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first-order transitions, lattice models, very nonlinear sigma models, 
liquid crystals, lattice gauge models
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\begin{document}

\title{Provable first-order transitions for nonlinear vector and gauge models with
continuous symmetries }
\author{Aernout C. D. van Enter\\Institute for Theoretical Physics, Rijksuniversiteit Groningen \\P.O. Box 800, 9747 AG Groningen, the Netherlands\\aenter@phys.rug.nl
\and Senya B. Shlosman\\CPT, CNRS Luminy, Case 907 \\F13288 Marseille Cedex 9 France, and \\IITP, RAS, Moscow 101477, Russia\\shlosman@cpt.univ-mrs.fr}
\maketitle

\begin{abstract}
We consider various sufficiently nonlinear vector models of ferromagnets, of
nematic liquid crystals and of nonlinear lattice gauge theories with
continuous symmetries. We show, employing the method of Reflection Positivity
and Chessboard Estimates, that they all exhibit first-order transitions in the
temperature, when the nonlinearity parameter is large enough. The results hold
in dimension $2$ or more for the ferromagnetic models and the $RP^{N-1}$
liquid crystal models and in dimension $3$ or more for the lattice gauge
models. In the two-dimensional case our results clarify and solve a recent
controversy about the possibility of such transitions. For lattice gauge
models our methods provide the first proof of a first-order transition in a
model with a continuous gauge symmetry.

\end{abstract}

\section{Introduction}

In this paper we prove a number of results showing that nearest neighbor
models with a sufficiently nonlinear, rotation-invariant, nearest-neighbour
interaction -- sufficiently nonlinear meaning that the nearest neighbour
interaction has the shape of a deep and narrow well -- show a first-order
transition in temperature. Part of our results have appeared in \cite{EntShl}.

We remind the reader that the first-order transitions occur when the free
energy density (or pressure) at some values of the thermodynamic parameters is
non-differentiable as a function of one of the parameters in the Hamiltonian.
In our examples this parameter will be the temperature. Equivalently, at these
parameter values different infinite-volume Gibbs measures exist which have
different expectation values for an observable dual to the
nondifferentiability parameter. In our examples this observable will be the
energy. For further aspects of Gibbs measure theory and the associated
thermodynamic formalism we refer to \cite{EFS, Geo1, GeoHagMae}.

An example of a model which has appeared in the literature and which can be
treated by our methods is given by the ferromagnetic Hamiltonian
\[
H=-J\sum_{{<}i,j{>}\in\mathbb{Z}^{2}}\left(  \frac{1+\cos\left(  \phi_{i}%
-\phi_{j}\right)  }{2}\right)  ^{p},
\]
with $p$ large.

Our results confirm earlier numerical work on this model \cite{DomSchSwe,
BloGuoHil}, which, however, has been contested by various authors. For some of
this literature, see \cite{vH,Kn,Wo,Mi,JMN,AF}.

Our analysis is not restricted to ferromagnets, but also applies to $RP^{N-1}$
(liquid-crystal) models (such as were first introduced by Lasher and Lebwohl
\cite{LasLeb,Las}) and to lattice gauge models (which are invariant under
local, as opposed to global, rotation symmetries).

We find that these nonlinear ferromagnetic, liquid-crystal and lattice gauge
models, (with either abelian or non-abelian symmetries) all have 1st order
transitions in the temperature.

The standard ferromagnetic ${N}$-vector models are either believed or
sometimes rigorously known to have 2nd order transitions in $d=3$ or higher, a
"Kosterlitz-Thouless" transition in $d=2,N=2,$ and no transition for $d=2$ and
higher $N$ \cite{convention}. In the XY-model $(N=2)$ for either $d=2$ or high
$d$ these results are rigorous, for the other models there is a consensus
based on both numerics and heuristic arguments.

In contrast, for the standard versions of the liquid crystal and lattice gauge
models, as well as for very non-linear ferromagnetic $\sigma$-models, both
numerics and high temperature series suggested the existence of 1st order
transitions, despite some theoretical and numerical analyses originally either
suggesting 2nd order transitions, no transitions at all, or
Kosterlitz-Thouless type transitions. Furthermore, the phase transition in the
3d liquid crystal models was observed to become more strongly first order when
a nonlinear term was added. For some of this literature, see e.g.
\cite{KunZum2,LasLeb,Las,EspTag,ABLS,Sol, CJR, Pes,Sav,ShSl,SokSta, DiG, RP,
Rom, Mag, MR, FPB, PP1,PP2} and references therein. Moreover, in the limit
where $N$ approaches infinity (the spherical limit) 1st order transitions were
found, in dimension $2$ or more \cite{KunZum1,SonTch,SokSta}. This spherical
limit result also holds for our nonlinear interactions in the ferromagnetic
case \cite{CarPel}. Whether such a first-order transition can also occur for
finite $N$ larger than $3$ in $d=2$, or whether it might be an artefact of the
spherical limit has for a long time been a matter of controversy (see for
example \cite{SokSta, SonTch}). In fact, Sokal and Starinets described the
existence of such a first-order transition as a \textquotedblleft
pathology\textquotedblright. 

Our result finally settles this question: first-order phase transitions for
models with continuous symmetry in $d=2$ can occur, despite the conjecture to
the contrary of \cite{SokSta}. Our results in $d=2$ are thus essentially in
agreement with the analysis of \cite{SonTch}. In contrast to what was
suggested in most earlier analyses, the symmetry or the low-temperature
properties of the model do not play a role of any great importance, and
neither do the nature of the topological excitations or the
spin-dimensionality. In fact, for our nonlinear choice of interaction the
spin-dimensionality $N$ does not need to be large and can be as small as $2$.
Also the lattice dependence of the phenomenon found in \cite{SonTch} seems
somewhat of an artefact which disappears if one varies the nonlinearity
parameter. The wide occurrence of first-order transitions in liquid-crystal
and lattice gauge models indicates that a proof in these type of models may be
of even more direct physical relevance than in the case of ferromagnets.

The main ingredient of our proofs is a similarity between such nonlinear
models and high-$q$ Potts models, which allows one to adapt proofs for Potts
models, such as were first developed in \cite{KotShl}, and based on
\cite{DS2}, to prove first-order transitions in the temperature parameter. We
remark that some similar results were found by L. Chayes \cite{CP}, also
making use of the Potts resemblance. See also \cite{AC}, where high
temperature uniqueness was proven, in almost the whole high-temperature
region. The arguments for showing the existence of first-order transitions,
such as have been used for Potts ferromagnets in $d$ at least $2$, apply to
the ferromagnetic and liquid-crystal models, those developed for Potts lattice
gauge models in $d$ at least $3$ \cite{KotShl}, apply to the lattice gauge models.

The fact that our proofs are insensitive to the nature of the phases between
which the transition takes place implies that one might have in the
ferromagnetic or liquid-crystal models a transition between a disordered
high-temperature phase and either a (ferromagnetically or nematically)
ordered, a Kosterlitz-Touless or a disordered phase at low temperatures.
Similarly one might find a transition either between a confining and a
nonconfining -- Coulomb-like -- phase or between two confining phases in the
lattice gauge models. Which one occurs in a particular case should depend on
dimension and{/}or symmetry of the system, but our methods do not provide
information on the low temperature regime, although in some cases known
methods may apply. Another consequence of our methods is that we show examples
where there are additional transitions between distinct ordered or distinct
Kosterlitz-Thouless low-temperature phases.

In particular, we emphasize that our proofs are also insensitive as to whether
the symmetry group of the lattice gauge model is abelian -- in which case it
is expected that in $4$ dimensions a transition between a confined and a
Coulomb-like phase occurs \cite{Guth, FS1} --, or nonabelian, in which case
both states are expected to be confining, (this is also expected in general in
$d=3$). For a heterodox discussion on the difference between what is to be
expected in abelian and nonabelian models, including some history of this
problem cf \cite{PatSei}.

\section{Notation and Results}

We consider a lattice $\mathbb{Z}^{d}$, and either spin models, in which the
random variables $\sigma_{i}$ live on the sites, or lattice gauge models,
where the variables live on the bonds (or links) of the lattice. The
parameters of our models are the spatial dimension $d$, the spin-dimension
$N$, and the nonlinearity parameter $p$. The \textquotedblleft
standard\textquotedblright\ versions of the models are obtained by taking
$p=1$.

For ferromagnetic models the variables are $N$-component unit vectors, living
on the sphere $\mathbb{S}^{N-1}$. We will present the argument in the
$2$-component case in detail, in the general case the proof is essentially the
same. In the $N=2$ case we also sometimes use angle variables $\phi_{i}$ to
denote the spins. The ferromagnetic models first considered in
\cite{DomSchSwe} and generalized to $N=3$ in \cite{BloGuoHil}, were given by
\begin{equation}
H=-J\sum_{{<}i,j{>}\in\mathbb{Z}^{2}}\left(  \frac{1+\cos\left(  \phi_{i}%
-\phi_{j}\right)  }{2}\right)  ^{p}.\label{21}%
\end{equation}


In fact the precise shape of the well-potential is not very important, as long
as it is narrow enough. For convenience we first present the argument for a
rectangular-well potential, and afterwards discuss the necessary adaptations
to treat general wells.

The property we will always need is Reflection Positivity (RP) \cite{RP}. This
will hold for all our examples, but, as is well known, it restricts us to
nearest-neighbour-cube interactions (C-interactions, in the terminology of
\cite{Geo}) and in particular prevents us to extend our proof to quantum spin models.

We denote our spin variables by $\sigma$, and we consider interactions $U$
which are nearest neighbor, and which contain only functions of inner products
between neighboring spins.

Thus the general form of our model is:%

\begin{equation}
H=-\sum_{{<}i,j{>}\in\mathbb{Z}^{d}}U(\sigma_{i}\cdot\sigma_{j}) \label{21a}%
\end{equation}


We will also use (by abuse of notation) the angle between neighboring spins as
the argument of the function $U$, as we will consider $U$-s which are rotation
invariant, and thus only depend on this angle.

When $U$ has a maximum at 1 and is a decreasing function of the cosine of the
angle $\phi$ between neighboring spins, the model is ferromagnetic.

We begin with the simplest case of\textbf{ }a square-well potential in
two-dimensions, with classical $XY$-spins.\textbf{ }The parameter
$\varepsilon$ describing the width of the well will play the same role of a
small parameter here as $\frac{1}{q}$ does in the $q$-state Potts model.

\begin{theorem}
Let $U(\phi)=1$ for $|\phi|\leq\varepsilon$, and $U=0$ otherwise, $d=2$ and
$N=2$. For $\varepsilon$ small enough this model has a first-order phase
transition in temperature. In particular, there exists a temperature where at
least two different Gibbs measures with different energy densities coexist.
\end{theorem}

\begin{proof}
We introduce projection $P_{b}^{o},$ which is the characteristic function of
the event that the bond $b$ is \textquotedblleft ordered\textquotedblright,
that is the angle between the two spins at the ends of the bond $b$ differ by
less than $\varepsilon,$ -- and $P_{b}^{d}$, the indicator of the event
\textquotedblleft$b$ is disordered\textquotedblright, that is the spins at the
ends of $b$ differ by more than $\varepsilon$. It is immediate that the
expectation of $P_{b}^{d}$ at high temperatures is close to one, and that the
expectation of $P_{b}^{o}$ is close to one at low temperatures. We need to
show that ordered and disordered bonds tend not to be neighbours at all
temperatures. Thus we need to estimate the expectation of $P_{b}%
^{d}P_{b^{\prime}}^{o}$, with $b,b^{\prime}$ two orthogonal bonds sharing the
same site. Once we have it, the proof follows, see \cite{Shl}.

For this it is sufficient to apply a chessboard estimate, following the
approach of \cite{DS2,KotShl}, going back to \cite{FL}, and described in e.g.
\cite{Shl,Geo}. We follow in particular \cite{Shl}. Our two-dimensional model
has RP property with respect to reflections in lines $\left\{  x\pm
y=k\right\}  ,$ $k=...,-1,0,1,...$ . We need to estimate the probability of
the occurrence of a \textquotedblleft universal contour\textquotedblright,
which in our case will be the set of configurations such that in a toroidal
volume $\Lambda$ consisting of $L^{2}$ sites, (where $L$ is a multiple of 4) a
quarter of the sites -- those belonging to diagonals chosen periodically at
distance 4 -- are surrounded by ordered bonds, and another quarter of the
sites, along the diagonals halfway, are surrounded by disordered bonds
(compare \cite{Shl}, figure 6). We have thus alternating diagonal strips of
ordered and disordered squares. Once we have obtained this estimate, we can
apply a Peierls-type contour argument.

To estimate this universal-contour probability, we have to estimate the
partition function over all configurations in which the universal contour
occurs, from above. We do it by noticing that for three quarters of the sites
one integrates over an interval of at most $\varepsilon$, and that half of the
($2\Lambda$) bonds contribute an energy $\beta,$ from which we obtain%
\[
Z_{L,univcont}(\beta,\varepsilon)\leq Cst\ e^{\beta L^{2}}{\varepsilon
}^{{\frac{3}{4}}L^{2}}e^{O(L)}.
\]
We estimate the full partition function from below by%
\[
Z_{L}(\beta,\varepsilon)\geq max[1,Z_{L}^{o}].
\]
The lower bound $1$ follows from the positivity of the function $U$.

In the bound for the ordered partition function $Z_{L}^{o},$ we can restrict
the integration over spin variables at each site, to the interval
$[-\frac{\varepsilon}{2},\frac{\varepsilon}{2}]$. This implies immediately
that
\[
Z_{L}^{o}\geq\varepsilon^{L^{2}}e^{2\beta L^{2}}%
\]
(which is larger than $1$ for $\beta\geq\beta_{0}=-\frac{ln\varepsilon}{2}$,
which in its turn gives an approximate value for the transition temperature),
so we obtain%
\[
\frac{Z_{L,univcont}(\beta,\varepsilon)}{Z_{L}(\beta,\varepsilon)}%
\leq\varepsilon^{O\left(  \frac{L^{2}}{4}\right)  }.
\]
This is immediate for  $\beta\geq\beta_{0}$, and for $\beta\leq\beta_{0}$ we
use the observation that $e^{\beta}\leq e^{\beta_{0}}=\varepsilon^{-\frac
{1}{2}}$.
\end{proof}

\textbf{Remark 1:}

In 2-dimensional models, the Mermin-Wagner theorem \cite{MerWag} and its
extension, the Dobrushin-Shlosman theorem \cite{DS1} (a recent version of
which also includes a non-continuous interaction such as the one under
consideration here \cite{ISV}), imply that all possible Gibbs measures are rotation-invariant.

\textbf{Remark 2:}

Of course, the above theorem can be extended to higher dimensions $d>2$. The
only difference in the proof would be that one can not use the RP in the
$45^{\circ}$ planes, so one uses instead RP\ in the coordinate planes and
their integer shifts, i.e. in the planes $\left\{  x_{i}=k\right\}  ,$
$i=1,2,...,d,$ $k=...,-1,0,1,...$ . That changes the definition of the
universal contour. As a result, the corresponding estimates becomes somewhat
weaker, but the conclusions of the theorem still hold. The generalization to
$N>2$ is immediate.

\bigskip

Our methods, combined with the technique of \cite{PecSh}, also imply that more
than one transition can occur, even infinitely many, at an infinite sequence
of lower and lower temperatures, between all paramagnetic ($d=2,n\geq3$), all
Kosterlitz-Thouless ($d=2,n=2$), or all ordered ($d\geq3$) phases.

Choose e.g. for the potential function $U$ a summable sum of characteristic
functions on fastly decreasing intervals : $U(x) = \sum_{n} 2^{-n}
1_{\varepsilon_{n}}(x)$, with $\varepsilon_{n} (= \varepsilon_{n-1}^{3})=
\varepsilon^{3^{n-1}}$, with the first $\varepsilon$ small enough.

Ground states for this interaction with wells in wells, which for obvious
reasons we will call the Seuss-potential \cite{Seuss} (see fig. 1), are
perfectly ordered, and the model is clearly ferromagnetic. At a sequence of
increasing inverse temperatures $\beta_{n}$ one has first-order transitions,
where one has at $\beta_{n}$ coexistence of Gibbs measures, one concentrated
on configurations with most bonds in well $n$, evenly distributed, and another
-- on configurations with most bonds in well $n+1$. The widths and depths of
the successive wells are chosen in such a way that the sequence of inverse
transition temperatures is growing like $\frac{3}{2}^{n}$.

\begin{theorem}
For $\varepsilon$ small enough, the above Seuss-model has an infinite set of
temperatures where first order transitions in the temperature occur, for all
$N \geq2, d \geq2$.
\end{theorem}

Our next step indicates how to generalize from rectangular wells to polynomial wells.

\begin{theorem}
Consider the model ($N=2,d=2$) with nearest-neighbor Hamiltonian%
\begin{equation}
H=-J\sum_{{<}i,j{>}\in\mathbb{Z}^{2}}\left(  \frac{1+\cos\left(  \phi_{i}%
-\phi_{j}\right)  }{2}\right)  ^{p}.
\end{equation}
For $p$ large enough this model has a first-order transition.
\end{theorem}

\begin{proof}
We employ the fact that for small difference angles $cos(\phi_{i}-\phi_{j})$
is approximately $1-O(\left[  \phi_{i}-\phi_{j}\right]  ^{2})$, and
furthermore that $lim_{p\rightarrow\infty}(1-\frac{1}{p})^{p}=\frac{1}{e}.$
This suggests to choose $\varepsilon(p)$ to be $\frac{1}{\sqrt{p}}$. The
difference with the rectangular-well model is that now the distinction between
ordered and disordered bonds becomes somewhat arbitrary, and we make a
slightly different choice, namely we call a bond ordered if%
\[
\left\vert \phi_{i}-\phi_{j}\right\vert \leq\varepsilon(p)\equiv\frac{C}%
{\sqrt{p}}.
\]


We will choose $C$ large, which implies that all disordered bonds have low
energy (close to zero), which we will need in the estimate on the upper bound
for the universal contour partition function.

We also introduce the notion of \textquotedblleft strongly
ordered\textquotedblright\ bonds, which have their energies close to maximal
energy: a bond $i,j$ is strongly ordered if%
\[
\left\vert \phi_{i}-\phi_{j}\right\vert \leq\frac{1}{C\sqrt{p}}.
\]
We will use them in estimating the ordered partition function $Z_{L}^{o}$ from below.

We then have, similarly to before, $Z_{L}(\beta,p)\geq max[1,Z_{L}^{o}]$. We
bound the ordered partition function $Z_{L}^{o}$ from below by integrating
over the spin variables at each site within the strongly ordered interval
$[-\frac{1}{C\sqrt{p}},\frac{1}{C\sqrt{p}}]$. There the energy is close to its
mininum, $-J$. Thus we obtain%
\[
Z_{L}(\beta,p)\geq\left(  \frac{1}{C\sqrt{p}}\right)  ^{L^{2}}e^{2\beta
J\left(  1-O\left(  \frac{1}{C^{2}}\right)  L^{2}\right)  }.
\]


For the $Z_{L,univcont}$ we obtain similarly as before,%
\[
Z_{L,univcont}(\beta,p)\leq e^{\beta JL^{2}[1+O(e^{-C^{2}})]\left(  \frac
{C}{\sqrt{p}}\right)  ^{\frac{3}{4}L^{2}+O(L)}}.
\]


The rest of the argument is identical to the one before, once we choose $C$
large enough. For example a choice $C=p^{\delta}$ for some small positive
$\delta$ will do.
\end{proof}

Again, generalizations to higher $N$ and $d$ are immediate.

\textbf{Remark 1}:

The transition, and our proof for it, persists if one applies a small external
field; thus it is immediately clear that no Lee-Yang circle theorem will hold,
in contrast to the standard ferromagnetic XY-models.

\textbf{Remark 2}:

We know that at low temperatures percolation of ordered bonds holds
\cite{Geo}; it follows from our results that the associated percolation
transition is also first order.

\bigskip

For liquid crystal $RP^{N-1}$ models one either considers variables -- usually
denoted $n_{i}$ -- which live on the projective manifold, obtained by
identifying a point on the $N$-sphere with its antipod, or equivalently one
can consider ordinary spins on the $N$-sphere, and divide out this
\textquotedblleft local gauge symmetry\textquotedblright\ afterwards. The last
approach is the route we will pursue, as it allows us to literally apply the
identical proof in the ferromagnetic and the liquid crystalline cases.

Thus we consider Hamiltonians of the form
\begin{equation}
H=-J\sum_{{<}i,j{>}\in\mathbb{Z}^{2}}\left(  \frac{1+\cos^{2}\left(  \phi
_{i}-\phi_{j}\right)  }{2}\right)  ^{p}.
\end{equation}
In the ferromagnetic case we called a bond \textquotedblleft
ordered\textquotedblright\ if the angle $\theta$ between two neighboring sites
is small enough. Here we call it ordered if the angle $\theta\mathrm{mod}\pi$
is small enough. Then the argument goes through without any changes. There is
a first-order phase transition for $p$ chosen large enough (in general the
values of $p$ for which the proof works depend on $N$ and $d$) between a
high-temperature regime, in which most nearest neighbor bonds are disordered,
and a low-temperature regime, in which most nearest neighbor bonds are
ordered. This holds for each dimension at least $2$, and whereas the
Mermin-Wagner theorem excludes nematic long-range order in $d=2$
\cite{MerWag}, in $d=3$ and higher long-range order it will occur
\cite{AngZag}. Between the ordered and the disordered phase(s) free energy
contours occur, whose probabilities are estimated to be uniformly small via a
contour estimate valid over a whole temperature interval. In the contour
estimate again use is made of the Reflection Positivity \cite{RP} of the model.

\begin{theorem}
For any nonlinear $RP^{N-1}$ model in dimension $2$ or more and $p$ high
enough, there is a first order transition, that is, there exists a temperature
at which the free energy is not differentiable in the temperature parameter.
In particular, there exists a temperature where at least two different Gibbs
measures with different energy densities coexist.
\end{theorem}

\smallskip

For lattice gauge models the variables are elements of a unitary
representation of a compact continuous gauge group, e.g. $U(1),\ SU(n)$, or
sums thereof \cite{Smi}. Here we present the argument in the simplest case of
a $U(1)$-invariant interaction in $3$ dimensions:%

\begin{equation}
H=-J\sum_{plaquettes\ P\in\mathbb{Z}^{3}}L\left(  U_{P}\right)  ,
\end{equation}
with $L(U_{P})=\left(  \frac{1+\cos\left(  \phi_{e_{1}}+\phi_{e_{2}}%
-\phi_{e_{3}}-\phi_{e_{4}}\right)  }{2}\right)  ^{p}.$ Here the $e_{i}$ denote
the 4 edges making up the plaquette $P$.

The effect of choosing the nonlinearity parameter $p$ high is again that the
potential, although it still has quadratic minima, becomes much steeper and
narrower. In this way one constructs in a certain sense a ``free energy
barrier'' between ordered and disordered phases.

The lattice gauge model proof becomes similar to the arguments from
\cite{KotShl}. When the product over the link variables is sufficiently close
to unity, we'll call the plaquette \textquotedblleft ordered\textquotedblright%
, \textquotedblleft disordered\textquotedblright\ otherwise. This distinction
corresponds to unfrustrated and frustrated plaquettes in the Potts case. We
will sketch the argument for the toy model where the potential $L(U)$ is
chosen to be $1$ if the sum $\phi_{P}$ of the oriented angles along the
plaquette $P$ is between $-\frac{\mathbb{\varepsilon}}{2}$ and $+\frac
{\mathbb{\varepsilon}}{2}$ and zero otherwise. The generalization to the
non-rectangular-well potentials can then be done in the same way as before.
The correspondence again is that ${\mathbb{\varepsilon}}$ is of order
$O(\frac{1}{\sqrt{p}})$.

Our strategy is to find bounds for free energy contours between ordered
phases, in which one has mainly cubes with 6 ordered plaquettes, and
disordered phases, in which most cubes have 6 disordered plaquettes. We need
thus to estimate the weights of contours consisting of cubes which are neither
ordered nor disordered. The number of possibilities for such cubes includes
the 7 possibilities given in \cite{KotShl}, plus we have now the additional
8th possibility of having cubes with one disordered plaquette and five ordered ones.

For the partition function $Z_{L}$ on a cube $B_{L}$ of size $L^{3}$ we use
the (quite rough) lower bound
\begin{equation}
Z_{L}\geq max(Z_{L}^{d},Z_{L}^{o}),
\end{equation}
where $Z_{L}^{d}$ (resp., $Z_{L}^{o}$) is part of the partition function,
calculated over all configurations which have all plaquettes disordered
(resp., mostly ordered). For the disordered partition function $Z_{L}^{d}$ we
obtain the lower bound $\left(  1-4\varepsilon\right)  ^{3L^{3}}$ (we take a
normalized reference measure, giving a weight $1$ to each link).

For the ordered partition function $Z_{L}^{o}$ we proceed as follows: we first
choose a set of bonds $T_{L}$ in $B_{L},$ which is a tree, passing through
every site. For example, we can put into $T_{L}$ all vertical bonds --
$z$-bonds -- except these connecting sites with $z$-coordinates $0$ and $1,$
plus all $y$-bonds in the plane $z=0,$ except these connecting the sites with
$y$-coordinates $0$ and $1,$ plus all $x$-bonds of the line $y=z=0,$ except
the one between the sites $\left(  0,0,0\right)  $ and $\left(  1,0,0\right)
.$ The site $\left(  0,0,0\right)  $ can be taken as a root of $T_{L}.$ Note
that the number of bonds in $T_{L}$ is $L^{3}-1.$ Therefore it is not
surprising (and easy to see) that for every edge configuration $\mathbf{\phi
}=\left\{  \phi_{b},b\in T_{L}\right\}  $ there exists a unique site
configuration $\mathbf{\psi}=\mathbf{\psi}_{\mathbf{\phi}}=\left\{  \psi
_{x},x\in B_{L}\right\}  ,$ such that the following holds:

\begin{enumerate}
\item Let $\mathfrak{g}^{\psi}$ denote the gauge transformation, defined by
the configuration $\psi.$ Then $\left(  \mathfrak{g}^{\psi}\circ\mathbf{\phi
}\right)  \Bigm|_{b}=1$ for every bond $b\in T_{L};$

\item $\psi_{\left(  0,0,0\right)  }=1.$
\end{enumerate}

\noindent For every family of bonds $S\subset B_{L}$ let us define a bigger
family $C\left(  S\right)  ,$ by the rules:

\begin{enumerate}
\item $S\subset C\left(  S\right)  ,$

\item for every four bonds $\left\{  b_{1},...,b_{4}\right\}  ,$ making a
plaquette, such that three of them are in $S,$ we have $\left\{
b_{1},...,b_{4}\right\}  $ $\subset C\left(  S\right)  .$
\end{enumerate}

\noindent Then we can consider also the sets $C^{2}\left(  S\right)  =C\left(
C\left(  S\right)  \right)  ,$ $C^{3}\left(  S\right)  ,$ and so on. Define
$\mathfrak{C}\left(  S\right)  =\cup_{k}C^{k}\left(  S\right)  .$ Note that
the number of plaquettes in $\mathfrak{C}\left(  T_{L}\right)  $ is
$3L^{3}-O\left(  L^{2}\right)  .$ We claim now that for every configuration
$\mathbf{\phi}_{T_{L}}=\left\{  \phi_{b},b\in T_{L}\right\}  $ one can specify
(in a continuous way) a collection of arcs

\noindent$\left\{  I_{b}=I_{b}\left(  \mathbf{\phi}_{T_{L}}\right)  \subset
S^{1},b\in\mathfrak{C}\left(  T_{L}\right)  ~\backslash~T_{L},\left\vert
I_{b}\right\vert =\frac{\varepsilon}{4}\right\}  ,$ such that for every
configuration $\mathbf{\phi}$ on $B_{L},$ which coincides with $\mathbf{\phi
}_{T_{L}}$ on $T_{L},$ and for which the values $\phi_{b}$ on the bonds
$b\in\mathfrak{C}\left(  T_{L}\right)  ~\backslash~T_{L}$ belong to the above
segments $I_{b},$ all the plaquettes that fall into $\mathfrak{C}\left(
T_{L}\right)  $ are non-frustrated. That would imply that
\[
Z_{L}^{o}\geq\left(  \frac{\varepsilon}{4}\right)  ^{2L^{3}}\exp\left\{
3J\left(  L^{3}-O\left(  L^{2}\right)  \right)  \right\}
\]
by Fubini's theorem. To see the validity of our claim, consider first the case
when the configuration $\mathbf{\phi}_{T_{L}}\equiv\mathbf{1}\in S^{1}$ (here
$1$ is the neutral element). Then the choice of the segments $I_{b}$ is easy:
$I_{b}\left(  \mathbf{1}\right)  =\left[  -\frac{\varepsilon}{8}%
,\frac{\varepsilon}{8}\right]  $ for every $b\in\mathfrak{C}\left(
T_{L}\right)  ~\backslash~T_{L}.$ For a general $\mathbf{\phi}_{T_{L}}$ let us
take the corresponding gauge transformation $\mathfrak{g}^{\mathbf{\phi
}_{T_{L}}}$ (which is the identity for $\mathbf{\phi}_{T_{L}}\equiv\mathbf{1}%
$), and we define our segments by
\[
I_{b}\left(  \mathbf{\phi}_{T_{L}}\right)  =\left(  \mathfrak{g}%
^{\mathbf{\phi}_{T_{L}}}\right)  ^{-1}I_{b}\left(  \mathbf{1}\right)  .
\]


This provides a lower bound%

\begin{equation}
Z_{L}\geq\max\left[  \left(  1-4\varepsilon\right)  ^{3L^{3}},\left(
\frac{\varepsilon}{4}\right)  ^{2L^{3}}\exp\left\{  3J\left(  L^{3}-O\left(
L^{2}\right)  \right)  \right\}  \right]  .
\end{equation}


This bound on the partition function as the maximum of the ordered and
disordered term is similar to the argument in \cite{EntShl}. It plays the same
role as the bound in terms of a fixed energy partition function given in
\cite{KotShl}.

To obtain our contour estimates, by Reflection Positivity we need to estimate
the partition functions of configurations constrained to have a
\textquotedblleft universal contour\textquotedblright. The estimates of the 7
types of universal contours mentioned in \cite{KotShl} are of a similar form
as in that paper with the number of Potts states $q$ replaced by $\frac
{1}{\varepsilon},$ up to some constant. The universal contour due to the new
case of cubes with one disordered plaquette consists of configurations in
which the horizontal plaquettes in every other plane are disordered, and all
the other ones are ordered. These configurations have a similar entropy
contribution to the partition function as the ordered configurations, but the
energy per cube is $\frac{5}{6}$ of that of a cube in the fully ordered
situation. For $\mathbb{\varepsilon}$ small enough (which corresponds to $p$
large enough) also such a contour is suppressed exponentially in the volume.
The combinatorial factor in the contour estimate changes by some finite
constant, which choosing $\mathbb{\varepsilon}$ small enough takes care of.

To summarize we have obtained the following result:



\begin{theorem}
For lattice gauge models with plaquette action $\left(  \frac{1+L(U_{P})}%
{2}\right)  ^{p}$, (where $L(U_{P})=Tr(U_{P}+U_{P}^{\ast})$) in dimension $3$
and more and $p$ high enough, there is a first order transition, that is there
exists a temperature at which the free energy is not differentiable in the
temperature parameter. In particular, there exists a temperature where at
least two different Gibbs measures with different energy densities coexist.
\end{theorem}

Here $U_{P}^{\ast}$ denotes the adjoint operator of $U_{P}$.

\section{Summary and discussion}

Our results provide a number of answers to questions which were raised before.
As we discussed in the introduction, the nonlinear two-dimensional
ferromagnetic models were studied numerically, and our results fully confirm
what was found in \cite{DomSchSwe,BloGuoHil}. Our work provides to our
knowledge the first case in which a first order transition for a lattice gauge
model with a continuous gauge symmetry group is rigorously obtained. Whereas
the example of the Potts lattice gauge model in $d=3$ or higher is between a
confining and a nonconfining phase \cite{KotLaaMesRui, LaaMesRui}, in our
theorem this is to be expected in $d=4$, with $U(1)$ symmetry only. For $d=3$
and also for $SU(n)$ in $d=4$ we conjecture that confined phases exist on both
sides of the phase transition.

Our proof only gives results for very high values of the nonlinearity
parameter $p$. We will discuss some further aspects of what may actually be
the $p$-values for which to expect first-order transitions, and what one might
hope to prove. The recent work of Biskup and Chayes \cite{BisCha} shows that
if a reflection positive model has a phase transition in mean field theory,
then also at sufficiently high dimension a first-order transition occurs. They
include in their discussion the $RP^{N-1}$ model for $N=3$, for which even for
the standard choice $p=1$ (so there is no strong nonlinearity in the
interaction), a first-order transition is derived. The mean field analysis of
\cite{BloGuo} indicates that a similar result for the ferromagnetic case holds
if $p=3$, and here a sufficiently strong nonlinearity is indeed needed. For
lattice gauge models on the other hand, also the standard (p=1) actions lead
to first-order transitions in mean-field theory (\cite{Zin}, section 34.4),
which indicates a first-order transition in sufficiently high dimension.

Similarly, if one believes that here the spherical ($N$ to infinity) limit is
not singular (which has been a matter of controversy itself), then for the
square lattice, $N$ large and $p$ larger than $6$ the sufficiently nonlinear
ferromagnet might have a first-order transition, while for the $RP^{N-1}$ case
on the square or triangular lattice even for $p=1$ a first order transition
occurs, although for the hexagonal lattice one presumably needs a higher value
of $p$ \cite{SokSta,SonTch}.

As mentioned before, numerical work suggests that in the standard ($p=1$)
Lebwohl-Lasher model with $N=3$ in $d=3$, as well as in the $U(1)$%
-lattice-gauge model in $d=4$, a first order transition should occur; however,
this seems far away from any provable result.

\textbf{Acknowledgement. }\textit{We thank in particular E.~Domany and
A.~Schwimmer who suggested to us to consider lattice gauge models, and also
L.~Chayes, D.~v.d. Marel, A.~Messager, K.~Netocn\'{y}, S.~Romano and A.~Sokal
for stimulating discussions and/or correspondence. S.S. acknowledges the
financial support of the RFFI grant 03-01-00444.}

\begin{thebibliography}{99}                                                                                               %


\bibitem {convention}The rigorous results about ferromagnetic $2$-component
models are described for example in \textquotedblleft Random Walks, Critical
Phenomena and Triviality in Quantum Field Theory\textquotedblright\ by
R.~Fern\'{a}ndez, J.~Fr\"{o}hlich and A.~D.~Sokal, Springer, 1992. A more
recent result on the Kosterlitz-Thouless point of the $2$-dimensional $XY$
model can be found in L. Chayes, \emph{Comm. Math.Phys.} 197, 623--640, 1998.

The description of critical (2nd-order transition) points and the behaviour of
their exponents as is expected in the physics literature can be found for
example in various contributions to Phase Transitions and Critical Phenomena,
Vol 6, Eds C.~Domb and M.~S.~Green, Academic Press, 1976.

\bibitem {ABLS}G.~Arnold, B.~Bunk,T.~Lippert and K.Schilling. \newblock
Compact QED under scrutiny: it's first order. \newblock arXiv:hep-lat{/}%
0210010, 2002.

\bibitem {AC}K.~S.~Alexander and L.~Chayes. \newblock Nonperturbative criteria
for Gibbsian uniqueness. \emph{Comm. Math. Phys.}, 189:447--464, 1997.

\bibitem {AF}G.~Alvarez and H.~Fort. \newblock Phase transitions driven by
vortices in two-dimensional superfluids and superconductors: From
Kosterlitz-Thouless to first order. \newblock {\em Phys. Rev. B}, 63:132504, 2001.

\bibitem {AngZag}N.~Angelescu and V.~A.~Zagrebnov \newblock A lattice model of
liquid crystals with matrix order parameter. \newblock{\em
J.Phys. A, Math. Gen.}, 15:L639--L643, 1982.

\bibitem {BisCha}M.~Biskup and L.~Chayes. \newblock Rigorous analysis of
discontinuous phase transitions via mean-field bounds. \newblock{\em
Comm. Math. Phys.} 238:53--93, 2003.
%mp-arc 02-325, arXiv Math.PR{/}0206232


\bibitem {BloGuo}H.~W.~J.~Bl\"ote and W.~N.~Guo. \newblock Phase transitions
in the two-dimensional O(3) model.
\newblock{\em Int.J.Mod. Phys. B},16:1891--1897, 2002.

\bibitem {BloGuoHil}H.~W.~J.~Bl\"ote, W.~N.~Guo and H.~J.~Hilhorst. \newblock
Phase transition in a Two-Dimensional Heisenberg Model. \newblock{\em
Phys. Rev. Lett.}, 88:047203, 2002.

\bibitem {CarPel}S.~Caracciolo and A.~Pelissetto. \newblock Two-dimensional
Heisenberg model with nonlinear interaction. \newblock{\em Phys. Rev. E}, 66:
016120, 2002.

\bibitem {CJR}M.~Creutz, L.~Jacobs and C.~Rebbi. \newblock Monte Carlo
computations in lattice gauge theories. \newblock{\em Phys. Rep.},
95:201--282, 1983.

\bibitem {CP}L.~Chayes, private communication.

\bibitem {DS1}R.~L.~Dobrushin and S.~B.~ Shlosman. Absence of breakdown of
continuous symmetry in two-dimensional models of statistical physics,
\textit{Comm. Math. Phys.} \textbf{42} (1975), 31--40.

\bibitem {DS2}R.~L.~Dobrushin and S.~B.~ Shlosman. \newblock Phases
corresponding to the local minima of the energy. \newblock{\em
Selecta Math. Soviet.} 1, 317--338, 1981.

\bibitem {DomSchSwe}E.~Domany, M.~Schick and R.~H.~Swendsen. \newblock First
Order Transition in an XY model with Nearest-Neighbor Interactions.
\newblock{\em Phys. Rev. Lett.} 52, 1535--1538, 1984.

\bibitem {EntShl}A.~C.~D. van Enter and S.~B.~Shlosman. \newblock First-Order
Transitions for n-Vector models in Two and More Dimensions: Rigorous Proof.
\newblock{\em Phys. Rev. Lett}, 89: 285702, 2002.

\bibitem {EspTag}D.~Espriu and L.~Tagliacozzo. \newblock Compact lattice U(1)
and Seiberg-Witten duality. \newblock arXiv:hep-th{/}0301086, 2003.

\bibitem {FL}J.~Fr\"ohlich and E.~H.~Lieb. \newblock Phase transitions in
anisotropic lattice spin systems. \emph{Comm.Math.Phys.},60:233--267, 1978.

\bibitem {FPB}A.~I.~Farinas Sanchez, R.~Paredes V., B.~Berche.
\newblock Evidence for a topological transition in nematic-to-isotropic phase
transition in two dimensions. \newblock arXiv:cond-mat{/}0301065, 2003.

\bibitem {FS1}J.~Fr\"ohlich and T.~Spencer. \newblock Massless Phases and
Symmetry Restoration in Abelian Gauge Theories.
\newblock{\em Comm.Math.Phys.},83:411--468, 1983.

\bibitem {Geo1}H.-O.~Georgii. \newblock{\em
Gibbs Measures and Phase Transitions}. \newblock Walter de Gruyter (de Gruyter
Studies in Mathematics, Vol.\ 9), Berlin--New York, 1988.

\bibitem {Geo}H.-O.~Georgii. \newblock Percolation for Low Energy Clusters and
Discrete Symmetry Breaking in Classical Spin Systems. \newblock{\em
Comm.Math.Phys.},81:455--473, 1981.

\bibitem {GeoHagMae}H.-O.~Georgii, O.~H{\"a}ggstr{\"o}m and C.~Maes. \newblock
The random geometry of equilibrium phases. \newblock Phase transitions and
critical phenomena (C. Domb and J.L. Lebowitz, Eds.), Academic Press, London,
to appear.

\bibitem {DiG}A~ Di Giacomo. \newblock Investigating QCD Vacuum on the
lattice. \newblock {\em Nucl.Phys. Proc. Suppl.} 108:21--28, 2002.

\bibitem {Guth}A.~H.~Guth. \newblock Existence proof of a nonconfining phase
in four-dimensional U(1) lattice gauge theory.
\newblock{\em Phys. Rev. D},21:2291--2307, 1980.

\bibitem {vH}J.~E.~van Himbergen. \newblock From continous to first-order
transition in a simple XY model. \newblock {\em Phys.Rev.Lett.}, 53:5--8, 1984.

\bibitem {ISV}D.~Ioffe, S.~B.~Shlosman and Y.~Velenik. \newblock 2D Models of
Statistical Physics with Continuous Symmetry: The Case of Singular
Interactions. \newblock {\em Comm. Math. Phys.}, 226:433--454, 2002.

\bibitem {JMN}A.~Jonsson, P.~Minnhagen and M.~N\'ylen. \newblock New critical
point for two-dimensional XY-type models. \newblock {\em Phys.Rev.Lett},
70:1327--1330, 1993.

\bibitem {Kn}H.~J.~F.~Knops. \newblock First-Order transition in the XY model.
\newblock{\em Phys. Rev. B}, 30:470--472, 1984.

\bibitem {KotShl}R.~Koteck\'y and S.~B.~Shlosman. \newblock First-order
transitions in large entropy lattice models. \newblock{\em Comm.Math. Phys.},
83:493--515, 1982.

\bibitem {KotLaaMesRui}R.~Koteck\'{y}, L.~Laanait, A.~Messager and J.~Ruiz.
\newblock The $q$-state Potts model in the standard Pirogov-Sinai Theory:
surface tensions and Wilson loops \newblock{\em J. Stat. Phys.}, 58:199--248, 1990.

\bibitem {KunZum1}H.~Kunz and G.~Zumbach. \newblock First-order phase
transitions in the two- and three-dimensional $RP^{n-1}$ and $CP^{n-1}$
models, in the large-$n$ limit. \newblock{\em J.Phys. A, Math. Gen.},
22:L1043--L1048, 1982.

\bibitem {KunZum2}H.~Kunz and G.~Zumbach. \newblock Topological phase
transitions in a two-dimensional nematic n-vector model: A numerical study.
\newblock{\em Phys. Rev. B}, 46:662--673, 1992.

\bibitem {LaaMesRui}L.~Laanait, A.~Messager and J.~Ruiz.
\newblock Discontinuity of the Wilson string tension in the $4$-dimensional
lattice pure gauge model. \newblock{\em Comm. Math. Phys.}, 126: 103--131, 1989.

\bibitem {Las}G.~Lasher. \newblock Monte Carlo Results for a Discrete-Lattice
Model of Nematic Ordering \newblock{\em Phys. Rev. A}, 5:1350--1354, 1972.

\bibitem {LasLeb}P.~A.~Lebwohl and G.~Lasher. \newblock Nematic-Liquid-Crystal
Order-A Monte Carlo Calculation. \newblock{\em Phys Rev A}, 6:426--429, 1972
and 7:2222, 1973.

\bibitem {Mag}D.~Magistrali. \newblock master's thesis, Pavia, 2002.

\bibitem {MerWag}N.~D.~Mermin and H.~Wagner. \newblock Absence of
Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Heisenberg
Models. \newblock{\em Phys. Rev. Lett.}, 17:1133--136, 1966.

\bibitem {Mi}F.~Mila. \newblock First-order versus Kosterlitz-Thouless
transition in a class of modified XY models. \newblock {\em Phys.Rev.B},
47:442--445, 1993.

\bibitem {MR}E.~Mondal and S~.K.~Roy. \newblock Finite-size scaling in the
planar Lebwohl-Lasher model. \newblock {\em Phys. Lett.A}312:397--410, 2003.

\bibitem {PatSei}A.~Patrascioiu and E.~Seiler. \newblock The Difference
between Abelian and Non-Abelian Models: Fact and Fancy.
\newblock arXiv:math-ph{/}9903038, 1999.

\bibitem {PecSh}E.~A.~Pecherski and S.~B.~Shlosman. \newblock Low-temperature
phase transitions in systems with one ground state. \newblock{\em
Theor. Math. Phys.}, 70: 325--330, 1987.

\bibitem {Pes}M.~Peskin. \newblock Mandelstam-'t Hooft duality in abelian
lattice models. \newblock{\em Ann. Phys.}, 113:122--152, 1978.

\bibitem {PP1}N.Priezjev and R.~A.~Pelcovits. \newblock Cluster Monte Carlo
Simulations of the Nematic-Isotropic Transition. \newblock \emph{Phys. Rev.
E}, 63:062702, 2001.

\bibitem {PP2}N.Priezjev and R.~A.~Pelcovits. \newblock Disclination loop
behavior near the nematic-isotropic transition. \newblock {\em Phys. Rev E},
64:031710, 2001.

\bibitem {PR}A.~Pal and S.~K.~Roy. \newblock Monte Carlo simulation of a
planar lattice model with $P_{4}$ interactions. \newblock {\em Phys.Rev.E},
67:011705, 2003.

\bibitem {Rom}S.~Romano. \newblock Computer simulation study of a nematogenic
lattice-gas model with fourth-rank interactions. \newblock {\em Int.J.Mod.
Phys. B}, 16:2901--2915, 2002.

\bibitem {RP}The method of reflection positivity was developed by F.~J.~Dyson,
J.~Fr\"ohlich, R.~B.~Israel, E.~H.~Lieb, B.~Simon and T.~S.~Spencer in a
series of papers. It is described in the last chapters of \cite{Geo} or in
\cite{Shl}.

\bibitem {Sav}R.~Savit. \newblock Topological Excitations in U(1) Invariant
Theories. \newblock{\em Phys. Rev. Lett.} 39:55--58, 1977.

\bibitem {Seuss}Dr.~ Seuss. \newblock The Cat in the Hat comes back. 1958.

\bibitem {Shl}S.~B.~Shlosman. \newblock The method of reflection positivity in
the mathematical theory of first-order phase transitions. \newblock{\em
Russian Mathematical Surveys}, 41: 83--145, 1986.

\bibitem {ShSl}P.~Shukla and T.~J.~Sluckin. \newblock Renormalisation group
study of a lattice model for the isotropic-nematic transition.
\newblock {J.Phys.A}, 18:93-100, 1985.

\bibitem {Smi}J.~Smit. \newblock Introduction to quantum fields on a lattice.
\newblock Cambridge Lecture Notes in Physics, 15, CUP, Cambridge, 2002. See in
particular Chapters 4 and 5.

\bibitem {SokSta}A.~D.~Sokal and A.~O.~Starinets. \newblock Pathologies of the
large-N limit for $RP^{N-1}, CP^{N-1}, QP^{N-1}$ and mixed isovector{/}%
isotensor sigma-models. \newblock{\em Nucl. Phys. B}, 601: 425--502, 2001.

\bibitem {Sol}S.~Solomon. \newblock Vortices and phase transitions in
nonabelian spin models. \newblock{\em Phys. Lett. B}, 100:492--496, 1981.

\bibitem {SonTch}O.~Tchernyshyov and S.~L.~Sondhi. \newblock Liquid-gas and
other unusual thermal phase transitions in some large-N magnets.
\newblock{\em
Nucl.Phys. B}, 639:429--449, 2002.

\bibitem {EFS}A.~C.~D. van Enter, R.~Fern{\'a}ndez, and A.~D. Sokal. \newblock
Regularity properties and pathologies of position-space renormalization-group
transformations: Scope and limitations of {G}ibbsian theory. \newblock{\em
J. Stat. Phys.}, 72:879--1167, 1993.

\bibitem {Wo}H.~Woldemariam. Augsburg master thesis, 2000.

\bibitem {Zin}J.~Zinn-Justin \newblock Quantum Field Theory and Critical
Phenomena, 4th edition. \newblock Clarendon Press, Oxford 2002.
\end{thebibliography}


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