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Contribution to the John T. Lewis memorial meeting, held in Dublin June 2005.
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first-order transitions, nonlinear n-vector models
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\title{First-order transitions for very nonlinear sigma models.}

\author{
  {\normalsize Aernout C.~D.~van Enter}        \\[-1mm]
  {\normalsize\it Centre for Theoretical Physics}   \\[-1.5mm]
  {\normalsize\it Rijksuniversiteit Groningen}         \\[-1.5mm]
  {\normalsize\it Nijenborgh 4}                \\[-1.5mm]
  {\normalsize\it 9747 AG Groningen}           \\[-1.5mm]
  {\normalsize\it THE NETHERLANDS}             \\[-1mm]
  {\normalsize\tt aenter@phys.rug.nl}        \\[-1mm]
\\ [-1mm]
  {\normalsize Senya B.~Shlosman}            \\[-1mm]
  {\normalsize\it Centre de  Physique Theorique}   \\[-1.5mm]
  {\normalsize\it CNRS}         \\[-1.5mm]
  {\normalsize\it Luminy, Case 907}                \\[-1.5mm]
  {\normalsize\it F13288, Marseille, Cedex 9}           \\[-1.5mm]
  {\normalsize\it FRANCE}             \\[-1mm]
  {\normalsize\tt shlosman@cpt.univ-mrs.fr}        \\[-1mm]
%\\ [-1mm]  \and
{\protect\makebox[5in]{\quad}}}
\pagenumbering{arabic}
\begin{document}
\maketitle \baselineskip=14pt \noindent {\bf Abstract.} In this
contribution we discuss the occurrence of first-order transitions in 
temperature in various short-range lattice models with a rotation symmetry. 
Such transitions turn out to be widespread under the condition 
that the interaction potentials are sufficiently nonlinear.

\bigskip

\noindent{\em In memory of John Lewis, who by his kind guidance and 
excellent scientific and professional sense has always been a 
stimulus and an inspiration.} 







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%\begin{document}





\maketitle
\section{Introduction}

According to the predictions of the 
Renormalisation Group (RG), ``universality'' holds. 
This means that in great generality 
the nature of the phase transition between high-temperature 
and low-temperature phases, and more 
in particular the corresponding critical exponents, will 
depend only on dimension, symmetry and the range
(short-range or long-range) of the interaction. The classical Landau mean field
theory similarly predicts that the nature of the sponataneously broken   
symmetry determines the order of the transition.     
Although in many cases such RG predictions have been confirmed, there are
some examples where, somewhat unexpectedly, 
first-order instead of the predicted second-order (or absence of any) 
transitions were observed, 
see e.g. \cite{BisChaCra,KunZum1,SokSta, SonTch}. 

In some cases, such as for the 
nearest-neighbour $q$-state Potts models, one might think that 
it is the nature of the broken (permutation) symmetry 
which seems to govern whether there is a 
first-order (at high $q$) or a second-order (at low $q$) 
transition. However, this is not really the case in general.
Indeed, as Onsager \cite{Fey} already knew, there seems to be no 
general method to predict whether a transition is first-order or second-order.
For example, an essentially short-range (large but more or less finite-range) 
2-dimensional 3-state Potts model with a first-order transition 
(as opposed to the nearest-neighbor second-order transition) 
was found in \cite{BisChaCra}. 

In this contribution, based on \cite{ES1, ES2}, we will present another 
class of models, which are invariant under rotation symmetries, 
but in having first-order transitions, differ from the 
``universal'', naively expected, second-order behaviour.  

For short-range ferromagnetic, $d$-dimensional, 
rotation-invariant n-vector models 
standard lore \cite{convention} predicts the following (universal) behaviour:

\smallskip

1) If $d=2$, there is at any temperature a unique Gibbs measure; 
for $n=2$ 
\noindent
( classical XY spins, in field theoretical language the 
``nonlinear sigma model'') at zero external field there is no spontaneous 
magnetization, but there is an infinite-order
transition between a low-temperature Kosterlitz-Thouless phase with slowly 
decaying correlations and a high-temperature phase with exponential 
correlation decay, for higher $n$ there is no phase transition.  

\smallskip

2) If $d=3$ or higher, there is a second-order transition between a magnetised
low-temperature phase and a high-temperature phase. If $d=3$, one 
has $n$-dependent critical exponents, in higher dimensions one obtains 
mean-field exponents. In an external field there is always 
a unique Gibbs measure.

\smallskip

Here we present a rather wide class of models in which this standard 
lore is violated.





% ----------------------------------\bibitem{Geo}

\section{ Notation and some background}
For  general background on the theory of Gibbs measures we refer
to \cite{EFS, Geo, GeoHagMae, Sin, Sim}. We will here mostly consider  spin
models, living on a finite-dimensional lattice $\bbZ^d$. The spins
will take  values in a circle ($n=2$) or be $n$-component unit vectors
on an $n$-sphere.


We will use small Greek letters $\si,\eta,\ldots$ to denote
spin configurations in finite or infinite sets of sites. The
nearest-neighbour Hamiltonians in a finite volume $\Lambda $, 
usually a $d$-dimensional torus, will be given by

\begin{equation}
H^{\Lambda}(\sigma) = \sum_{\langle i,j \rangle \subset
\Lambda} U( \sigma_i . \sigma_j) + 
\sum_{ i    \in \Lambda} h.\sigma_i, 
%+ \sum_{\langle i,j \rangle
%\atop i \in \Lambda, 
%j \in \Lambda^c} J'(i,j)\, \sigma_i \eta_j
\end{equation}

Associated to these 
Hamiltonians $H^{\Lambda}(\sigma)$ are Gibbs measures
\begin{equation}
\mu^{\Lambda}(d \sigma)= {\frac{1}{Z^{\Lambda}}} \exp  [-H^{\Lambda}(\sigma)]\mu_{0}^{\Lambda}(d \sigma)
\end{equation}

Here $\mu_{0}$ denotes the rotation-invariant product measure. 
The choice of the function $U$ of the inner product between 
the spins at neighbouring sites -- which take values on the interval 
$[-1,1]$ -- will determine our model. 
For our proofs we will always need that the interactions 
are {\em Reflection Positive}, which restricts us to interactions 
living on the unit cube, or interactions which are long-range of 
a particular form \cite{Geo,RP}, and we are restricted to studying the 
infinite-volume limits of the above periodic-boundary-condition measures. 

\noindent
The choice 
\begin{equation}
U(x)=-x
\end{equation}

\noindent
provides the standard classical XY and Heisenberg models.
Equivalently, as a function $W$ of the difference 
angle $\theta$ between neighbouring spins, this means choosing
\begin{equation}
W(\theta)= U(cos (\theta))= -cos(\theta)  
\end{equation}






\section{Results, and some remarks on proofs.}


In this section we describe our results on the nonstandard $n$-vector models. 

We start with the case of of zero field $h=0$ and $d=2, n=2$,
so we have classical $XY$ spins in two dimensions. This 
seems to be the first case which was considered in the literature 
\cite{DomSchSve} as an example of the 
phenomenon we display, although their arguments -- which we here prove to be correct -- were later contested 
\cite{Kno,vHim}.
The original choice of \cite{DomSchSve} was

\begin{equation}
W(\theta) = -{({1 \over 2}(1 + cos \theta))}^{p}
\end{equation}

\noindent
for p large enough.
 
We concentrate mostly on  a mathematically slightly simpler but essentially 
similar choice of model, which was introduced by \cite{AC}: 

\begin{equation}
W(\theta) = -1{[-\varepsilon, +\varepsilon]}(\theta)
\end{equation}

\noindent
where the parameter $\varepsilon$ will be chosen small.

Both these potentials have the form of a deep (depth $=1$) 
and narrow (width  $=O({\frac{1}{ \sqrt p}})$, 
cf width $=2 \varepsilon$) well, compared to the 
standard, rather shallow-well, cosine shape.

The second model is a square well (or a top hat) potential, in 
which  the distinction between being in or out of the well is 
unambiguous, in the first model there is a slight 
arbitrariness therein. 

First we notice that by the Mermin-Wagner theorem  
\cite{MerWag,DS2,IofShlVel, Pfi}, all Gibbs measures 
are rotation invariant, so that the spontaneous magnetisation 
is necessarily zero.
This does not prevent, however, the presence of multiple Gibbs measures (as 
was known already from  the model of \cite{Shl2}, 
where a discrete, chiral, symmetry was shown to be broken).

Our first result about the square-well model says that:

\smallskip

\noindent
{\bf Theorem}

\noindent
{\em For $\varepsilon$ small enough, 
there is a transition temperature where two  
Gibbs measures, an ``ordered'' one and a ``disordered'' one, coexist.
In the ordered state most bonds are ordered, in the sense that the two spins
at its ends have a  difference angle smaller than $\varepsilon$ 
(they are in the well), in the disordered state the opposite is true.}
 

\smallskip

{\bf Remarks about the proof}: The proof is a fairly straightforward 
application of the Reflection Positivity, chessboard estimates method, 
such as was developed by Dyson, Fr\"ohlich, Israel, Lieb, Simon \cite{RP}.  
One obtains a Peierls-type contour estimate by comparing any contour 
probability with and estimating the probability of a ``universal contour'', 
in which in our case at distance $2$ one finds alternating diagonals of 
``good'' ordered and ``good'' disordered sites (see \cite{Shl1}).  

Following \cite{DS1, KS}, we employ free-energy contours, rather than 
the more usual energy contours. Such a contour consists of a appropriately 
connected set of ``bad'' sites. A site is defined to be ``good'' if the 
bonds ending at it are either all ordered or all disordered, ``bad'' otherwise.
The free energy of a bad site will turn out to be 
approximately $\frac{3}{4}$ that of a good site, once 
$\varepsilon$ is small enough.  

To estimate the universal contour, one  has to integrate over all 
configurations such that the prescribed arrangement of 
ordered and disordered bonds, which has a positive density (one half) 
of bad sites, occurs. The proof follows a similar intuition as the proof 
of \cite{KS} for the large-$q$ Potts model. This similarity with the 
Potts problem was already remarked upon  in  \cite{DomSchSve}. 
The correspondence is that $\varepsilon$, (and the same holds for
$ \frac{1}{\sqrt p}$), correponds to $\frac{1}{q}$. 
To perform the estimate we observe that the total partition function
of an $N$-by-$N$ square satisfies
\begin{equation}
Z_N \geq max [1, {({\frac{1}{2\pi}} \varepsilon \ exp 2 \beta)}^{N^2}].
\end{equation}
Indeed, on the one hand, we use that the potential is positive, and on the 
other hand we get a lower bound by taking the integral 
at each site over the interval 
$[-\frac{1}{2}\varepsilon,  \frac{1}{2}\varepsilon]$, 
so that each bond is ordered.  

The restricted partition function $Z_{univcont}$ which is the
integral over all configurations compatible with 
the universal contour satisfies 

\begin{equation}
Z_{univcont} \leq {(2 \varepsilon)}^{\frac{1}{4} N^2} 
exp \beta({\frac{1}{2}N^2)} exp O(N) .  
\end{equation}

These estimates are valid on a temperature interval.
We conclude that somewhere on this interval there is a first-order 
transition in temperature  at some $\beta_{t}$, between an ordered and a 
disordered Gibbs measure, as was first numerically found in \cite{DomSchSve}.
The value of $\beta_{t}$ is approximately given by
$2\beta_{t}= - ln \varepsilon$.

The non-square-well model can be treated in a very similar way (see 
\cite{ES1,ES2} and also \cite{BiKo}). The ordered Gibbs measure 
has a polynomial spin-spin correlation decay of Kosterlitz-Thouless type.

\bigskip

{\bf Generalisations:}

1) The same method of proof works if either the spin dimensionality $n$ or the 
dimension $d$ of the lattice is larger than $2$ (or both). For the case of 
Heisenberg  spins ($n=3$) in $d=2$, the first-order transition was first 
found in \cite{BloGuoHil, BloGuo}. In this case  the low-temperature phase, as well as the high-temperature phase,  
presumably has exponential decay of the spin-spin correlations. 
For the $n \to \infty$ spherical limit see also \cite{CarPel, CarMoPel}. 

For $d \geq 3$, the Mermin-Wagner theorem 
does not apply anymore, and the low-temperature phase now displays 
a spontaneous magnetisation. 

\smallskip

2) In a small external field there still is a first-order transition between 
an ordered and a disordered phase, which now we expect to be both pure phases
(extremal Gibbs measures) also in higher dimensions.

\smallskip

3) Instead of a single well, one can also consider potentials having the shape 
of repeated wells in wells (or a hat-in-a-hat-in-a-hat...), 
which give rise to possibly infinitely many transitions.
A choice of such a Seuss (\cite{Seu}) potential is  $U(x) = - \sum_{n} 2^{-n}
1_{\varepsilon_{n}}(x)$ with ${\varepsilon_{n}}(= \varepsilon_{n-1}^{3})=
\varepsilon^{3^{n-1}}$, with the first $\varepsilon$ small enough. Such 
transitions  in which one keeps jumping in deeper and deeper wells, can 
occur either between nonmagnetised-nonmagnetised,  magnetised-nonmagnetised 
or magnetised-magnetised Gibbs measures, and the nonmagnetised measures 
may display either exponential or Kosterlitz-Thouless decay.     

\smallskip

4) Instead of ferromagnetic models the argument also works for nematic
liquid crystal $RP^{n}$ models \cite{Las, LasLeb}, 
in which one considers interactions of the form $U(x)=-x^{2p}$ 
for which there are two minima on the interval $[-1,+1]$. 
Here even for $p=1$, and $n=3$, first-order transitions were found 
numerically (see e.g. \cite{KunZum1,KunZum2,Ro}) in $d=3$, whereas the 
occurrence of a transition in the limit $n \to \infty$ in $d=2$ has 
been a  matter of controversy \cite{SokSta, SonTch}. Additional 
numerical references are  mentioned in \cite{ES2}.
Further models of this type, with a larger number of 
sharp minima, combined with a term causing  a chiral symmetry breaking 
as in \cite{Shl2}, are considered in \cite{MeNa}.   

\smallskip

5) Similarly as for the Potts gauge model of \cite{KS}, we can prove the 
existence of a first-order transition in various nonlinear lattice 
gauge models with continuous symmetries. In some of such models too
first-order transitions were concluded  on the base of numerical data 
(see e.g. \cite{EspTag,ABLS,Smi}). 
In \cite{ES2} we provide the first occasion where a first-order 
transition for a lattice gauge model in the presence of a continuous 
symmetry can be proven.






\section{Conclusions and Comments}

In many cases, some of them also of direct physical interest, 
first-order transitions occur instead of the second-order transition which a 
naive universality argument would predict. Such first-order transitions tend  
to occur more easily when either the nonlinearity parameter $p$, the 
spin-dimensionality $n$ -- the sperical limit --, or the dimension $d$  
--the mean-field limit -- \cite{BisCha} of the lattice is large.  

In this paper we have presented a number of models where we can prove these 
transitions for sufficiently strong nonlinearities.
Furthermore, if one adds an external field, the transition persists.
Although the two phases on both sides of the transition can have different
characters (breaking other symmetries, having polynomial or exponential decay
of the spin-spin correlations, etc), it can be shown that no ``intermediate''
phases exist (there is a ``forbidden gap'' for the energy variable 
\cite{BiKo}). 

The method of Reflection Positivity we use has the disadvantage that one is 
limited in the  interactions one can take, in the sense that they need to be 
defined on the unit cube. As compared to the more robust Pirogov-Sinai 
contour methods, RP methods have the advantage, however, that they are more 
generally applicable in that we need no information about the phases on both 
sides of the transition, whereas for Pirogov-Sinai methods the phases need to 
be pure and will typically have some kind of exponential decay of correlations.
On the other hand, with RP methods we cannot obtain results for interface 
or surface properties, or about the completeness of the phase diagram.  

If one varies the nonlinearity parameter in $d=2$,
one moves towards a critical point where the second-order transition is 
expected to have Ising characteristics, in higher dimensions varying the 
nonlinearity parameter will lead one to a tricitical point.   

Although our proofs require a fairly large nonlinearity (that is, a large
value of $p$, or a small value of $\varepsilon$), either by numerical methods
or in the large-$n$ or large-$d$ limit we expect and sometimes know 
that the type of first-order transitions we studied are occurring, especially
for the liquid-crystal models,  for much smaller values of $p$,$n$, 
and{/}or $d$. 


\bigskip

\noindent 
{\em Acknowledgements} During this work we had very stimulating input from 
various colleagues, in particular L.~Chayes who developed independently a 
number of these ideas, and E.~Domany and A.~Schwimmer who urged us to also 
consider lattice gauge models. Some useful remarks on the manuscript by 
C.~K\"ulske are gratefully acknowledged.



%\addcontentsline{toc}{section}{\bf References}
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%\noindent {\em Acknowledgements}:


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