site $x$ belongs
first to the components ${\zeta_B}$, then to ${\zeta_C}$ and then once again to ${\zeta_B}$,
the operators $\Phi^q_{\llC_j}$ must be such that,
at the end of the intermediate period, they transform the configuration at
the site $x$ to what it was before they started acting.
\end{itemize}
With these properties we can prove the following result.
\begin{proposition}\label{p.ppp}
Consider two families ${\cal B}$ and ${\cal C}$ of operators, and
an operator
$\Phi^q_{\bf \llD} := \Phi^q_{{\llD}_{n+m}} \,\cdots\,\Phi^q_{{\llD}_1}$,
%\end{document}
defined as above, where $({\llD_{n+m}}\,\cdots{\llD_1})$ is a permutation of the sequence
$({\llB}_1,\ldots, {\llB}_n,
{\llC}_1,\ldots, {\llC}_m)$.
Assume that properties (P1) and (P2) stated above are satisfied and that
the operators $\sphiq{i}$ are monomials of even degree in
the creation and annihilation operators, i.e.,
\begin{equation}
[{\bf c}^*_{x{\sigma}},\sphiq{i}] = 0 \quad
\hbox{and}\quad
[{\bf c}_{x{\sigma}},\sphiq{i}]=0
\label{ffix.5}
\end{equation}
whenever $(x,{\sigma})$ does not occur in ${\llB}_i$.
Then, for any vector $|\Gamma_0^p\rangle$, there exists a
complex number $w_{\cal C}$ such that
\begin{equation}
\langle \Gamma_0^p |
\Phi^q_{{\llD}_{n+m}}\,\cdots\,\Phi^q_{{\llD}_1}
|\Gamma_0^p\rangle \;=\; w_{\cal B}\,w_{\cal C}\;.
\label{ffix.10}
\end{equation}
\end{proposition}
\proof
We first note that the result is obviously true if the operator
$\Phi^q_{\bf {\llD}} \equiv \prod_{i=1}^{n+m} \Phi^q_{{\llD}_i} $ is
identically zero. Hence we may assume that this is not the case.
In particular, this implies that the creation and annihilation operators corresponding to a pair $(x,{\sigma})$ necessarily occur alternately in the
sequence of operators constituting $\Phi^q_{\bf {\llD}}$.
This observation allows us to restrict our attention to a
situation in which no pair $(x,\sigma)$ occurs in {\it both} sets, ${\cal B}$ and ${\cal C}$. Indeed, if $(x,\sigma)$ occurs in ${\cal B}$ {\it
and} ${\cal
C}$, then by (P2) there is an even number of occurrences in the latter
between two subsequent occurrences in the former. Therefore the
occurrences of $(x,\sigma)$ in ${\cal C}$ can be grouped into pairs
not having any intermediate occurrence in ${\cal B}$. In this
situation, if we use the anticommutation rules to move
the first element of each pair to a position immediately preceding the
second element, the commutation through intermediate operators in
${\cal B}$ does not produce any non-trivial phase because of \reff{ffix.5}.
Hence we obtain a product of the form ${\bf c}_{x{\sigma}}{\bf
c}^*_{x{\sigma}}$, (or ${\bf c}^*_{x{\sigma}}{\bf c}_{x{\sigma}}$), which we can
replace by the identity operator {\bf 1} by using the anticommutation
relation. This is because the opposite order of the operators would necessarily
yield zero. In this way we obtain
%\end{document}
\begin{equation}
\Phi^q_{\bf {\llD}} |\Gamma_0^p\rangle \;=\; \epsilon^{x\sigma}_{\cal C}\,
{\widetilde\Phi}^q_{\bf \llD} | \Gamma_0^p\rangle\;,
\label{ffix.105}
\end{equation}
%\end{document}
where $\epsilon^{x\sigma}_{\cal C}$ is a phase and the symbol `
\raisebox{-0.5ex}{$\widetilde{~}$} ' has
been used to indicate that ${\cal C}$ no longer consists of creation or
annihilation operators corresponding to the pair $(x,{\sigma})$.
%\end{document}
We should point out that the operators constituting ${\widetilde\Phi}^q_{\bf {\llD}}$ also satisfy properties (P1)
and (P2) and the commutation relations \reff{ffix.5}. Note that the operators in
${\cal B}$ are {\it unchanged}. We can repeat this procedure
for all the remaining pairs which occur in both sets, ${\cal B}$ and ${\cal C}$.
At the end we obtain
\begin{equation}
\Phi^q_{\bf \llD} |\Gamma_0^p\rangle = \epsilon_{\cal C}\,{\widehat\Phi}^q_{\bf \llD} | \Gamma_0^p\rangle\;,
\label{ffix.25}
\end{equation}
where $\epsilon_{\cal C}$ is the product of the phases, $\epsilon^{x\sigma}_{\cal C}$,
for all pairs occurring in ${\cal B}$ {\it and} in ${\cal C}$, and the symbol `
\raisebox{-0.5ex}{$\widehat{~}$} ' is used to indicate that the operators in ${\cal C}$ have been
stripped of creation and annihilation operators corresponding to these pairs.
Hence we no longer have any creation or annihilation operator in ${\cal C}$ corresponding to a
pair $(x,{\sigma})$ occurring in ${\cal B}$, and we can move the remaining operators
in ${\cal
C}$ through the operators in ${\cal B}$, without producing any further phases.
This results in an equation
\begin{equation}
\Phi^q_{{\llD}_{n+m}}\,\cdots\,\Phi^q_{{\llD}_1}
|\Gamma_0^p\rangle \;=\; \epsilon_{\cal C}\,
\widehat \Phi^q_{{\llC}_m}\,\cdots\,\widehat \Phi^q_{{\llC}_1}
\, \sphiq{n} \,\cdots\, \sphiq{1}
| \Gamma_0^p\rangle\;,
\label{ffix.15}
\end{equation}
which, given \reff{ffix.22}, yields
\begin{eqnarray}
\langle \Gamma_0^p |
\Phi^q_{{\llD}_{n+m}}\,\cdots\,\Phi^q_{{\llD}_1}
|\Gamma_0^p\rangle &=& w_{\cal B}\,\epsilon_{\cal C}
\, \langle \Gamma_0^p |
\widehat \Phi^q_{{\llC}_m}\,\cdots\,\widehat \Phi^q_{{\llC}_1}
|\Gamma_0^p\rangle \nonumber\\
&\bydef& w_{\cal B}\, w_{\cal C}\;.\qed
\label{ffix.100}
\end{eqnarray}
\bigskip
This proof can be extended to surfaces with more than two connected
components. The situation is more complicated for
anyons. In particular, for $d=2$, it is necessary to demand that
the component formed by the $\sphiq{i}$'s is not only
disjoint but is also not linked with the one formed by the action of
the $\Phi^q_{\llC_i}$'s. For the sake of generality and in preparation
for further studies, we shall consider this extra condition of non-linking as part of our definition of independent contours.
%\end{document}
\section{Contour expansions}
\subsection{Contour expansion for the partition functions}
To construct the partition function corresponding to a boundary condition $s_p$, (defined in \reff{fi.55}), contours are added by summing over the ``spatial'' degrees of freedom
and integrating over the ``time'' axis. We shall denote this
sum-integral operation by a combined symbol: if $g$ is a
complex-valued function on quantum contours, then
\begin{eqnarray}
\intsum_{\zeta} g(\zeta) &\bydef& 1 \,+\, \sum_{n\ge 1}
\, \sum_{(\llB_1,\ldots,\llB_n)} \,
\sum_{(\Gamma_0,\ldots,\Gamma_n)}
\indic[{\rm (i), (ii)}]\nonumber\\
&& {}\times
\int_0^\beta d\tau_1 \cdots \int_0^\beta d\tau_n \,
\indic \left[\beta \ge {\textstyle \sum_{i=1}^n} \tau_i\right]
\,g(\zeta)\;.
\label{s.13}
\end{eqnarray}
We sum each $\llB_i$ over all
quantum bonds, and each $\Gamma^p_i$ over all possible families of compatible classical contours (with exterior $p$-contours). By $\indic [E]$ we mean the indicator function of the
event $E$; in particular, $\indic[{\rm (i), (ii)}]$ in \reff{s.13} vanishes
unless the sections $\Gamma_i$ satisfy the following conditions [see Definition \ref{qcont}]:
\begin{itemize}
\item[(i)]
$0\;\le\;\Bigl||\Gamma_i|-|\Gamma_{i-1}|\Bigr| \;\le\; |B_i|\;$.
\item[(ii)] $\Gamma_0^p=\Gamma_n^p$
\end{itemize}
%More generally, if $F=\{F_k\}_{k\ge0}$ is a family of functions on
%$k$-tuples of contours (not necessarily with the same exterior label)
%with each $F_k(\zeta_1,\ldots,\zeta_k)$ {\em symmetric}\/ upon
%permutations of its arguments, and ${\cal P}$ some property of
%contours, we shall often denote
%\begin{equation}
%\intsum_{\{\zeta_k\}\;{\rm satisf.}\;{\cal P}} F(\{\zeta_k\})
%\;\bydef\; 1 \,+\, \sum_{k\ge 1} {1\over k!}
%\intsum_{\zeta_1\;{\rm satisf.}\;{\cal P}}\cdots
%\intsum_{\zeta_k\;{\rm satisf.}\;{\cal P}}
%F_k(\zeta_1,\ldots,\zeta_k)\;.
%\end{equation}
We are interested in the ``sums'' corresponding to partition
functions for piecewise-cylindrical finite regions $V$ in
$d+1$ dimensions.
For such regions we define the
volume, $|V|$, to be the sum of the volumes of the constituent cylindrical
regions, where, however, in consistency with our choice of units for
areas in $\zed^d$, the areas of the bases of the cylinders are measured in
units of the sampling plaquette. Similarly, we obtain the area of
the internal boundaries, $\partial V$, by adding the surface areas of these
piecewise-cylindrical regions to the area of the bases. Again, we use sampling-plaquette units in $\zed^d$.
>From now on, the symbol $V$ will indicate a {\em
piecewise-cylindrical}\/ region of $\zed^d\times[0,\beta]$.
The partition function for such a region $V$, with a spatial boundary
condition $s_p$, is (formally) defined by the series
\begin{eqnarray}
\lefteqn{
\Xi_p(V) \;=\; \exp\Bigl[-\int_V \,
e_x (s_p)\Bigr]}\nonumber\\[10pt]
&& {}\times \intsum_{\scriptstyle\{\zeta_k\}\subset\hat V\atop{
\atop{\scriptstyle\;{\rm compatible}}}}
\biggl[\prod_k w(\zeta_k)\biggr]
\biggl[\prod_{u=1}^P \exp\Bigl\{-\int_{L_u}\,
\left[e_x(s_u)-e_x(s_p)\right]\Bigr\}\biggr], \label{s.14}
\end{eqnarray}
where the exterior contours of each compatible family are $p$-contours.
The weights $w$ are given by expression
\reff{s.10}, and the region $\widehat V$ is obtained from $V$
by adding to each spatial section the plaquettes touching $V$, i.e.,
\begin{equation}
\widehat V = \widehat\Lambda \times [0,\beta],
\label{vhat}
\end{equation}
where $\widehat\Lambda$ is as defined in \reff{fix.10}. The region $L_u=L_u\bigl(\{\zeta_k\}\bigr)$ consists
of the
set of points in $V$ that are either
$u$-correct or that belong to a $u$-quantum contour.
We shall call such a series a {\em contour expansion}\/ of the
partition function $\Xi_p(V)$.
For $V$ of the form $\Lambda\times[0,\beta]$ we recover expression
\reff{fi.55}. We shall use the letter $V$ for space-time regions, $\Lambda$
for spatial ones, and the abbreviation $\Xi_p(\Lambda)\bydef\Xi_p(\Lambda\times[0,\beta])$.
We shall be interested in the quantity
\begin{equation}
f_p(V) \;\bydef\; {-1\over |V|} \log\Xi_p(V)\;,
\label{f.free3}
\end{equation}
whenever the series \reff{s.14}
converges to a nonzero value, and in the limit
\begin{equation}
f \;\bydef\; \lim_{V\nearrow\zed^d\times[0,\beta]} f_p(V)\;,
\label{fff.1}
\end{equation}
whenever it exists.
Note that we have used the same symbols in the above definitions as those used in \reff{f.free1} and \reff{f.free2} to denote the free energy densities. This is in anticipation of the fact that, in the regimes analyzed in this paper, both
definitions agree for regions of the form $V=\Lambda\times[0,\beta]$.
%%%% DONE UPTO HERE %%%%%%%
\subsection{Contour expansion for the quantum expectations}
\label{scexp}
An expansion for the expectations
\begin{equation}
{\tr^{s_p}_\Lambda {\bf A}\, e^{-\beta {\bf H}_\Lambda}\over
\tr^{s_p}_\Lambda e^{-\beta {\bf H}_\Lambda}}
\;\bydef\; {\Xi^{\bf A}_p(\Lambda)\over \Xi_p(\Lambda)}
\label{ffs.14}
\end{equation}
can be constructed by expanding the numerator and the denominator. For the
latter we have the previously developed expansion, \reff{s.14}. A similar expansion can be
obtained for the numerator by proceeding as follows: We expand
${\bf A} e^{-\beta{\bf H}_\Lambda}$ with the help of the
iterated-Duhamel formula \reff{fi.52} and perform steps (a)--(d) of
Section \ref{sduh}. Let us assume, without loss of generality, that
${\bf A} \equiv {\bf A}_D \in{\cal A}_D$ for some finite $D\subset\zed^d$. Moreover, for
{\it fermions}, we can also assume that ${\bf A}$ is an even monomial in creation
and annihilation operators. Therefore we can view the operator ${\bf A}$ as
giving rise to an extra ``quantum bond'' ${\underline{D}}$.
We then obtain an expansion for
$\Xi^{\bf A}_p(\Lambda)$ which is exactly of the same form \reff{fi.56}, but where the terms of the sum are labelled by sequences of the form
\begin{equation}
\Upsilon^p_{\bf A} \;=\; \left(\Gamma_0^p, \llB_1,\Gamma_1^p,\tau_1, \ldots,
\llB_n,\Gamma_n^p,\tau_n, {\underline{D}},\Gamma_0^p \right)
\label{fff.30}
\end{equation}
with weights
\begin{eqnarray}
w(\Upsilon^p_{\bf A}) &=& \langle \Gamma^p_0| {\bf A} | \Gamma^p_n \rangle
\langle \Gamma^p_{n} | -\phiq{\llB_n} |
\Gamma^p_{n-1} \rangle \cdots \langle \Gamma^p_{1} | -\phiq{\llB_1} |
\Gamma^p_{0} \rangle \nonumber\\
&& {}\times e^{-\left(\beta-\sum\tau_i\right) E(\Gamma^p_0)} \,
e^{-\tau_1 E(\Gamma^p_1)} \,\cdots \, e^{-\tau_n E(\Gamma^p_n)}\;.
\label{fff.35}
\end{eqnarray}
Note that the factor $\langle \Gamma^p_0| {\bf A} | \Gamma^p_n \rangle$ plays
a role analogous to $\langle \Gamma^p_0| -\phiq{\llB_{n+1}}| \Gamma^p_n \rangle$,
with $\llB_{n+1}= {\underline{D}}$.
To obtain a contour expansion, we must exhibit factorization of the expansion
for $\Xi^{\bf A}_p(\Lambda)$ over the components of $\Upsilon^p_{\bf A}$.
The ``time'' integrals involved are the same as for the partition function $\Xi_p(\Lambda)$,
hence the factorization illustrated in Section \ref{ssint} remains
valid. For bosons the weights factorize as before. For fermions too, a
factorization of the weights can be exhibited if we use the following
manipulations: We group into a single entity all the components of
$\Upsilon^p_{\bf A}$ whose
sections at ``time'' $\beta$ intersect the set ${\underline{D}}$. In this way,
we obtain a special component $\zeta_{\bf A}$ corresponding to the action of ${\bf A}$. We shall refer to $\zeta_{\bf A}$ as the quantum contour {\it associated with} ${\bf A}$. It is defined by a sequence
of the form \reff{fff.30} and has a weight given by \reff{fff.35}. An example
of such a contour is given in Figure \ref{factA}, below.
\begin{figure}[htb]
\vspace{13cm}
\caption{A quantum contour, $\zeta_{\bf A}$, {\it associated with} a local
observable ${\bf A} \in {\cal{A}}_D$ where $D$ is a finite subset of the lattice.}
\label{factA}
\end{figure}
\medskip
All the other components of $\Upsilon^p_{\bf A}$ are quantum contours defined
as in the previous section, i.e., their weights do not involve the operator
${\bf A}$. The factorization result of Proposition \ref{p.ppp}
applies. [In Figure \ref{fact}, an example of a surface for
which this proposition is valid is given. Note that the component $\zeta_C$ in
this figure is a contour associated with a local observable.].
The weight $w(\Upsilon^p_{\bf A})$ can hence be written as a product
\begin{equation}
w(\Upsilon^p_{\bf A})= \Bigl[\prod_{\{\zeta_k\}} w(\zeta_k) \Bigr]\,w(\zeta_{\bf A}),
\end{equation}
where
\begin{equation}
\Bigl(\bigcup_{k}\zeta_k\Bigl)\,\bigcup \zeta_{\bf A} = \Upsilon^p_{\bf A}.
\end{equation}
Factorization implies that we can expand the numerator of
the expectations \reff{ffs.14} in the form
\begin{eqnarray}
\lefteqn{
\Xi^{\bf A}_p(V) \;=\; \exp\Bigl[-\int_V \,
e_x(s_p)\Bigr] \,
\intsump_{\zeta_{\bf A}\subset\hat V} \,w(\zeta_{\bf A})}\nonumber\\
&& {}\times
\intsum_{\scriptstyle\{\zeta_k\}\subset\hat V\atop{
\atop{\scriptstyle\;{\rm compatible}}}}
\biggl[\prod_k w(\zeta_k)\biggr]
\,\biggl[\prod_{u=1}^P \exp\Bigl[-\int_{L_u}\,
\left[e_x(s_u)-e_x(s_p)\right]\Bigr]\biggr]
\nonumber\\
&&\qquad {}\times
\indic\Bigl[\Bigl\{\zeta_{\bf A}, \{\zeta_k\}\Bigr\}
\hbox{ compatible; exterior contours are $p$-contours}\Bigr]\;.
\label{fex.1}
\end{eqnarray}
The starred sum-integral refers to an expression of the form \reff{s.13}, but where in condition (ii) we demand
\begin{equation}
{\bf A} |\Gamma^p_n> \;\propto\; |\Gamma^p_{n+1}>, \quad {\rm with} \quad \Gamma^p_{n+1}=\Gamma^p_0.
\label{fex.2}
\end{equation}
The weights $w(\zeta_{\bf A})$ satisfy a Peierls bound:
\begin{equation}
|w(\zeta_{\bf A})| \;\le\; {\lambda^{|B(\zeta_{\bf A})|}}
\, \exp[-J|\zeta_{\bf A}|_\perp] \,\frac{\|A\|}{\lambda^{|D|}}\;,
\label{s.11.11}
\end{equation}
where
\begin{equation}
|\zeta_{\bf A}|_\perp := |\Gamma_0|
\Bigl(\beta-\sum_{i=1}^n\tau_i\Bigr) +
|\Gamma_1| \tau_1 + \cdots + \tau_n |\Gamma_n|\;,
\end{equation}
\begin{equation}
|B(\zeta_{\bf A})| := |B_1|+\cdots+|B_n| + |D|
\end{equation}
and $\|\cdot\|$ denotes the usual operator norm.
\section{Cluster expansion for the symmetric or the single-phase
regime} \label{scss}
\subsection{The result}
\label{reslt}
Having formulated a convenient low-temperature expansion in terms of
contours, we must address the task of proving its convergence for some open set of values of
$\beta$ and $\lambda$. The main mathematical complications
arise from the requirement of compatibility of the contours in expression \reff{s.14}. Compatibility is a highly non-local condition (two
arbitrarily far away nested contours can be rendered incompatible by a
mismatch between the labels of their exterior and interior configurations).
In this section we analyze the simpler case in which compatibility
reduces to non-linking (or non-intersection), i.e., when the labels of the
configurations outside the support of the contours become irrelevant. The results of
this section form the basis for the full-fledged theory which is to be discussed
in Section \ref{sps}. We are concerned with an expansion of the form
\begin{equation}
\Xi_p(V) \;=\;
\intsum_{\scriptstyle\{\zeta^p_k\}\subset\widehat V\atop
\scriptstyle\;{\rm non-linked}} \prod_k w(\zeta^p_k) \;,
\label{fff.40}
\end{equation}
with weights as in \reff{s.10} satisfying the quantum Peierls condition
\reff{s.11}. This type of expansion is obtained when there is a
single ground state for an open set of parameters $\underline \mu$, or,
more generally, for values of $\underline \mu$ for which the ground
states are related by some symmetry operation. In the latter case, the removal of any contour of a compatible family leads to another
compatible family of contours. These have been among the first situations treated by contour arguments \cite{dob65.f,gri64.f}. Among
our examples of Section \ref{sexamples}, this symmetric situation occurs at the coexistence point
$h^{\rm stagg}=0$ for the Fisher antiferromagnet, with $K=0$ and
$|h|<2$ [Figure \ref{fisher} (b)], and at the coexistence point $\mu_{\rm
stagg}=0$ for its lattice-gas
transcription, [i.e., Example 2], with $K=0$ and
$\mu <2$ [Figure \ref{example} (b)].
There is a well established technology to analyze (the log of)
``volume-exclusion'' expansions like \reff{fff.40}, namely the methods of cluster-expansion \cite[Chapter 4]{rue78}. The method involves some
standard combinatorics (reviewed in Section \ref{smore} below) and a
bound on the sum of the weights of contours containing a
fixed point.
Let
\begin{equation}
f_p \;=\; -\lim_{ V\nearrow\szed^d\times[0,\beta]}
{1\over |V|} \log\Xi_p( V)\;,
\label{s.20}
\end{equation}
In Sections \ref{sclp} and \ref{ssres} we shall prove the following key result.
%
\begin{theorem}\label{tresult}
There exist strictly positive constants $\widetilde J$ and
$\varepsilon_0$ such that, for each $\beta$ and $\lambda$ in the region
\begin{equation}
\max \bigl(e^{-\beta\tilde J}\,,\, \lambda\bigr) \;<\;
\varepsilon_0,
\end{equation}
the cluster expansion \reff{fff.40} converges
absolutely for all piecewise-cylindrical regions $V$. The
free-energy density $f_p$ exists in this region and is jointly analytic
in $e^{-\beta\tilde J}$ and $\lambda$. Moreover it satisfies
\begin{equation}
f_p \;=\; \OO(\varepsilon_0)\;,
\label{g.1}
\end{equation}
and
\begin{equation}
\Bigl| | V| f_p + \log \Xi_p( V) \Bigr| \;\le\;
|\partial V|\, \OO(\varepsilon_0)\;.
\label{g.2}
\end{equation}
\end{theorem}
\medskip
\begin{corollary}\label{cresult}
There exists a positive constant $\widetilde\varepsilon_0$ such that, in the
region
\newline
$\max
\bigl(e^{-\beta\tilde J}\,,\, \lambda\bigr) < \widetilde\varepsilon_0$, the limit
\begin{equation}
\lim_{\Lambda\nearrow\zed^d}\,
{\Xi_p^{\bf A}(\Lambda) \over \Xi_p(\Lambda)}
\;=:\; \langle {\bf A} \rangle_{\beta\,\lambda}
\end{equation}
exists and is jointly analytic as a
function of $e^{-\beta}$ and $\lambda$, for any local observable ${\bf A}$. Moreover
\begin{equation}
\lim_{\scriptstyle \beta\to\infty \atop \scriptstyle\lambda\to 0}\,
\langle {\bf A} \rangle_{\beta\,\lambda}
\;=\; \langle s_p | {\bf A} | s_p \rangle\;.
\label{fst.0}
\end{equation}
\end{corollary}
This proves the stability of all the phases occurring in the present situation. In particular, this proves that the phase
diagrams of Figures \ref{fisher} (b) and \ref{example} (b) remain {\em undeformed}
at low temperatures and under small quantum perturbations.
\medskip
\noindent
{\it Remark}: In the arguments that follow we shall impose conditions of the form ``$J$ sufficiently large'' and ``$\lambda$
sufficiently small''. As long as the temperature is sufficiently low and
$\lambda$ is sufficiently small, we can choose $J$ large enough simply by a rescaling. This is because the Hamiltonian
${\bf H}_\Lambda$, which depends on the parameters $J$ and $\lambda$, appears in
the partition function in the form $\beta\,{\bf H}_\Lambda$ which we can write as
\begin{equation}
\beta({\bf H}^{\rm cl}_\Lambda + {\bf V}_\Lambda) \;=\;
{\beta\over\beta'}\, (\beta' {\bf H}^{\rm cl}_\Lambda+ \beta'
{\bf V}_\Lambda)\;.
\label{s.11.1}
\end{equation}
Upon rescaling, $\beta'J\to J$, we see that a large $\beta'$ leads to a
large $J$. The product $\beta'\lambda$, which is the rescaled perturbation
parameter, is small if $\lambda$ is sufficiently small. Once $\beta'$ is
fixed we adjust $\beta$ such that $\beta/\beta'$ is sufficiently large.
\subsection{Combinatorics of the cluster expansion}
\label{smore}
We summarize some results on cluster expansions,
adapted to the case where contours are discrete in all dimensions, except one. We can extend the conventional
proofs for the convergence of cluster expansions (applicable when the
compatibility relation corresponds to non-intersection) to the more general
situation when the contours are either disjoint {\it or} non-linked.
For the usual proofs the
reader may consult for instance \cite{sei82}, \cite{bry84} or
\cite{pfi91}.
\medskip
We use the notation $\zeta\incomp\widetilde\zeta$, see \cite{kotpre86}, to denote the condition that $\supp\zeta$
either intersects or is linked with $\supp\widetilde\zeta$. In the following, we shall use the term linking to refer to both intersection and linking.
A {\em cluster}\/ is a {\em finite}\/ family
$\{\zeta_1, \cdots ,\zeta_n\}$ of contours that cannot
be decomposed into two non-linked subfamilies. To simplify our notation we have omitted the superscripts of the
contours $\zeta$ referring to their exterior configurations.
The following theorem gives the condition for the convergence of a cluster
expansion.
\begin{theorem}\label{tclus1}
If
\begin{equation}
C\;\bydef\; \sup_{\widetilde\zeta}{1\over |\widetilde\zeta|}
\,\intsum_{\zeta{\incomp\; }\widetilde\zeta}
|w(\zeta)|\, e^{|\zeta|} \;<\;1\;,
\label{f.n.10}
\end{equation}
then the expansion \reff{fff.40} is absolutely convergent and $\log\Xi(V)$
has an absolutely convergent expansion of the form
\begin{equation}
\log\Xi(V) \;=\; \sum_{N\ge 1} {1\over N!}
\intsum_{\zeta_1\subset \widehat V}\cdots
\intsum_{\zeta_N\subset \widehat V} w^\trunc(\{\zeta_1,\ldots,\zeta_N\})
\label{s.15}
\end{equation}
(the {\em cluster expansion}\/), where $w^\trunc$ is a
%$V$-independent
function on families of contours with the properties that
\begin{equation}
w^\trunc(\{\zeta_1, \cdots ,\zeta_N\})\;=\; 0 \quad
\hbox{if $\{\zeta_1, \cdots ,\zeta_N\}$ is not a cluster}\;,
\label{s.17}
\end{equation}
and satisfying the bound
\begin{equation}
\sup_{\vec x, t}\intsum_{\{\zeta_1,\ldots,\zeta_N\}\ni (\vec
x,t)} {|w^\trunc(\zeta_1,\ldots,\zeta_N)| \over
|\zeta_1\cup\cdots\cup\zeta_N|}\;\le\;
\alpha_N
\label{s.17.1}
\end{equation}
where $\alpha_N$ is a constant of the order of
$$\sup \prod_{i=1}^N |w(\zeta_i)|,$$
the supremum being taken over all sets of $N$ contours (not
necessarily compatible).
\end{theorem}
\smallskip
[The notation $\{\zeta_1,\ldots,\zeta_N\}\ni (\vec x,t)$ means that
$(\vec x,t)$ belongs to the union of the supports of the quantum contours $\zeta_1,\ldots,\zeta_N$.]
>From this and the periodicity of the
contour ensemble one obtains an expression for the quantity $f$ defined in \reff{fff.1}.
\begin{corollary}\label{ccluster}
If
\begin{equation}
C\;\bydef\; \sup_{\widetilde\zeta}{1\over |\widetilde\zeta|}
\,\intsum_{\zeta{\incomp\; }\widetilde\zeta}
|w(\zeta)|\, e^{|\zeta|} \;<\;1\;,
\label{s.19}
\end{equation}
then the quantity $f$, defined in \reff{fff.1}, exists, and is given by
\begin{eqnarray}
f &=& -\sum_{N\ge 1}\, \frac{1}{|W|} \sum_{(\vec x,0) \in W}
\intsum_{\{\zeta_1,\ldots,\zeta_N\}\ni (\vec x,0)} {w^\trunc(\zeta_1,\ldots,\zeta_N) \over
|\zeta_1\cup\cdots\cup\zeta_N|} \nonumber\\
&=& \alpha',
\label{s.21}
\end{eqnarray}
where $W$ is a fundamental cell of the configuration and $\alpha'$ is a constant of order $C$.
Moreover,
\begin{equation}
\Bigl| | V| f + \log \Xi( V) \Bigr| \;\le\;
|\partial V|\, \OO(C)\;.
\label{s.22}
\end{equation}
\end{corollary}
The contour ensembles we are considering in this paper are all {\em
periodic}\/, rather than just translation-invariant. That is why in
\reff{s.17.1} we have taken a supremum over sites and in \reff{s.21} we have
summed over a fundamental cell $W$ of the configuration.
Nevertheless, the relevant estimates will be done majorizing the
contour weights via the Peierls condition \reff{s.11}. This majorizing
ensemble is then {\em translation-invariant}\/, and hence this supremun over
sites is superfluous in the key estimate that follows.
The limit $f$ can be interpreted as a free energy of the ensemble of
contours \reff{fff.40}. The coefficient on the RHS of
\reff{s.22} can be interpreted as a surface tension representing a
finite-volume correction.
\subsection{The key estimate}
\label{skey}
In view of Corollary \ref{ccluster} and
the freedom of rescaling discussed at the end of Section \ref{reslt}, we see
that the following lemma is the key step needed to prove Theorem
\ref{tresult}.
\begin{lemma}\label{lkey}
There exist $\lambda_0 >0$ and $\beta_0 <\infty$ such that, for
$\lambda\le\lambda_0$ and $\beta\ge\beta_0$,
\begin{equation}
\intsum_{\supp \zeta\ni (\vec 0,0)} \lambda^{|B(\zeta)|} \,
\exp[-J|\zeta|_\perp]
\;\le\; \OO(e^{-\beta J/2}) + \OO(\lambda)\;.
\label{s.23}
\end{equation}
\end{lemma}
\smallskip
\proof We use the fact that the integrand on the LHS of \reff{s.23}
depends on the sections $\Gamma_i$ and the quantum bonds
$\llB_i$ {\em only}\/ through their sizes, to write it as a sum of
contributions of ``entropy'' and ``energy'' factors.
We first decompose the LHS as
\begin{equation}
\intsum_{\supp \zeta\ni (\vec 0,0)} \lambda^{|B(\zeta)|} \,
\exp[-J|\zeta|_\perp]
\;=\; S^0 + S^{> 0}\;,
\label{f.imp}
\end{equation}
where $S^0$ is the contribution due to contours without transition
(perfectly cylindrical contours), and $S^{>0}$ is the rest. The bound
on $S^0$ is exactly as in the usual Peierls argument:
\begin{eqnarray}
S^0 &\le& \sum_{l\ge 1} \card\{\Gamma \,:\, |\Gamma|=l,
\supp\Gamma\ni\vec 0\}\,\,e^{-\beta Jl}\nonumber\\
&=& \OO(e^{-\beta J})\;.
\label{f.imp.2}
\end{eqnarray}
Regarding $S^{>0}$, we have:
\begin{eqnarray}
S^{>0}&\le& \sum_{n\ge 1} \, \sum_{(j_1,\ldots,j_n)}\,
\lambda^{j_1+\cdots +j_n}\,
\sum_{\scriptstyle (l_1,\ldots,l_n)
\atop \scriptstyle |l_i-l_{i-1}|\le j_i \; (l_0\equiv l_n)}
N(j_1,l_1,\ldots,j_n,l_n) \nonumber\\
&& {}\times\;
\int_0^\beta d\tau_1 \cdots \int_0^\beta d\tau_n \,
\indic \left[\beta \ge {\textstyle \sum_{i=1}^n}\tau_i\right]
\nonumber\\
& & \ \quad\exp\{-J\left[l_0 (\beta-{\textstyle \sum_{i=1}^n}\tau_i) +
l_1\tau_1+ \cdots + l_{n} \tau_n \right]\}\;,
\label{s.26}
\end{eqnarray}
where
\begin{equation}
N(j_1,l_1,\ldots,j_n,l_n) \;\bydef\;
\card\Bigl\{ (\Gamma_0,\ldots,\Gamma_n) \,:\,
\hbox{(a) and (b) below}\Bigr\}\;:
\label{s.27}
\end{equation}
\begin{itemize}
\item[(a)] $|\Gamma_i|=l_i$; $\Gamma_0=\Gamma_n$.
\item[(b)] There exists a sequence
$(\llB_1,\ldots,\llB_n)$ of quantum
bonds with $|\llB_i|=j_i$, such that there is a contour $\zeta$ formed by
the sections $\Gamma_i$ and the bonds $\llB_i$, with $\supp\zeta\ni (\vec
0,0)$. Conditions necessary for this property to be satisfied are:
\begin{itemize}
\item[(b.1)]
$\Gamma_i$ is obtained from $\Gamma_{i-1}$ by acting with some
$\phiq{\llB_i}$, with $|\llB_i|=j_i$, on $|\Gamma_{i-1}\rangle$.
\item[(b.2)] There exists one section $\Gamma_i$ or one quantum bond
$\llB_i$ such that $\vec 0\in\Gamma_i$ or $\vec 0\in \llB_i$.
\end{itemize}
%
\end{itemize}
We can distinguish a contribution due to ``entropy'' [the factor
$N(j_1,\ldots,l_n)$], and another due to ``energy'' (the exponential
and the powers of $\lambda$). To prove \reff{s.23}, we must show that
``energy'' overwhelms ``entropy''.
The two contributions on the RHS of
\reff{s.23} arise from two different types of quantum contours, namely the
long contours and the short contours defined in Section 3.3. A {\em long}\/ contour extends all the way from ``time'' zero to $\beta$, i.e., {\em none
of the sections $\Gamma_i$ are empty}\/. We shall denote the set of
such contours by $\ql$. A short contour has a ``time''-height strictly smaller than
$\beta$. It appears and disappears under the action of the quantum
interactions $\phiq{\llB_1}, \ldots, \phiq{\llB_n}$. Hence, under the action of {\it one} of these $n$
interactions, the section size of the
contour reduces to zero, i.e., there is one (and only one) value of $i$, [$1\le
i \le n$], for which
$$
\phiq{\llB_i}: \Gamma_{i-1} \longmapsto \Gamma_i, \quad {\rm with}\,\,\,|\Gamma_{i-1}| \ne 0 \,\,\,{\rm and}\,\,\, |\Gamma_i| = 0.
$$
The
set of short contours will be denoted by $\qs$.
\medskip
\noindent
{\em Bound for long contours}\/.
We start with the {\em entropy
bound}\/, that is the bound on $N(j_1,l_1,\ldots,j_n,l_n)$.
By condition (b.2) above,
\begin{equation}
N(l_0,\ldots,l_n) \;\le\; (l_{\rm max} + j_{\rm max})
\,\widetilde N(j_1,l_1,\ldots,j_n,l_n)\;,
\label{s.28}
\end{equation}
where $l_{\rm max}=\max_i l_i$, $j_{\rm max}=\max_i j_i$, and
$\widetilde N(j_1,l_1,\ldots,j_n,l_n)$ is the number of ``pinned''
contours, that is, contours with the given section and quantum bond
sizes for which $(\vec 0,0)$ is the first point (e.g.\ in
lexicographic order) of its support\, [$\vec 0 \in \Gamma_i, \,or \, \vec 0 \in B_i$].
To evaluate $\widetilde N$ we imagine that we ``construct'' the
quantum contour by starting from a section with minimal size $l_{min}$:
\begin{eqnarray}
\lefteqn{
\widetilde N(j_1,l_1,\ldots,j_n,l_n) \;\le\; }\nonumber\\
&& \sum_{\Gamma \in {\rm CC}(l_{\rm min},j_1,\ldots,j_n)}
{\cal N}_{\Gamma\to\Gamma}(j_{i_{\rm min}+1}, l_{i_{\rm
min}+1},\ldots, j_{i_{\rm min}-1}
l_{i_{\rm min}-1},j_{i_{\rm min}})\;.
\nonumber\\
\quad
\label{s.28.1}
\end{eqnarray}
Here, $i_{\rm min}$ satisfies $l_{i_{\rm min}}=\min_i l_i\bydef l_{\rm
min}\,$,
\begin{verse}
${\rm CC}(l,j_1,\ldots,j_n)\;\bydef\;\{\Gamma \,\colon\, |\Gamma|=l$,
and $\Gamma$ is a section of a quantum contour with $n$
transitions given by the actions of
$\phiq{\llB_1},\ldots, \phiq{\llB_n}\}$
\end{verse}
and
\begin{verse}
${\cal N}_{\Gamma_0\to\Gamma_n}(j_1,l_1,\ldots, j_{n-1},l_{n-1},j_n)
\;\bydef\;$
number of ways of choosing sections $\Gamma_1,\ldots,\Gamma_{n-1}$ of
areas $l_1,\ldots,l_{n-1}$ and quantum bonds
$\llB_1,\ldots,\llB_n$ of areas
$j_1,\ldots,j_n$ such that two consecutive sections
$\Gamma_i$ and $\Gamma_{i-1}$ of the
sequence $\Gamma_0,\Gamma_1,\ldots,\Gamma_{n-1},\Gamma_n$ are obtained from
each other by acting with $\phiq{\llB_i}$.
\end{verse}
By induction one can see that
\begin{equation}
{\cal N}_{\Gamma_0\to\Gamma_n}(j_1,l_1,\ldots, j_{n-1},l_{n-1},j_n)
\;\le\; (a^{2d})^n \prod_{i=1}^n (l_i+j_i)\,j_i\, {c_d}^{j_i}
\label{s.28.2}
\end{equation}
where $c_d$ is a dimension-dependent constant (the one for the
K\"onigsberg bridge lemma). The proof of this
fact is presented at the end of this subsection. Thus
\begin{eqnarray}
\lefteqn{
\widetilde N(j_1,l_1,\ldots,j_n,l_n) \;\le\; }
\nonumber\\
&&\card\Bigl({\rm CC}(l_{\rm min},j_1,\ldots,j_n)\Bigr)
(a^{2d})^n \prod_{i=1}^n (l_i+j_i)\,j_i\, c_d^{j_i}\;.
\label{s.29}
\end{eqnarray}
The bound on $\card\Bigl({\rm CC}(l_{\rm min},j_1,\ldots,j_n) \Bigr)$ must take care
of the fact that the section of area $l_{\rm min}$ need not be
connected [see discussion following \reff{s.9.3}], but its components
cannot be too dispersed . More precisely, property
\reff{s.9.3} implies that for each $l_{\rm min}$ there is a {\em
connected}\/ set formed by a number of plaquettes ranging from
$l_{\rm min}$ to $l_{\rm min}+j_1+\cdots+j_n$.
Therefore (by the K\"onigsberg bridge lemma), there exists a constant
$c_d\ge1$, depending only on the spatial dimension, such that
\begin{eqnarray}
\card\Bigl({\rm CC}(l_{\rm min},j_1,\ldots,j_n)\Bigr)&\le&
c_d^{l_{\rm min}} + \cdots + c_d^{l_{\rm min}+j_1+\cdots+j_n}
\nonumber\\
&\le& (j_1+\cdots+j_n+1)\,c_d^{l_{\rm min}+j_1+\cdots+j_n}\;.
\label{s.30}
\end{eqnarray}
Substituting \reff{s.28}, \reff{s.29} and \reff{s.30} in \reff{s.26} we
obtain the bound
\begin{eqnarray}
S^{>0}_{\ql}&\le& \sum_{n\ge 1} (a^{2d})^n \,
\sum_{(j_1,\ldots,j_n)\,:\, j_i\ge 1}\,(j_1+\cdots+ j_n+1)\,
\lambda^{j_1+\cdots +j_n} \nonumber\\
&&\sum_{\scriptstyle (l_1,\ldots,l_n)\,:\, l_i\ge 1
\atop \scriptstyle |l_i-l_{i-1}|\le j_i \; (l_0\equiv l_n)}
(l_{\rm max} + j_{\rm max})\,c_d^{l_{\rm min}}\,
\Bigl[\prod_{i=1}^n (l_i+j_i)\,j_i\, (c_d)^{2j_i}\Bigr]\,
R(l_1,\ldots,l_n)\;,\nonumber\\
\quad
\label{s.32}
\end{eqnarray}
with
\begin{eqnarray}
\lefteqn{R(l_1,\ldots,l_n) \;\bydef\;
\int_0^\beta d\tau_1 \cdots \int_0^\beta d\tau_n \,
\indic \left[\beta \ge {\textstyle \sum_{i=1}^n}\tau_i\right]}
\nonumber\\
&&{}\times \exp-J\left[l_0 (\beta-{\textstyle \sum_{i=1}^n}\tau_i) +
l_1\tau_1+ \cdots + l_{n} \tau_n \right]\;.
\label{s.32.1}
\end{eqnarray}
In order to obtain a bound on this last integral, the {\em energy bound}, we
proceed as follows: We use the bound
\begin{equation}
e^{-J\alpha\, l_i} \;\le \; e^{-J\alpha\, l_{\rm min}/2}
e^{-J\alpha\, l_i/2}\;,
\label{s.33}
\end{equation}
for $\alpha = \tau_1,\ldots,\tau_n,
(\beta-{\textstyle \sum_{i=1}^n}\tau_i)$, to extract an overall factor
$e^{-\beta J l_{\rm min}/2}$ outside the integral on the RHS. of \reff{s.32.1}. The remaining
integral is the same as the original one, but with $J$ replaced by $J/2$. By
neglecting the indicator function and the term proportional to $l_0$ in the
exponent, and extending the limits of integration to infinity,
one obtains
\begin{equation}
R(l_1,\ldots,l_n) \;\le\; \left({2\over J l_1}\right) \cdots
\left({2\over J l_n}\right) \, e^{-\beta J l_{\rm min}/2}\;,
\label{s.34}
\end{equation}
which, by \reff{s.32}, implies that
\begin{eqnarray}
S^{>0}_{\ql}&\le& \sum_{n\ge 1} \left({2 a^{2d}\over J}\right)^n \,
\sum_{(j_1,\ldots,j_n)\,:\,j_i\ge 1}\,(j_1+\cdots+ j_n+1)\,
\lambda^{j_1+\cdots +j_n} \nonumber\\
&&\sum_{\scriptstyle (l_1,\ldots,l_n)\; l_i\ge 1
\atop \scriptstyle |l_i-l_{i-1}|\le j_i \; (l_0\equiv l_n)}
(l_{\rm max} + j_{\rm max})\,
\left(c_d\,e^{-\beta J/2}\right)^{l_{\rm min}}\,
\Bigl[\prod_{i=1}^n (1+j_i)\,j_i\, (c_d)^{2j_i}\Bigr]\;.
\nonumber\\
\quad
\label{s.35}
\end{eqnarray}
The sum over the $l_i,\,1\le i\le n$, can be written purely in terms of $l_{\rm min}$
($\ge 1$) and $j_1,\ldots,j_n$. Indeed, for each $l_i$, there are only
$2j_i+1$ possible values for $l_{i+1}$. Hence, once $l_{\rm min}$ is given, the
sum over the remaining $l_i$'s yields an extra factor $\prod_i(2j_i+1)$. One
now notes that the maximum size, $l_{\rm max}$, of a section of the contour satisfies the bound
\begin{equation}
l_{\rm max} \;\le \; l_{\rm min} + j_1+\cdots + j_n .
\label{s.36}
\end{equation}
This is because the section of ``area'' $l_{\rm max}$ is obtained from the
section of ``area'' $l_{\rm min}$ by the action of {\it at most} $n$ quantum
interactions, $\phiq{\llB_1},\ldots,
\phiq{\llB_n}$, the latter corresponding to quantum bonds ${\llB_1},\ldots,
{\llB_n}$ of sizes $j_1,\cdots,j_n$.
Also,
\begin{equation}
j_{\rm max} \;\le \; j_1+\cdots + j_n\;.
\end{equation}
Thus
\begin{eqnarray}
S^{>0}_{\ql}&\le& \sum_{n\ge 1} \left({2 a^{2d}\over J}\right)^n
\sum_{(j_1,\ldots,j_n)\,:\,j_i\ge 1}\,(j_1+\cdots+ j_n+1)
\Bigl[\prod_{i=1}^n (1+2j_i)^2\,j_i\, (c_d)^{2j_i}\Bigr]
\lambda^{j_1+\cdots +j_n} \nonumber\\
&&{}\times \sum_{l_{\rm min}\ge 1}(l_{\rm min}+2j_1+\cdots+2j_n)\,
\left(c_d\,e^{-\beta J/2}\right)^{l_{\rm min}}\nonumber\\
&=& \OO(e^{-\beta J/2})\;.
\label{s.37}
\end{eqnarray}
\medskip
\noindent
{\em Bound for the short contours}\/.
There are two types of short contours. There is a ``collapsed'' type, corresponding to actions of all the $\phiq{\llB}$ that do not alter the configuration. By the exponential bound \reff{f.n.0}, the
contribution of these fluctuations to the expansion \reff{s.23}
is simply given by
\begin{eqnarray}
c\, \sum_{B\ni\vec 0}\, \lambda^{|B|} &\le& c\,\sum_{j\ge 1}
(c_d\lambda)^j \nonumber\\
&\le& \OO(\lambda)
\label{col.1}
\end{eqnarray}
The remaining short contours have a finite extension in the
``time''-direction. We shall assume, without loss of generality, that the quantum interaction $\phiq{\llB_1}$ reduces the section size of
the contour to
zero, i.e., we assume that $\Gamma_1 = \emptyset$ and hence $l_1 = 0$.
%Let us assume that it is the quantum interaction $\phiq{\llB_1}$ which reduces th%e section size of
%the contour to
%zero, i.e., we assume that $\Gamma_1 = \emptyset$ and hence $l_1 = 0$. [The
%other cases can be bounded analogously and will be taken into account be
%adding a factor $n$ at the end.]
%and exactly one empty section (if they had more, they would split into
%two contours). Thus, $n\ge 2$. Let us assume that $\Gamma_1=\emptyset$. Hence% $l_1 = 0$.
%; the other cases can be
%bounded analogously so we will just add a factor $n$ at the end.
The entropy bound can be obtained in a way similar to that for long contours, but with the following modifications:
\begin{itemize}
\item{$l_{\rm min}$ is the minimum of the sizes of the {\em non-empty}\/
sections, $l_2,\ldots,l_n (=l_0)$.}
\item{the ``area'' $l_1$ on the RHS of \reff{s.29} is zero.}
\item{The entire contour (which includes the sections of areas
$l_{\rm min}$ and $l_{\rm max}$), corresponds to the actions of
$\phiq{\llB_1},\ldots,\phiq{\llB_n}$. }
\end{itemize}
Hence
\begin{equation}
l_{max}, j_{max} \le j_1 + \ldots + j_n,
\end{equation}
which implies that
\begin{equation}
l_{max} + j_{max} \le 2(j_1 + \ldots + j_n).
\end{equation}
Moreover, in place of the bound \reff{s.30}, we have that
\begin{equation}
\card\Bigl({\rm CC}(l_{min},j_1,\ldots,j_n)\Bigr) \le (j_1 + \ldots + j_n) c^{(j_1 + \ldots + j_n)}
\end{equation}
With the above modifications, we obtain that
\begin{eqnarray}
\lefteqn{N(j_1,0,j_2,l_2,\ldots,j_n,l_n)}\nonumber\\
&\le& 2{(j_1 + \ldots + j_n)}^{2}\,c_d^{j_1+\cdots+j_n}\,
(a^{2d})^n\, \prod_{i=1}^n (l_i+j_i)\,j_i\, c_d^{j_i}\;.
\label{s.39}
\end{eqnarray}
In calculating the energy bound, we make use of the inequality
$e^{-J\alpha l_i} \le e^{-J /2}\,e^{-J\alpha l_i/2}$ for $i \ne 1$. Proceeding
as in the case of long contours, we obtain the bound
\begin{eqnarray}
R(0,l_2,\ldots,l_n) &\le& \left({2\over J l_2}\right) \cdots
\left({2\over J l_n}\right) \int_0^\beta d\tau_1\,
e^{-(\beta-\tau_1) J /2}\nonumber\\
&\le&
\left({2\over J l_2}\right) \cdots
\left({2\over J l_n}\right)
\left({2\over J }\right) \;,
\label{s.40}
\end{eqnarray}
where the last inequality results from extending the limit of integration to
infinity. Hence
%\begin{eqnarray}
%S^{>0}_{\qs} &\le& \OO(\lambda) +
%\sum_{n\ge 2} n\,\left({2 a^{2d}\over J}\right)^n
%\sum_{(j_1,\ldots,j_n)\;:\,j_i\ge 1}\,2(j_1+\cdots+ j_n)^2
%\nonumber\\
%&&\qquad\qquad {}\times
%\Bigl[\prod_{i=1}^n (1+2j_i)^2\,j_i\, (c_d)^{2j_i}\Bigr]
%\lambda^{j_1+\cdots +j_n} \nonumber\\
%&=&\; \OO(\lambda) + \OO(\lambda^2)\;.\qed
%\label{s.42}
%\end{eqnarray}
\begin{eqnarray}
S^{>0}_{\qs} &\le& \OO(\lambda) +
\sum_{n\ge 2} n \left({2 a^{2d}\over J}\right)^n
\sum_{(j_1,\ldots,j_n)\;:\,j_i\ge 1}\,2(j_1+\cdots+ j_n)^2
\nonumber\\
&&\qquad\qquad {}\times
\Bigl[\prod_{i=1}^n (1+2j_i)^2\,j_i\, (c_d)^{2j_i}\Bigr]
\lambda^{j_1+\cdots +j_n} \nonumber\\
&\le&\; \OO(\lambda) + \OO(\lambda^2)\;.\qed
\label{s.42}
\end{eqnarray}
\medskip
\noindent
{\it Proof of the claim \reff{s.28.2}}\/. The proof follows by induction in $n$.
The induction step is carried out as follows:
\begin{eqnarray}
\lefteqn{
{\cal N}_{\Gamma_0\to\Gamma_n}(j_1,l_1,\ldots,j_{n-1}, l_{n-1},j_n)
\;=\;} \nonumber\\
&&\sum_{\Gamma_{n-1}\in {\rm CC}_{l_{n-1},j_n}(\Gamma_n)}
{\cal N}_{\Gamma_0\to\Gamma_{n-1}}(j_1,l_1,\ldots,j_{n-2},
l_{n-2},j_{n-1}) \;,
\label{s.43}
\end{eqnarray}
where
\begin{verse}
${\rm CC}_{l_{n-1},j_n}(\Gamma_n)\;\bydef\;\{\Gamma \,:\,
|\Gamma|=l_{n-1}$,
and $\Gamma$ differs from $\Gamma_n$ through the action of some $\phiq{\llB_n}$
with $|\llB_n|=j_n\}$.
\end{verse}
By the inductive hypothesis
\begin{equation}
{\cal N}_{\Gamma_0\to\Gamma_n}(j_1l_1,\ldots,j_{n-1}, l_{n-1}) \;\le\;
\card\Bigl({\rm CC}_{l_{n-1},j_n}(\Gamma_n)\Bigr)
\,(a^{2d})^{n-1} \prod_{i=1}^{n-1} (l_i+j_i)\,j_i\, c_d^{j_i}\;.
\label{s.44}
\end{equation}
We have to consider two cases:
(i) $l_{n-1}\ge l_n$. In this case, the bond $B_n$ must intersect
$\supp \Gamma_{n-1}$. The number of such possibilities is bounded by the
product of the number of sites in $\Gamma_{n-1}$ ($=l_{n-1}a^d$),
the number of sites in $B_{n-1}$ ($=j_{i-1}a^d$) and the number of
bonds $\llB_n$ with $|B_n|=j_n$. The latter is less than or equal
to $c_d^{j_n}$, for some constant $c_d$ depending on the
dimension $d$.
(ii) $l_{n-1}m$. In this case we can replace the constant
$\varepsilon_0$ in \reff{g.1}
and \reff{g.2} by
$(\varepsilon_0)^m$.
\subsection{Stability of phases in the symmetric regime}
\label{ssres}
Here we prove Corollary \ref{cresult}. Our proof is based on the original
Peierls argument, but uses the cluster expansion technique to get around the
fact that weights may be negative and imaginary. We choose an observable ${\bf A}\in {\cal A}_D$.
>From the discussion of Section \ref{scexp} we have that
\begin{equation}
{\Xi_p^{\bf A}(\Lambda) \over \Xi_p(\Lambda)} \;=\;
\intsump_{\scriptstyle\zeta_{\bf A}\subset\widehat V}\,w(\zeta_{\bf A})
\, U_V (\zeta_{\bf A}),
\label{fst.1}
\end{equation}
where
\begin{equation}
U_V (\zeta_{\bf A}) \;\bydef\;
\frac {{\displaystyle\intsum\nolimits}_{\hspace{-0.5em}\scriptstyle\{\zeta_k\}
\subset\widehat V\atop\scriptstyle
{\rm non-linked}}
\biggl[\prod_k w(\zeta_k)\biggr]
\, \indic\Bigl[\zeta_k \hbox{ non-linked to } \zeta_{\bf A}\Bigr]}
{{\displaystyle\intsum\nolimits}_{\hspace{-0.5em}
\scriptstyle\{\zeta_k\}\subset\widehat V}
\biggl[\prod_k w(\zeta_k)\biggr]},
\label{fst.5}
\end{equation}
for $V=\Lambda\times[0,\beta]$. The quotient on the RHS of \reff{fst.5} is
amenable to the cluster-expansion technology: The
logarithm of the numerator produces clusters for
which no contour is linked with $\zeta_{\bf A}$, while the logarithm of the
denominator produces all clusters, with no restrictions. The quotient
corresponds to the clusters with at least one contour linked with $\zeta_{\bf A}$. Hence,
\begin{equation}
\log U_V (\zeta_{\bf A})\;=\;
\sum_{N\ge 1}\, \intsum_{\scriptstyle
\{\zeta_1,\ldots,\zeta_N\}\,\incomp\, \zeta_{\bf A}
\atop\scriptstyle \zeta_j\subset\widehat V}
w^\trunc(\zeta_1,\ldots,\zeta_N) \;,
\label{fst.10}
\end{equation}
and we have the bound
\begin{equation}
|U_{\zeta_{\bf A}}(V)| \;\le\; \exp\Bigl[|\zeta_{\bf A}|
\OO(\varepsilon_0)\Bigr]
\label{fst.15}
\end{equation}
{\em uniformly in}\/ $\Lambda$, and, due to the Peierls bound \reff{s.11.11},
\begin{equation}
|w(\zeta_{\bf A}) U_V(\zeta_{\bf A})| \;\le\; \frac{\|{\bf A}\|}{\lambda^{|D|}}\,
\widetilde\lambda^{|B(\zeta_{\bf A})|}
\, \exp[-\widetilde J|\zeta_{\bf A}|_\perp],
\label{ffs.11.11}
\end{equation}
again uniformly in $\Lambda$. We use the notation $\widetilde\lambda=\lambda e^{\OO(\varepsilon_0)}$
and $\widetilde J= J - \OO(\varepsilon_0)$.
As a consequence, expression \reff{fst.1} can be treated by
cluster-expansion methods, as the key estimate \reff{s.23} is
valid. The ``dominant'' contribution in \reff{fst.1} arises from terms for which
$\zeta_{\bf A}$ corresponds to $\langle s_p | {\bf A} | s_p \rangle$ and has a
support
$$ {\rm {supp}}\,\zeta_{\bf A} = D.$$
We obtain that
\begin{equation}
{\Xi_p^{\bf A}(\Lambda) \over \Xi_p(\Lambda)}
\;=\; \langle s_p | {\bf A} | s_p \rangle \, + S_1(\beta,\lambda)\,+\, S_2(\beta,\lambda)\;,
\label{fst.25}
\end{equation}
where $S_1(\beta,\lambda)$ is the contribution of short contours which intersect the
set $D$ at the ``time'' $\beta$, whereas $S_2(\beta,\lambda)$ is the contribution of
long contours intersecting $D$ at ``time'' $\beta$. The key estimate given in Lemma 6.5, and the fact that all the components of
$\zeta_{\bf A}$ must intersect some site in $D$, imply that
$S_1(\beta,\lambda)$ and $S_2(\beta,\lambda)$ are bounded by
$$
|D| \frac{\|{\bf A}\|}{\lambda^{|D|}}\,\OO(\varepsilon_0)\;.
$$
The expression \reff{fst.25} and the above bound together prove Corollary \ref{cresult}. They prove that
the quotients $\Xi_p^{\bf A}(\Lambda) / \Xi_p(\Lambda)$ are analytic
functions of $(e^{-\beta\tilde J}, \widetilde\lambda)$ in a region
\begin{equation}
\max(|e^{-\beta \tilde J}|,|\widetilde\lambda|) \;<\,
\widetilde\varepsilon
\label{f.reg1}
\end{equation}
where $\widetilde\varepsilon$ is a constant independent of $V$.
Each term of these series converges to the corresponding term of the
series \reff{fst.1} {\it without} the condition $\zeta_{\bf
A}\subset\widehat V$. In addition, the finite-volume series and this
infinite-volume limit are majorized by the same absolutely convergent
series. By dominated convergence, the limit $\Lambda\nearrow\zed^d$
of the series is the series of the limits, throughout the region \reff{f.reg1}, which, therefore,
is also the region of analyticity of the limit. We conclude that
the quantum expectations satisfy
\begin{eqnarray}
\langle {\bf A} \rangle_{\beta\,\lambda} &=&
\lim_{\Lambda\nearrow\zed^d}\, {\Xi_p^{\bf A}(\Lambda) \over \Xi_p(\Lambda)}
\nonumber\\
&=& \langle s_p | {\bf A} | s_p \rangle \,+\,
|D| \frac{\|{\bf A}\|}{\lambda^{|D|}}\,\OO(\varepsilon_0)\;.
\end{eqnarray}
This proves equation \reff{fst.0}.
\subsection{Differentiability of the expansion}
\label{sdiff}
Let us reintroduce the parameters $\underline \mu$ in the
interaction and hence in the weights $w$, and discuss the
consequences of the smoothness hypothesis (H1.2) (Section \ref{shyp}).
This hypothesis implies that the derivatives of the weights satisfy a
Peierls condition analogous to the one obeyed by the weights themselves, except for a
factor proportional to $|\zeta|$. This factor can be absorbed in a rescaling
of $J$ and $\lambda$; so we can assume that
\begin{equation}
\left| {\partial w_{\underline \mu}(\zeta) \over \partial \mu_i} \right|
\;\le\; \widetilde\lambda^{|B(\zeta)|}\, e^{-\beta\tilde J}.
\label{ffi.1}
\end{equation}
Therefore, from the preceding results, we conclude that the series formed
by the derivatives of the weights converge uniformly
in a small interval around each $\underline \mu\in {\cal O}$ (and
absolutely in $e^{-\beta \tilde J}$ and $\lambda$). This implies that
the series for the partition function can be differentiated term by
term, and thus it and its logarithm are differentiable functions of
$\underline \mu$. Using the cluster expansion we obtain
\begin{equation}
\left|{\partial \over \partial \mu_i}\log \Xi_p(V)\right|
\;\le\; |V|\, \OO(\varepsilon_0)
\label{ffi.5}
\end{equation}
and, writing $\Xi_p=\exp(\log\Xi_p)$,
\begin{equation}
\left|{\partial \over \partial \mu_i} \Xi_p(V)\right|
\;\le\; |V|\, |\Xi_p(V)|\, \OO(\varepsilon_0)\;.
\label{ffi.10}
\end{equation}
%\end{document}
These observations will be useful in Section \ref{stabd} below.
%\end{document}
%*************************
\section{Pirogov-Sinai theory for quantum perturbations}
\label{sps}
\subsection{Overview. The initial trick}
We now turn to the proof of the main Theorem \ref{t.bout} in the
general (non-symmetric) situation. This involves dealing with the
contour expansions \reff{s.14} [which we repeat in \reff{ss.14} below for the
reader's convenience] with a non-local compatibility
condition among contours (the matching of the labels of nested
interiors and exteriors). Following the standard approach, going back
to the original work of Pirogov and Sinai \cite{pirsin75,pirsin76},
the theory is constructed in two parts: First
(Section \ref{stabp} below), a criterion is established to determine
the stable phases for a {\em fixed}\/ interaction; second (Section
\ref{stabd}), the stability of the phase diagram as a whole is
determined. The parameters $\underline \mu$ play a role only in the
second part (and hence will not be displayed in the first
part).
In our treatment, we closely follow the
excellent presentation of Borgs and Imbrie \cite{borimb89}.
%---~the first to allow the contour weights to be signed and even
%complex-valued~--- which is based on the reformulation due to Minlos
%Zahradn{\'\i}k \cite{zah84}.
Our proofs are basically a transcription
of those in \cite{borimb89}, except for some small adaptations and
simplifications. The starting point of our proof is the formal expression \reff{s.14} for the
partition functions $\Xi_p(V)$ and, more generally, the expressions
\reff{fex.1} for $\Xi^{\bf A}_p(V)$. For the convenience of the
reader we repeat \reff{s.14} here:
\begin{eqnarray}
\lefteqn{
\Xi_p(V) \;=\; \exp\Bigl[-\int_V \,
e_x(s_p)\Bigr]}\nonumber\\[10pt]
&& {}\times \intsum_{\scriptstyle\{\zeta_k\}\subset\widehat V\atop{
\atop{\scriptstyle\;{\rm compatible}}}}
\biggl[\prod_k w(\zeta_k)\biggr]
\biggl[\prod_{u=1}^P \exp\Bigl\{-\int_{L_u} \,
\left[e_x(s_u)-e_x(s_p)\right]\Bigr\}\biggr]
\;, \nonumber\\
\ \label{ss.14}
\end{eqnarray}
where the exterior contours of each compatible family are $p$-contours. Both
quantities, the energies $e_x(s_u)$ and the weights $w(\zeta)$, are
complex-valued, and the latter satisfy the quantum Peierls bound
\reff{s.11}.
We follow the procedure, introduced by Minlos and Sinai
\cite{minsin67,minsin68}, to eliminate the inconvenient
compatibility condition in \reff{ss.14}. We first resum \reff{ss.14}
(formally!) over the contours in the interior of the exterior contours,
%--- insertion ---------
%that is we (formally) resum $\Xi_p(V)$ inside $\int_(\zeta^p)$, where
%$\zeta^p$ is an exterior contour of the set $\
\begin{eqnarray}
\lefteqn{
\Xi_p(V) \;=\; \exp\Bigl[-\int_V \,
e_x(s_p)\Bigr]}\nonumber\\[10pt]
&& {}\times \intsum_{\scriptstyle\{\zeta_k^p\}\subset\widehat V
\atop\scriptstyle\;{\rm exterior}}
\biggl[\prod_k w(\zeta_k^p)
\prod_{u=1}^P \Xi_u\Bigl(\inter_u(\zeta_k^p)\Bigr)
\exp\Bigl\{\int_{\inter_u(\zeta_k^p)} \;
e_x(s_p)\Bigr\}\biggr]
\;, \nonumber\\
\ \label{d.6}
\end{eqnarray}
and then multiply and divide the RHS by
$\Xi_p\Bigl(\inter_u(\zeta_k^p)\Bigr)$
to obtain
\begin{eqnarray}
\lefteqn{
\Xi_p(V)\, \exp\Bigl[\int_V \,
e_x(s_p)\Bigr]\;=\;}\nonumber\\[10pt]
&& \intsum_{\scriptstyle\{\zeta_k^p\}\subset\widehat V
\atop \scriptstyle\;{\rm exterior}}
\biggl[\prod_k W(\zeta_k^p)
\prod_{u=1}^P \Xi_p\Bigl(\inter_u(\zeta_k^p)\Bigr)
\exp\Bigl\{\int_{\inter_u(\zeta_k^p)} \;
e_x(s_p)\Bigr\}\biggr]
\;, \nonumber\\
\ \label{d.7}
\end{eqnarray}
with the new weights
\begin{equation}
W(\zeta^p) \;\bydef\; w(\zeta^p)
\prod_{u=1}^P {\Xi_u\Bigl(\inter_u(\zeta^p)\Bigr) \over
\Xi_p\Bigl(\inter_u(\zeta^p)\Bigr)}\;.
\label{d.8}
\end{equation}
One can now repeat the same procedure for each factor,
\begin{equation}
\Xi_p\Bigl(\inter_u(\zeta_k^p)\Bigr)
\exp\Bigl\{\int_{\inter_u(\zeta_k^p)} \;
e_x(s_p)\Bigr\},
\end{equation}
in \reff{d.7}. This iteration finally yields the expression
\begin{eqnarray}
\Xi_p(V) &=& \exp\Bigl\{-\int_V \;
e_x(s_p)\Bigr\}
\intsum_{\scriptstyle\{\zeta_k^p\}\subset\widehat V
\atop\scriptstyle{\rm non-linked}} \prod_k W(\zeta_k^p)\nonumber\\[-5pt]
&&\ \label{d.9}\\[-5pt]
&\bydef& \exp\Bigl\{-\int_V ;
e_x(s_p)\Bigr\}\; \widehat \Xi_p(V)
\nonumber
\end{eqnarray}
Identity \reff{d.9} yields an alternative formal expression
for the partition function $\Xi_p$ defined in \reff{ss.14}. If either one of
these two expansions converges uniformly, then so does the other. In
\reff{d.9}, however, the contours are only required to be
non-linked, rather than compatible in the sense used in \reff{ss.14}.
Hence, at least from a combinatorial point of view, \reff{d.9} is simpler
to deal with. Moreover, the factor $\widehat \Xi_p(V)$ is of the form
\reff{fff.40} and hence is amenable to the
cluster-expansion techniques of Section \ref{sbasic}.
However, one is confronted with the following problem: Whereas the original weights
$w(\zeta^p)$ satisfied the quantum Peierls condition \reff{s.11}, there is
no a priori bound on the new weights $W(\zeta^p)$, defined in \reff{d.8}. Hence, in order to prove the
convergence of the expansions in \reff{ss.14} and \reff{d.9}, we have to
devise a method to control the new weights. This is done in the following
section. The same considerations apply to $\Xi^{\bf A}_p$.
\subsection{Criterion for the stability of phases}
\label{stabp}
A sufficient condition for the stability of the $s_p$-phase is the
absolute convergence of the expansions for ${\rm log}\Xi_p(V)$, (with
$\Xi_p(V)$ as in \reff{d.9}) and of the analogous expansions for ${\rm log}\Xi^{\bf A}_p$.
>From the discussions in Sections \ref{sbasic} and \ref{scss} and the
similarity of the expansion \reff{d.9} of $\widehat \Xi_p(V)$ to that of $\Xi_p(V)$ in \reff{fff.40}, we conclude that
it suffices to check that the new weights $W(\zeta^p)$ satisfy a quantum
Peierls condition \reff{s.11}, provided $J$ is large enough and $\lambda$
is small enough. Definition \reff{d.8} of these weights implies in turn
that a sufficient condition for the Peierls condition to be valid is
that there exist some constant $z$ such that
\begin{equation}
\left| {\Xi_u\Bigl(\inter_u(\zeta^p)\Bigr) \over
\Xi_p\Bigl(\inter_u(\zeta^p)\Bigr)}\right| \;\le\;
\exp\bigl[ z|\partial \inter_u(\zeta^p)| \bigr],
\label{d.10}
\end{equation}
for all $u$ and all contours $\zeta^p$. Hence, by choosing a sufficiently
large $J$ and a small $\lambda$, we may attempt to ensure that the new weights $W(\zeta^p)$
are exponentially damped. The constant $z$ will be fixed once and for
all. Inspired by \cite{borimb89} we choose $z=4\alpha$, where
\begin{equation}
\alpha \;=\; 1 + a^d \max_{1\le u\le P} |e(s_u)|\;.
\label{d.ex.1}
\end{equation}
(If the configurations $s_u$ all have period $1$, i.e., are constant, $\alpha=1$ suffices.)
%[a careful look to the arguments below shows that this ``4'' is in
%fact $1+3\OO(\varepsilon_0)$, where
%``$\OO(\varepsilon_0)$'' comes from the cluster expansion
%of (truncated) free energies].
In the expansion of $\Xi^{\bf A}_p$ we also have to
consider the special surface $\zeta^p_{\bf A}$ which is a quantum contour {\it
associated with} ${\bf A}$ (defined in Section 3.6). To cover all cases, we
prove a bound of the form
\begin{equation}
\left| {\Xi_u(V) \over \Xi_p(V)}\right| \;\le\;
\exp\bigl[ 4\alpha|\partial V| \bigr]
\label{d.10.1}
\end{equation}
if $s_p$ is stable, for all regions $V\subset\zed^d\times[0,\beta]$.
We interpret \reff{d.10.1} as the condition for the stability of a space-time
region $V$. In particular, the bound \reff{d.10} is used to define stable contours.
Following \cite{zah84}, we require
\begin{definition}
\mbox{}
\begin{itemize}
\item[(i)] A region $V\subset\zed^d\times[0,\beta]$ is $p$-stable if
$\Xi_p(V)\neq 0$ and \reff{d.10.1} is satisfied for all $u$.
\item[(ii)] A $p$-contour $\zeta^p$ is stable if each
$\inter_u(\zeta^p)$ is $p$-stable, for $1\le u\le P$.
\end{itemize}
\end{definition}
It is evident that the weights $W(\zeta)$ of {\it stable} contours satisfy the
quantum Peierls condition. Hence, if we restrict the sums in \reff{d.9} to
stable $p$-contours then we can apply the cluster expansion technology of
Section \ref{sbasic}. This observation motivates us to define {\it truncated contour
partition functions}, as in \cite{zah84}:
\begin{eqnarray}
\Xi'_p(V) &\bydef& \exp\Bigl\{-\int_V \;
e_x(s_p)\Bigr\}
\intsum_{\scriptstyle\{\zeta_k^p\}\subset \widehat V
\atop{\scriptstyle{\rm stable}
\atop\scriptstyle {\rm non-linked}}} \prod_k W(\zeta_k^p)\nonumber\\[-5pt]
&&\ \label{d.11}\\[-5pt]
&\bydef& \exp\Bigl\{-\int_V \;
e_x(s_p)\Bigr\}\, \widehat \Xi'_p(V)\;,
\nonumber
\end{eqnarray}
The cluster expansion for the truncated partition function $\Xi'_p(V)$
converges absolutely. In particular, by Theorem \ref{tresult} we have that,
for $\beta$ large and $\lambda$ small, the {\em
truncated contour free energies}\/
\begin{equation}
f'_p \;\bydef\; -\lim_{V\nearrow\szed^d\times[0,\beta]} {1\over |V|} \log\Xi'_p(V)
\label{d.12}
\end{equation}
exist, and are of the form
\begin{equation}
f'_p \;=\; e(s_p) + \widehat f'_p,
\label{d.12.1}
\end{equation}
where $e(s_p)$ is the specific energy \reff{nil.1}, and the remainder $\widehat
f'_p$ is given by the cluster expansion \reff{s.21}, with $w$
replaced by the new weights $W$. Note that
\begin{equation}
\widehat f'_p \;=\; \OO(\varepsilon_0)\;,
\label{d.14.-1}
\end{equation}
by \reff{g.1}, where $\varepsilon_0$ is the constant appearing in Theorem \ref{tresult}, and thus
\begin{eqnarray}
\Bigl| |V| f'_p + \log \Xi'_p(V) \Bigr| &\le&
\Bigl| - \int_V \; e_x(s_p) + e(s_p) |V| \Bigr| +
\OO(\varepsilon_0) |\partial V| \nonumber\\
&\le& \alpha |\partial V|\;.
\label{d.14}
\end{eqnarray}
This bound is precisely the reason for our choice of $\alpha$. We also
observe that
\begin{eqnarray}
\Bigl| \int_V \; e_x(s_p) -f'_p |V| \Bigr|
&\le& \Bigl| - \int_V \; e_x(s_p) + e(s_p) |V| \Bigr|
+ |\widehat f'_p| \,|V|\nonumber\\
&\le& \alpha |V|\;.
\label{d.14.1}
\end{eqnarray}
We see that if, for a given boundary condition $s_p$, {\em all}\/
contours are stable then $f_p=f'_p$. More generally, if {\em all}\/
regions $V$ are $p$-stable we have that $(\Xi^{\bf A}_p)'=\Xi^{\bf
A}_p$ and the primed quantum expectations equal the unprimed
ones. (The prime indicates that the summation is over stable contours only.) The key observation of Pirogov-Sinai theory, in the
formulation due to Zahradn{\'\i}k \cite{zah84}, is that all regions $V$ are
$p$-stable if and only if the truncated free energy corresponding to the
boundary condition $s_p$ is {\it minimal}. Let
\begin{equation}
a_p \;\bydef\; \re f'_p - \min_{s_u\in\kk} \re f'_u\;.
\label{d.15}
\end{equation}
Then the stability criterion can be stated as follows:
\begin{theorem}
\label{tfunf}
If $a_p=0$ the $s_p$ phase is stable. Moreover, there is a region of
$\bbbc^2$ of the form $\max(|e^{-\beta \tilde J}|,|\lambda|)
< \varepsilon_0$ where the free-energy density and all the quantum
expectations are analytic functions of $e^{-\beta}$ and $\lambda$.
\end{theorem}
The key lemma needed in the proof of this theorem is the following:
\begin{lemma}\label{lfun}
%
The following statements are equivalent:
\begin{itemize}
%
\item[(i)] $a_p=0$.
%
\item[(ii)] All regions $V$ are $p$-stable.
%
\end{itemize}
%
\end{lemma}
\medskip
\noindent{\it Proof.\ }
We first prove that (ii)$\implies$(i), assuming that (i)$\implies$(ii)
holds. For this purpose, we consider a boundary condition $s_v$ for
which $a_v=0$. For each $V$, we have that $\Xi_p(V)=\Xi'_p(V)$, by
assumption, and $\Xi_v(V)=\Xi'_v(V)$ holds because (i)$\implies$(ii). Therefore
\begin{equation}
\exp\Bigl[4\,\alpha \OO(|\partial V|)\Bigr] \;\ge\;
\left| {\Xi_v(V) \over \Xi_p(V)}\right| \;=\;
\exp\Bigl[a_p\,|V| + \gamma
\,|\partial V|\Bigr]\;,
\label{d.16}
\end{equation}
where $\gamma$ is a constant of order $\varepsilon_0$. The leftmost
inequality expresses $p$-stability of the region $V$, while the equality on the
right is a consequence of \reff{d.14}. If $a_p>0$ the equation in \reff{d.16} leads to a contradiction for regions $V$ with
diverging volume-to-surface-area ratio. Thus we conclude that $a_p = 0$.
\medskip
\noindent{\it Proof of (i)$\implies$(ii).\ }
In order to understand the steps and definitions that follow, it is useful to inspect the ratio of partition
functions corresponding to different boundary conditions. From \reff{d.14} we have that, for any $s_v,
s_q\in\kk$,
\begin{equation}
\left| {\Xi'_v(V) \over \Xi'_q(V)}\right| \;=\;
\exp\Bigl[-(a_v-a_q)\,|V| +
\delta\,|\partial V|\Bigr]\;,
\label{d.17}
\end{equation}
where $\delta$ is bounded by the constant $\varepsilon_0$. From \reff{d.17} we conclude that if $a_q=0$ then
\begin{equation}
\left| {\Xi'_v(V) \over \Xi'_q(V)}\right| \;\le\;
\exp (|\partial V|),
\label{d.19}
\end{equation}
for large $\beta$ and small $\lambda$. Hence, in this case, the proof would be
complete if the truncated partition functions in \reff{d.19} could be replaced
by the untruncated ones and
\begin{equation}
\left| {\Xi_v(V) \over \Xi_q(V)}\right| \;\le\;
\exp ({\rm {const}}|\partial V|).
\end{equation}
More generally, for regions $V$ for which
\begin{equation}
a_q\,| V| \;\le\; |\partial V|\;,
\label{d.20}
\end{equation}
we have from \reff{d.17} that, for large $\beta$ and small $\lambda$,
\begin{equation}
\left| {\Xi'_v(V) \over \Xi'_q(V)}\right| \;\le\;
\exp (2\, |\partial V| ).
\label{d.21}
\end{equation}
As a first step, we would like
to show that the primes in \reff{d.19} and \reff{d.21} can be removed
for regions satisfying \reff{d.20}. If condition \reff{d.20} were
inherited by subregions of $V$ then we could prove inductively, from
\reff{d.21}, that $\Xi_p(V)=\Xi'_p(V)$. However, it is not true that the bound
\reff{d.20} remains valid for arbitrary subregions of $V$. Therefore it is convenient to resort to a sufficient
condition that has this hereditary feature. For this purpose, we introduce the
notion of {\it small regions} and {\it small contours}, adopting the definitions of \cite{borimb89}. For a
piecewise-cylindrical region $V$ of spatial sections $V_1,\ldots, V_n$,
we define the spatial diameter of $V$ as follows:
\begin{equation}
\sdiam V \;=\; \max_i \diam V_i\;.
\label{d.sdiam}
\end{equation}
\begin{definition}\mbox{}
\begin{itemize}
\item[(i)] A region $V$ is $q$-small if
\begin{equation}
a_q \sdiam \widehat V \;\le \; 1\;.
\label{d.22.r}
\end{equation}
\item[(ii)] A contour $\zeta$ is $q$-small if
\begin{equation}
a_q \sdiam \zeta \;\le \; 1\;;
\label{d.22.c}
\end{equation}
otherwise the contour is called $q$-{\em large}\/,
\end{itemize}
[where $\widehat V$ is as defined in \reff{vhat}].
\end{definition}
It is clear that smallness is inherited by subregions. Moreover, the bound
\reff{d.20} is valid for $q$-small regions, because
\begin{eqnarray}
a_q \,|V| &\le& a_q \sdiam V \, |\partial V|\nonumber\\
&\le& |\partial V| \;.
\label{d.23}
\end{eqnarray}
In particular, all contours inside a $q$-small region are $q$-small
contours. [It is for the sake of this property that we used $\widehat V$
in \reff{d.22.r}]. The hypothesis that $a_p=0$ implies that {\em all}\/
regions are $p$-small. As a consequence, the proof of the implication
(i)$\implies$(ii) is completed by
proving the following lemma \cite{zah84}:
\begin{lemma}\label{lstab}
For all $q$, $q$-small regions are $q$-stable.
As a consequence, all $q$-contours contained in $q$-small regions are
stable.
\end{lemma}
%Our previous remarks hint that the proof of this lemma is
%comparatively easy for regions so small that \reff{d.22.r} is true for
%all $q$ [see lines following \reff{d.21}], but for
%bigger regions one must deal with large contours in their interior.
%An argument due to Zahradn{\'\i}k, however, shows roughly that these
%large contours tend to pile up towards the boundary of
%the region available, and hence their contribution is a harmless
%exponential of the size of this boundary.
We prove this lemma by induction in the spatial diameters of the regions.
\newline
Let us assume that, {\em for all}\/ $u$, $a_u \sdiam V \le 1$ implies that
$\Xi_u (\widetilde V)\neq 0$ and
\begin{equation}
\left| {\Xi_v(\widetilde V) \over \Xi_u(\widetilde V)}\right| \;\le\;
\exp\bigl[ 4\alpha|\partial V| \bigr],
\label{d.25}
\end{equation}
for all $v$ and for all regions $\widetilde V$, contained in $V$,
with spatial diameter less than or equal to $m$. We pick some $s_q \in {\cal K}$
and some $q$-small region $\widehat V$ of spatial diameter $m+1$, and
prove the bound \reff{d.25}, with $u=q$. All contours $\zeta^q$
in this region are $q$-small, hence their interiors are $q$-small and
of spatial diameter strictly smaller than $m+1$. By
the inductive hypothesis such interiors satisfy \reff{d.25}, and hence
these contours are stable, yielding
\begin{equation}
\Xi'_q(V) \;=\; \Xi_q(V) \;.
\label{d.26}
\end{equation}
We remark that if $a_v=0$ then the proof is complete. This is because if
$a_v=0$ {\it all} regions are $v$-small, and, consequently, all $v$-contours
in $\widehat V$ are stable. This implies that $\Xi'_v(V) \;=\; \Xi_v(V)$,
which, along with \reff{d.26} and \reff{d.21}, implies that \reff{d.10.1} is
true.
Let us now consider a boundary condition $s_v$ for which $a_v \ne 0$.
To estimate $\Xi_v(V)/\Xi_q(V)$ we resort to maneuvres that are justified,
a-posteriori, by the proof of uniform
convergence. We start with expression \reff{d.6} for the partition
function of an ensemble of mutually exterior contours with exterior
configuration $s_v$, in a space-time volume $\widehat V$, and resum the
contribution of $v${\em-small exterior contours}\/. This yields
\begin{equation}
{\Xi_v(V) \over \Xi_q(V)} \;=\;
\intsum_{\scriptstyle\{\zeta_k^v\}\subset \widehat V
\atop{\scriptstyle v-{\rm large}
\atop\scriptstyle {\rm exterior}}}
{\Xi_v^{\rm small}(\exter) \over \Xi_q(V)}
\prod_k w(\zeta_k^v)\,
\exp\Bigl\{- \int_{\supp(\zeta_k^v)} \;
e_x(s_v)\Bigr\}\;
\Xi\Bigl(\inter(\zeta_k^v)\Bigr)\;.
\label{d.27}
\end{equation}
Here ``$\exter$'' is the region outside the $v$-large exterior contours
$\{\zeta^v_k\}$, the label ``small'' indicates a restriction to
configurations where all the exterior contours are $v$-small, and
$\Xi\bigl(\inter(\zeta^v)\bigr)\bydef\prod_{\tilde v}
\Xi_{\tilde v}\bigl(\inter_{\tilde v}(\zeta^v)\bigr)$.
If we multiply and divide the RHS of \reff{d.27} by
\begin{equation}
\Xi_q\bigl(\inter\bigr) \;\bydef\; \prod_k\prod_{\tilde v}
\Xi_q\bigl(\inter_{\tilde v}(\zeta^v_k)\bigr)\;,
\end{equation}
we obtain
\begin{equation}
{\Xi_v(V) \over \Xi_q(V)} \;=\;
\intsum_{\scriptstyle\{\zeta_k^v\}\subset \widehat V
\atop{\scriptstyle v-{\rm large}
\atop\scriptstyle {\rm exterior}}}
{\Xi_v^{\rm small}(\exter)\, \Xi_q(\inter) \,
\exp\Bigl\{-{\textstyle \sum_k\int_{\supp(\zeta_k^v)}} \;
e_x(s_v)\Bigr\} \over \Xi_q(V)} \prod_k Y(\zeta_k^v) \;,
\label{d.28}
\end{equation}
with
\begin{equation}
Y(\zeta_k^v) \;\bydef\; w(\zeta_k^v)\,
\prod_{\tilde v} {\Xi_{\tilde v}\bigl(\inter_{\tilde v}(\zeta^v_k)\bigr)
\over \Xi_q\bigl(\inter_{\tilde v}(\zeta^v_k)\bigr)}\;.
\end{equation}
We observe that, by the inductive hypothesis,
\begin{equation}
{\Xi'_v}^{\rm small}(\exter) \;=\; \Xi_v^{\rm small}(\exter)
\label{d.29}
\end{equation}
Identities \reff{d.26} and \reff{d.29} allow us to apply the finite-volume
bound \reff{d.14} to all the factors in \reff{d.28}, except $\prod_k Y(\zeta_k^v)$. We then obtain
\begin{eqnarray}
\lefteqn{
\left|{\Xi_v^{\rm small}(\exter)\, \Xi_q(\inter) \,
\exp\Bigl\{-{\textstyle \sum_k\int_{\supp(\zeta_k^v)}} \;
e_x(s_v)\Bigr\}
\over \Xi_q(V)}\right|
}\nonumber\\
&\le& \exp\Bigl[ - \re({f'_v}^{\rm small}-f'_q)\,
|V\setminus\inter| \,+\, 2 \alpha \,|\partial V|\Bigr]\,
\prod_k e^{(2d+1)\alpha|\zeta_k^v|}\;.
\label{d.30}
\end{eqnarray}
We have used \reff{d.14.1} and the geometrical bound
$|\partial \exter| + |\partial \inter| \le |\partial
V| + 2d\sum_k|\zeta^v_k|$.
We now use the $q$-smallness, inequality \reff{d.23}, of $V$ to bound
\begin{eqnarray}
- \re({f'_v}^{\rm small}-f'_q)\, |V\setminus\inter| &=&
(-a_v^{\rm small} + a_q) \, |V\setminus\inter|\nonumber\\
&\le& -a_v^{\rm small}|V\setminus\inter| + |\partial V|\;.
\label{d.31}
\end{eqnarray}
Furthermore, the quantum Peierls condition \reff{s.11} and
the inductive hypothesis \reff{d.25} for $u=q$ (combined with the bound
$|\partial \inter(\zeta_k^v)| \le 2d|\zeta_k^v|$) imply that
\begin{equation}
|Y(\zeta_k^v)| \;\le\; \lambda^{|B(\zeta_k^v)|}\,
e^{-J|\zeta_k^v|_\perp}\, e^{8d\alpha|\zeta_k^v|}\;.
\label{d.31.1}
\end{equation}
Substituting \reff{d.30}, \reff{d.31} and \reff{d.31.1} in \reff{d.28}, we get
the bound
\begin{eqnarray}
\left|{\Xi_v(V) \over \Xi_q(V)} \right| &\le&
e^{3\alpha |\partial V|}
\intsum_{\scriptstyle\{\zeta_k^v\}\subset \widehat V
\atop{\scriptstyle v-{\rm large}
\atop\scriptstyle {\rm exterior}}}
e^{-a_v^{\rm small}|V\setminus\inter|}
\,\prod_k \lambda^{|B(\zeta_k^v)|}\,
e^{-J|\zeta_k^v|_\perp}\, e^{(10d+1)\alpha|\zeta_k^v|}
\nonumber\\[10pt]
&:=& e^{3\alpha |\partial V|}
\intsum_{\scriptstyle\{\zeta_k^v\}\subset \widehat V
\atop{\scriptstyle v-{\rm large}
\atop\scriptstyle {\rm exterior}}}
e^{-a_v^{\rm small}|V\setminus\inter|}
\,\prod_k w^* (\zeta_k^v)\;.
\label{d.32}
\end{eqnarray}
%The weights $w^*$ can be required to satisfy the quantum Peierls condition by
%choosing a sufficiently large $J$ and small $\lamda$ (via the rescaling (3.44)%)?
To show that $e^{4\alpha |\partial V|}$ is an upper bound for \reff{d.32}, and
hence complete the proof of the lemma, it is convenient to follow \cite{zah84}
and consider the quantity
\begin{equation}
\widetilde \Xi_v^{\rm large}(V) := \intsum_{\scriptstyle\{\zeta_k\}\subset \widehat V
\atop{\scriptstyle non-linked}}\prod_k w^* (\zeta_k)\,e^{2d|\zeta_k|},
\end{equation}
where the label ``large'' indicates restriction to
configurations where all the exterior contours are $v$-large. This quantity can be
interpreted as the partition function of an ensemble of contours having
weights
\begin{equation}
\widetilde w(\zeta) \;\bydef\; w^*(\zeta)\, e^{2d|\zeta|}
\label{d.33}
\end{equation}
and confined to a space-time volume $ \widehat V$. It is evident that, for $\varepsilon_0$
small enough, the contour
weights $\widetilde w(\zeta)$ satisfy the quantum Peierls condition, and hence
the cluster expansion converges. Moreover, if $\widetilde f_v^{\rm large}$ is the corresponding free
energy density (defined as in \reff{s.20}) then it follows from Theorem 4.1 that
\begin{equation}
[\widetilde \Xi^{\rm large}_v(V)]^{-1} \;\le\;
e^{\widetilde f^{\rm large}_v|V|}\,
e^{\OO(\varepsilon_0)|\partial V|}\;,
\label{d.34.1}
\end{equation}
%-------------------------------
%In particular, if $V$ is given by $\inter(\zeta_k^v) := \sum_{\tilde
%v}\inter_{\tilde v}(\zeta_k^v)$, then b
%---------------------------------
%--------- to remove ---------------------%
%--------------------------------------------
We claim that
\begin{equation}
a_v^{\rm small} \;\ge\; -\widetilde f_v^{\rm large}\;.
\label{d.36}
\end{equation}
Indeed, by \reff{d.12.1} and Theorem \ref{tresult},
\begin{equation}
a_v^{\rm small} \;=\; a_v + \OO(C^{\rm large})\;.
\label{d.37}
\end{equation}
[see discussion after \reff{g.4}].
Moreover, for every $v$-large contour
\begin{eqnarray}
a_v \,(l_{\rm min}+j_1+\cdots+j_n) &\ge& a_v\,\sdiam\zeta
\nonumber\\
&>& 1\;,
\label{d.38}
\end{eqnarray}
where the first inequality follows from (the important) property
\reff{s.9.3}, and the second one is just the definition of largeness.
Therefore, by the final comment in Section \ref{sclp},
\begin{equation}
C^{\rm large} \;=\;
\OO\Bigl((\varepsilon_0)^{1/a_v}\Bigr)\;.
\label{d.39}
\end{equation}
By the same argument,
\begin{equation}
\widetilde f_v^{\rm large}\;=\;
\OO\Bigl((\varepsilon_0)^{1/a_v}\Bigr)\;.
\label{d.40}
\end{equation}
Hence
\begin{equation}
a_v^{\rm small} + \widetilde f_v^{\rm large} \;\ge\;
a_v + \OO\Bigl((\varepsilon_0)^{1/a_v}\Bigr)
\label{d.41}
\end{equation}
which is non-negative for $\beta$ large and $\lambda$ small, proving
\reff{d.36}.
For future purposes, we summarize the rest of the argument in the following
lemma which shows that \reff{d.36} causes the sum-integral in
\reff{d.32}
to yield at most a contribution exponential in the boundary.
By substituting the bound \reff{fsx.1.1}, shown below, into the RHS of \reff{d.32}, we obtain
the bound \reff{d.25}. This completes the inductive proof. \qed
\begin{lemma}\label{lzbi}
Consider weights $w^*(\zeta)$ satisfying a Peierls bound \reff{s.11}, and let $\widetilde f$ denote the contour free energy for the weights
$\widetilde w(\zeta) = w^*(\zeta)\, e^{2d|\zeta|}$ (well defined if
$\varepsilon_0$ is small enough). Then, for
$g\ge -\widetilde f$,
\begin{equation}
\intsum_{\scriptstyle\{\zeta_k\}\subset V
\atop\scriptstyle {\rm exterior}}
e^{-g|V\setminus\inter|} \,\prod_k w^* (\zeta_k)
\;\le\; \exp\Bigl[\OO(\varepsilon_0)|\partial V|\Bigr]\;.
\label{fsx.1.1}
\end{equation}
\end{lemma}
\proof (This is Lemma 3.2 of \cite{borimb89}. The proof given there
applies verbatim; we transcribe it for the sake of completeness.)
Multiply and divide the LHS of \reff{fsx.1.1} by $\widetilde \Xi_v^{\rm
large}(\inter) $. Using the analogue of \reff{d.34.1} for the region $\inter :=
\bigcup_{{\tilde v}, k}\inter_{\tilde v}(\zeta_k)$, and the bound $|\partial\inter| \le 2d
\sum_k |\zeta_k|$ we obtain
\begin{equation}
[\widetilde \Xi (\inter)]^{-1} \;\le\;
e^{\widetilde f|\inter|}\,
\prod_k e^{ 2d \sum_k |\zeta_k|}
\label{d.34}
\end{equation}
for $\varepsilon_0$ small enough.
Thus the LHS of \reff{fsx.1.1} satisfies
\begin{equation}
{\rm LHS} \;\le\;
\intsum_{\scriptstyle\{\zeta_k\}\subset \widehat V
\atop\scriptstyle {\rm exterior}}
e^{-g|V\setminus\inter|} \,e^{\widetilde f |\inter|}
\,\widetilde \Xi^{\rm large}(\inter)
\,\prod_k \widetilde w (\zeta_k)\;,
\end{equation}
and, since $-g\le \widetilde f$, we have that
\begin{eqnarray}
{\rm LHS} &\le& e^{\tilde f |V|}\,
\intsum_{\scriptstyle\{\zeta_k\}\subset \widehat V
\atop\scriptstyle {\rm exterior}}
\widetilde \Xi^{\rm large}(\inter)
\,\prod_k \widetilde w (\zeta_k)\nonumber\\
&\le& e^{\tilde f |V|}\, \widetilde \Xi^{\rm large}(V)\;.
\end{eqnarray}
Hence, by the analogue of \reff{d.34} for the region $V$,
\begin{equation}
{\rm LHS} \;\le\; e^{\OO(\varepsilon_0)|\partial V|}\;.
\qed
\end{equation}
\subsection{Stability of phase diagrams}
\label{stabd}
Finally we are in a condition to prove Theorem \ref{t.bout}.
The proof of this theorem involves two steps:
\begin{description}
\item[{\em Step 1:}]
Prove that the exponential damping of the original
weights $w$ and their derivatives implies an analogous damping of the
{\em new}\/ weights $W$ (for small contours) and of their derivatives.
\item[{\em Step 2:}]
Prove that these differentiability properties of the weights $W$ imply
that at low temperature and small $\lambda$, the contour free energies
$f'_p$ are so close to the energy densities $e(s_p)$ that
the manifolds defined by
\begin{equation}
\re f'_{p_1} = \cdots = \re f'_{p_k} < \re f'_{p_{k+1}}, \cdots ,
\re f'_{p_n}\;,
\label{sd.3}
\end{equation}
[i.e., \ ${\cal S}^{(\beta,\lambda)}_{\{s_{p_1},\ldots,s_{p_k}\}}$ defined in
\reff{sd.3}] are close to those defined in terms of the energies $e(s_p)$
[i.e., \ ${\cal S}^{(\infty,0)}_{\{s_{p_1},\ldots,s_{p_k}\}}$], and have differentiability properties similar to those of the weights $W$.
\end{description}
The proof of step 2, given step 1, is, {\em in principle}\/, an
exercise in implicit-function theorem technology. However, it is somewhat
subtle in cases, as the one we are interested in here, where the weights may
fail to be positive or (even) real. As
pointed out in \cite{borimb89}, there may appear zeroes of the
partition functions that destroy the continuity of the excess free
energies $a_v$.
In the sequel we shall only prove step 1; the proof of step 2 is a
straightforward adaptation of the argument given in \cite[Section
6]{borimb89} (replacing ``$\diam$'' by ``$\sdiam$'').
\begin{theorem}
Assume that there is a non-empty open set ${\cal O}\subset\IR^{P-1}$ such
that, for $\underline \mu\in{\cal O}$, the quantities
$e_{\underline\mu\,x}(s_v)$ and $w_{\underline \mu}$
are continuously differentiable functions of ${\underline{\mu}}$ and, moreover,
\begin{equation}
\left|w_{\underline \mu}(\zeta)\right| \;,\;
\left|{\partial w_{\underline \mu}(\zeta) \over \partial \mu_i}
\right| \;\le\; \lambda^{|B(\zeta)|}\,e^{-J|\zeta|_\perp},
\label{sd.1}
\end{equation}
for all contours $\zeta$, and
\begin{equation}
\left| e_{\underline\mu\,x}(s_v) \right| \;\le\; \alpha -1
\quad ; \quad
\left|{\partial e_{\underline\mu\,x}(s_v) \over \partial \mu_i}
\right| \;\le\; 1
\label{sd.2}
\end{equation}
for all $s_v\in{\cal K}$, $1\le i\le P-1$, for some $\alpha<\infty$.
Then there exists a constant $\widetilde J>0$ such that if
$\varepsilon_0 =
\max \bigl(e^{-\beta\tilde J}, \lambda\bigr)$ is sufficiently small,
we have that, for all $\underline \mu\in{\cal O}$ and all $q$-{\em
small} contours $\zeta^q$,
\begin{equation}
\left|W_{\underline \mu}(\zeta^q)\right| \;,\;
\left|{\partial W_{\underline \mu}(\zeta^q) \over \partial \mu_i}
\right| \;\le\; \widehat\lambda^{|B(\zeta^q)|}\,e^{-\hat J|\zeta^q|_\perp}
\label{sd.4}
\end{equation}
with $\widehat\lambda = \lambda e^{15 d\alpha}$ and
$\widehat J= J-15d\alpha$.
\end{theorem}
\noindent
{\it Remark}: Conditions \reff{sd.1} and \reff{sd.2} express the uniform boundedness
requirement of Hypothesis (H1.2). We can always rescale $J$, $\lambda$ and
the parameters $\mu_i$ such that there are no further constants in these
bounds.
\bigskip
\proof
Pick a small $q$-contour $\zeta^q$. The bound on $W(\zeta^q)$ is
immediate because of the stability of small $q$-contours (Lemma
\ref{lstab}). For the derivatives we would like to use the Leibnitz
formula,
\begin{eqnarray}
{\partial W(\zeta^q) \over \partial \mu_i} &=&
{\partial w(\zeta^q) \over\partial\mu_i}
\prod_{v=1}^P {\Xi_v\Bigl(\inter_v(\zeta^q)\Bigr) \over
\Xi_q\Bigl(\inter_v(\zeta^q)\Bigr)}\nonumber\\
&&{}+w(\zeta^q) \sum_{v\colon v\neq u} {\partial \over \partial \mu_i}
\left({\Xi_v\Bigl(\inter_v(\zeta^q)\Bigr) \over
\Xi_q\Bigl(\inter_v(\zeta^q)\Bigr)}\right)\,
\prod_{\tilde v\colon \tilde v\neq v}
{\Xi_{\tilde v}\Bigl(\inter_{\tilde v}(\zeta^q)\Bigr) \over
\Xi_q\Bigl(\inter_{\tilde v}(\zeta^q)\Bigr)}\;,
\nonumber\\
\quad
\label{sd.5}
\end{eqnarray}
and find suitable bounds for each term. This approach requires two
arguments: First, one must show that each of the factors is differentiable and, second, one must exhibit bounds on the derivatives.
The stability of $q$-small contours ---~ already proven above
~--- implies the correct
bounds for all factors $\bigl|\Xi_v\bigl(\inter_v(\zeta^q)\bigr) /
\Xi_q\bigl(\inter_v(\zeta^q)\bigr)\bigr|$, and the hypotheses take
care of the differentiability of $w(\zeta^q)$ and of the bounds on these
quantities and on $|\partial w(\zeta^q)/\partial \mu_i|$. What remains is to prove
the differentiability of the ratios
\begin{equation}
{\Xi_v\Bigl(\inter_v(\zeta^q)\Bigr) \over
\Xi_q\Bigl(\inter_v(\zeta^q)\Bigr)}
\end{equation}
and to find a bound on their derivatives.
It is easy to treat $\Xi_q$, because it only contains $q$-small, and
hence stable, contours, and we can apply the results of Section
\ref{sdiff}. For the numerators, however, we need to take into account
$v$-large, and hence possibly unstable, contours. It is imperative,
at this point, to work with the quotient $\Xi_v/\Xi_q$. In fact,
proceeding very much like in the proof of Lemma \ref{lstab}, one shows the following:
\medskip
\noindent
{\em Claim.}
If $V$ is $q$-small then, for any $v$, the quotient $\Xi_v(V)/\Xi_q(V)$
is differentiable, and
\begin{equation}
\left|{\partial \over \partial \mu_i}
\left({\Xi_v(V) \over \Xi_q(V)}\right)\right|
\;\le\; 5\, |V| \, e^{4\alpha|\partial V|}\;.
\label{sd.8}
\end{equation}
\smallskip
It is clear that this claim implies the proposed inequality
\reff{sd.4}. Indeed, inserting the hypotheses
\reff{sd.1}--\reff{sd.2}, the stability condition \reff{d.25} and the
claimed inequality \reff{sd.8} ---~for $V=\inter_v(\zeta^q)$~---
into \reff{sd.5}, we obtain
\begin{eqnarray}
\left|{\partial W(\zeta^q) \over \partial \mu_i} \right|
&\le& \lambda^{|B(\zeta^q)|}\,e^{-J|\zeta^q|_\perp}\,
e^{4\alpha|\partial\inter(\zeta^q)|}
\,\Bigr[1+5|\inter\bigl(\zeta^q\bigr)|\Bigr]
\nonumber\\
&\le& \lambda^{|B(\zeta^q)|}\,e^{-J|\zeta^q|_\perp}\,
e^{8d\alpha|\inter(\zeta^q)|} \, e^{7d|\zeta^q|}
\nonumber\\
&\le& \lambda^{|B(\zeta^q)|}\,e^{-J|\zeta^q|_\perp}\,e^{15d\alpha|\zeta^q|} \;,
\label{sd.9}
\end{eqnarray}
where we have used the bounds
$$|\partial\inter(\zeta^q)|\le 2d|\zeta^q| \quad ; \quad |\inter(\zeta^q)|
\le |\partial\inter(\zeta^q)|^2 $$
and $ 1+5x^2 \le \exp[7x/2]$. As $|\zeta| \le |B(\zeta)|+|\zeta|_\perp$, this last
bound implies \reff{sd.4}.
\medskip
To prove the claim we proceed, once again, by induction in the spatial
diameter of $V$. We assume that \reff{sd.8} is true for all $u$ and
for all $u$-small regions $\widetilde V$ with spatial diameter $m$ or
less. We shall now prove \reff{sd.8} for some $q$ and some $q$-small
region $V$ of diameter $m+1$.
We start with the resummed expression \reff{d.28} which we repeat for
the reader's convenience:
\begin{equation}
{\Xi_v(V) \over \Xi_q(V)} \;=\;
\intsum_{\scriptstyle\{\zeta_k^v\}\subset \widehat V
\atop{\scriptstyle v-{\rm large}
\atop\scriptstyle {\rm exterior}}}
{\Xi_v^{\rm small}(\exter)\, \Xi_q(\inter) \,
\exp\Bigl\{-{\textstyle \sum_k\int_{\supp(\zeta_k^v)}} \;
e_x(s_v)\Bigr\}
\over \Xi_q(V)}
\prod_k Y(\zeta_k^v) \;.
\label{sd.10}
\end{equation}
We shall use the product rule to calculate the $\mu_i$-derivative of the RHS. For the partition functions on the RHS we can use the
cluster-expansion technology, because only stable contours are
involved (Lemma \ref{lstab} for $\Xi^{\rm small}$, and the inductive
hypothesis for $\Xi_q$). Thus, we conclude differentiability and a bound
analogous to \reff{ffi.10}:
\begin{eqnarray}
\left|{\partial\over\partial\mu_i}\Xi_v^{\rm small}(\exter) \right|
&\le& \Biggl |\biggl[
\exp\Bigl\{-\int_{\exter} \; e_x(s_v)\Bigr\}\biggr]
\, {\partial\over\partial\mu_i}\widehat \Xi_v^{\rm small}(\exter)\Biggr|
\nonumber\\
&& \qquad {} +
\Biggl| {\partial\over\partial\mu_i}\biggl[-
\exp\Bigl\{\int_{\exter} \; e_x(s_v)\Bigr\}\biggr]
\, \widehat \Xi_v^{\rm small}(\exter)\Biggr|
\nonumber\\[10pt]
&\le& \Bigl|\Xi_v^{\rm small}(\exter)\Bigr| \,|\exter| \,+\,
\int_{\exter} \;
\left|{\partial e_x(s_v)\over\partial\mu_i}\right| \;
\Bigl|\Xi_v^{\rm small}(\exter)\Bigr|\nonumber\\[10pt]
&\le& 2\,\left|\Xi_v^{\rm small}(\exter)\right|\, |\exter|\;.
\label{sd.11}
\end{eqnarray}
The first inequality makes use of \reff{d.9}, the second one is due to
hypothesis \reff{sd.2}, and the third one follows from \reff{ffi.10}. Similarly,
\begin{equation}
\left|{\partial\over\partial\mu_i}
\left({\Xi_q(\inter)\over\Xi_q(V)}\right)\right|
\;\le\; 2\,\left|{\Xi_q(\inter)\over\Xi_q(V)}\right|\,
|V\setminus \inter|\;.
\label{sd.12}
\end{equation}
Moreover, from the differentiability of the energy-densities
and the bound \reff{sd.2},
\begin{eqnarray}
\lefteqn{
\left|{\partial\over\partial\mu_i}
\exp\Bigl\{-{\textstyle \sum_k\int_{\supp(\zeta_k^v)}} \;
e_x(s_v)\Bigr\}\right|}\nonumber\\
&&\;\le \; \sum_k|\zeta_k^v|\, \left|
\exp\Bigl\{-{\textstyle \sum_k\int_{\supp(\zeta_k^v)}} \;
e_x(s_v)\Bigr\}\right|\;.
\label{sd.13}
\end{eqnarray}
Finally, for the weights
\begin{equation}
Y(\zeta^v_k) \;=\; w(\zeta^v_k)
\prod_{u=1}^P {\Xi_u\Bigl(\inter_u(\zeta^v_k)\Bigr) \over
\Xi_q\Bigl(\inter_u(\zeta^v_k)\Bigr)}
\end{equation}
we use the hypotheses made on the original weights $w$ and the inductive
hypothesis. They imply that each factor is differentiable, and we can
use the bound:
\begin{equation}
\left|{\partial Y(\zeta^v_k) \over \partial \mu_i}
\right| \;\le\; \left|{\partial w(\zeta^v_k) \over \partial
\mu_i}\right| \, \prod_{u=1}^P
\left|{\Xi_u\Bigl(\inter_u(\zeta^v_k)\Bigr) \over
\Xi_q\Bigl(\inter_u(\zeta^v_k)\Bigr)}\right|
\,+\, w(\zeta^v_k)
\sum_{u=1}^P \left|{\partial \over \partial\mu_i}
\left({\Xi_u\Bigl(\inter_u(\zeta^v_k)\Bigr) \over
\Xi_q\Bigl(\inter_u(\zeta^v_k)\Bigr)}\right)\right|\;.
\end{equation}
Using the $q$-stability of the interior regions and the inductive
bound we obtain the upper bound
\begin{eqnarray}
\left|{\partial Y(\zeta^v_k) \over \partial \mu_i} \right|
&\le& \lambda^{|B(\zeta^v_k)|} e^{-J |\zeta^v_k|_\perp}\,
\Bigl[ e^{4\alpha|\partial \inter\zeta^v_k|} +
5 | \inter\zeta^v_k| e^{4\alpha|\partial \inter\zeta^v_k|}
\Bigr]\nonumber\\
&\le& \widehat\lambda^{|B(\zeta^v_k)|}
\,e^{-\hat J|\zeta^v_k|_\perp} \;=:\; W^*(\zeta^v_k) \;.
\label{sd.14}
\end{eqnarray}
[To obtain the last line we proceeded as in \reff{sd.9}.]
With the bounds \reff{sd.11}--\reff{sd.14} and the already known
bounds \reff{d.30}--\reff{d.31}, expression \reff{sd.10} yields
\begin{eqnarray}
\lefteqn{
\left|{\partial\over\partial\mu_i}
{\Xi_v(V) \over \Xi_q(V)}\right| \;\le\; e^{2\alpha|\partial V|}
\intsum_{\scriptstyle\{\zeta_k^v\}\subset \widehat V
\atop{\scriptstyle v-{\rm large}
\atop\scriptstyle {\rm exterior}}}
\exp\Bigl[ - \re({f'_v}^{\rm small}-f'_q)\,|V\setminus\inter|\Bigr]}
\nonumber\\[10pt]
&& {}\times \prod_k W^*(\zeta_k^v)\, e^{(2d+1)\alpha|\zeta_k^v|}
\,\Bigl[2\bigl(|\exter|+|V\setminus\inter|\bigr)\,+\,
\sum_k\bigl(|\zeta_k^v|+1\bigr)\Bigr]\;.
\label{sd.15}
\end{eqnarray}
The square bracket is bounded by
\begin{equation}
4\bigl|V\setminus\inter\bigr| + \sum_k 1 \;\le\; 5|V|\;,
\label{sd.16}
\end{equation}
and the remaining sum is bounded by $e^{\alpha|\partial V|}$, by
Lemma \ref{lzbi}.\qed
\section*{Acknowledgments} It is a pleasure to thank Claudio Albanese
and Luc Rey-Bellet for comments and criticism. R.F.\ is grateful to
Elliott Lieb, Nicolas Macris, Bruno Nachtergaele and Charles-Edouard
Pfister for very valuable discussions and suggestions, and to the
Institut f\"ur Theoretische Physik at ETH-H\"onggerberg and the
Department of Physics at Princeton University for hospitality while
this work was being performed. The research of R.F.\ was supported in
part by the Fonds National Suisse and by a John Simon Guggenheim Fellowship.
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\end{document}