Content-Type: multipart/mixed; boundary="-------------0604260716620" This is a multi-part message in MIME format. ---------------0604260716620 Content-Type: text/plain; name="06-132.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-132.keywords" generalized XY models, chessboard estimates, annealed site dilution ---------------0604260716620 Content-Type: application/x-tex; name="vers15.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="vers15.tex" %\documentclass[12pt]{article} %=================================================================== \documentclass[a4paper,12pt]{article} \textheight=25cm \textwidth=16.5cm \topmargin=-2.0cm %\topmargin=-10mm \oddsidemargin=0mm %================================================================== %\textheight=23cm %\textwidth=16cm %\topmargin=-2cm %\oddsidemargin -0.2cm % ---------------------------------------------------------------- \usepackage[centertags]{amsmath} 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%\newcommand{\dom}{\text{dom}\,} \newcommand{\support}{\text{supp}\,} \newcommand{\triple}{|\hspace{-0.6mm}|\hspace{-0.6mm}|} \newcommand{\id}{\mathrm{d}} \newcommand{\idv}{\id p \id q\;} \newcommand{\dL}{\partial\La} \newcommand{\sm}{_{\text{small} }} \DeclareMathOperator{\dist}{dist} \DeclareMathOperator{\Inn}{Int} \DeclareMathOperator{\inn}{int} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\Ran}{Ran} \DeclareMathOperator*{\limi}{\ul{lim}} \DeclareMathOperator*{\lims}{\ol{lim}} \DeclareMathOperator{\co}{con} \DeclareMathOperator{\free}{free} \DeclareMathOperator{\card}{card} \DeclareMathOperator{\var}{Var} \DeclareMathOperator{\cn}{Con} \DeclareMathOperator{\dom}{Dom} \DeclareMathOperator{\cst}{const} \newcommand{\print}[1]{${#1} \quad {\mathcal {#1}} \quad {\mathfrak {#1}} \quad {\mathbb {#1}} \quad {\boldsymbol {#1}}$ \newline} \DeclareMathOperator{\prb}{Prob} %\DeclareMathOperator{\Exp}{\mathsf{E}} \DeclareMathOperator{\sign}{Sign} \DeclareMathOperator{\pref}{Pref} %\newtheorem{lemma}{Lemma}[section] %\newtheorem{proposition}{Proposition}[section] %\newtheorem{theorem}{Theorem}[section] %\newtheorem{definition}{Definition}[section] %\newtheorem{corollary}{Corollary}[section] \newcommand{\bde}{\begin{definition}} \newcommand{\ede}{\end{definition}} \newcommand{\beq}{\begin{equation}} \newcommand{\eeq}{\end{equation}} \newcommand{\ben}{\begin{enumerate}} \newcommand{\een}{\end{enumerate}} \newcommand{\ble}{\begin{lemma}} \newcommand{\ele}{\end{lemma}} \newcommand{\bpr}{\begin{proof}} \newcommand{\epr}{\end{proof}} \newcommand{\bet}{\tilde\be} \newcommand{\lra}{\leftrightarrow} \newcommand{\comp}{\sim} \newcommand{\inc}{\not\sim} \newcommand{\st}{\star} \renewcommand{\pa}{\partial} \renewcommand{\1}{\mbox{{\bf 1}}} \DeclareMathOperator{\cov}{Cov} \newcommand{\kn}{\textbf{\emph{KN\ }}} % --------------------------------------------------------------- \title{First-order transitions \\for some generalized XY 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 Silvano Romano} \\[-1mm] {\normalsize\it Unit\`a di Ricerca CNISM e Dipartimento di Fisica ``A.Volta''} \\[-1.5mm] {\normalsize\it Universit\`a di Pavia} \\[-1.5mm] {\normalsize\it via A. Bassi 6} \\[-1.5 mm] {\normalsize\it I-27100, Pavia} \\[-1.5mm] {\normalsize\it ITALY} \\[-1mm] {\normalsize\tt Silvano.Romano@pv.infn.it} \\[-1mm] \\ [-1mm] {\normalsize Valentin A. Zagrebnov} \\[-1mm] {\normalsize\it Universit\'{e} de la M\'{e}diterran\'{e}e} \\[-1.5mm] {\normalsize\it Centre de Physique Th\'{e}orique} \\[-1.5mm] {\normalsize\it Luminy, Case 907} \\[-1.5mm] {\normalsize\it F-13288, Marseille, Cedex 9} \\[-1.5mm] {\normalsize\it FRANCE} \\[-1mm] {\normalsize\tt zagrebnov@cpt.univ-mrs.fr} \\[-1mm] %\\ [-1mm] \and %{\protect\makebox[5in]{\quad}}} %\\ [-1mm] \and {\protect\makebox[5in]{\quad}}} \pagenumbering{arabic} \begin{document} \maketitle \baselineskip=14pt \noindent {\bf Abstract.} In this note we demonstrate the occurrence of first-order transitions in temperature for some recently introduced generalized XY models, and also point out the connection between them and annealed site-diluted {(lattice-gas)} continuous-spin models. \bigskip \newtheorem{theorem}{Theorem} % Numbering by sections \newtheorem{lemma}{Lemma} % Number all in one sequence \newtheorem{proposition}[lemma]{Proposition} \newtheorem{corollary}[lemma]{Corollary} \newtheorem{definition}[theorem]{Definition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{claim}[theorem]{Claim} \newtheorem{observation}[theorem]{Observation} \def\proof{\par\noindent{\it Proof.\ }} \def\reff#1{(\ref{#1})} \let\zed=\bbbz % \def\zed{{\hbox{\specialroman Z}}} \let\szed=\bbbz % \def\szed{{\hbox{\sevenspecialroman Z}}} \let\IR=\bbbr % \def\IR{{\hbox{\specialroman R}}} \let\R=\bbbr % \def\IR{{\hbox{\specialroman R}}} \let\sIR=\bbbr % \def\sIR{{\hbox{\sevenspecialroman R}}} \let\IN=\bbbn % \def\IN{{\hbox{\specialroman N}}} \let\IC=\bbbc % \def\IC{{\hbox{\specialroman C}}} \def\nl{\medskip\par\noindent} \def\scrb{{\cal B}} \def\scrg{{\cal G}} \def\scrf{{\cal F}} \def\scrl{{\cal L}} \def\scrr{{\cal R}} \def\scrt{{\cal T}} \def\pfin{{\cal S}} \def\prob{M_{+1}} \def\cql{C_{\rm ql}} %\def\bydef{:=} \def\bydef{\stackrel{\rm def}{=}} %%% OR \equiv IF YOU PREFER \def\qed{\hbox{\hskip 1cm\vrule width6pt height7pt depth1pt \hskip1pt}\bigskip} \def\remark{\medskip\par\noindent{\bf Remark:}} \def\remarks{\medskip\par\noindent{\bf Remarks:}} \def\example{\medskip\par\noindent{\bf Example:}} \def\examples{\medskip\par\noindent{\bf Examples:}} \def\nonexamples{\medskip\par\noindent{\bf Non-examples:}} \newenvironment{scarray}{ \textfont0=\scriptfont0 \scriptfont0=\scriptscriptfont0 \textfont1=\scriptfont1 \scriptfont1=\scriptscriptfont1 \textfont2=\scriptfont2 \scriptfont2=\scriptscriptfont2 \textfont3=\scriptfont3 \scriptfont3=\scriptscriptfont3 \renewcommand{\arraystretch}{0.7} \begin{array}{c}}{\end{array}} \def\wspec{w'_{\rm special}} \def\mup{\widehat\mu^+} \def\mupm{\widehat\mu^{+|-_\Lambda}} \def\pip{\widehat\pi^+} \def\pipm{\widehat\pi^{+|-_\La\bibitem{mi} mbda}} \def\ind{{\rm I}} \def\const{{\rm const}} \bibliographystyle{plain} %\begin{document} \newpage \maketitle \section{Introduction} In some recent papers \cite{RZ,CRMP,MPMM,MPCR} a class of \textit{generalized} XY models was introduced and studied. These models are ferromagnetic and, in the simplest case, restricted to a nearest-neighbour XY - type interaction. In contrast to the %\textcolor{green}{ \textit{plane rotator} %} %\textit{rotator} interaction, they involve $3$-component spins (just as the classical Heisenberg model does) and they possess an $O(2)$ symmetry with respect to the X and Y components. This part is multiplied by a product of single-site terms, depending on the third (Z) component only, which is raised to an exponent $p > 0$. In the following, this factor will be simply called ``single-site term''. {For $p=1$ one recovers the standard ferromagnetic XY model, in which the interaction is defined by the scalar product of the nearest neighbour spin projections on the XY - plane.} {With the help of some correlation inequalities} it was found {\cite{RZ}} that in $d=2$ there is a transition between a low-temperature Berezinski\v\i-Kosterlitz-Thouless (BKT) phase and a high-temperature phase, whereas, in $d \ge 3$ dimensions, the existence of magnetic order at low temperatures was established. The nature of this transition, however, was left open in these works. In $d=2$ one might expect either a BKT scenario (an infinite-order transition between the BKT phase and a high-temperature phase), which was found when this exponent $p$ is small, or a first-order one between the BKT phase and the high-temperature phase , and in $d=3$ there may be an ordinary second-order transition (again found for small $p$), or a first-order transition. Recall that the first-order transition means coexistence of different infinite-volume Gibbs measures. Here this implies a jump in the energy density, in $d \geq 3$ (but not in $d=2$) accompanied by a jump in the magnetisation. \smallskip In this {note} we point out that, by a minor adaptation of the reflection positive chessboard-estimates analysis for free-energy contours (which goes back to \cite{DS1, KS} and which was recently applied to study magnetic transitions for \textit{nonlinear} classical vector spin models in \cite{ES1,ES2,ES3,BiKo,MeNa}), the occurrence of a first-order transition for sufficiently \textit{large} values of the exponent $p$ can be proven. Such a transition was already suspected to occur in $d=2$, based on the numerical data of \cite{MPCR}. In the last part of our paper we compare this result with further numerical data. \smallskip We remark that with some small modifications our analysis covers also the case of the first-order {instead of e.g. a BKT infinite-order transition} in some \textit{annealed} diluted lattice-gas models (see \cite {CKS}, \cite{GZ}, Sect. 2.4, and \cite{CSZ}), and even in some \textit{continuum} magnetic systems \cite{GTZ}. In this case the role of the nonlinearity is played by terms involving either the lattice-gas particle occupation numbers \cite {CKS, GZ, CSZ}, or the particle density \cite{GTZ}. This becomes in particular evident, when we consider the model of \cite{RZ} in the \textit{square-ditch} approximation, see (\ref{ditch-Ham}). Then the generalized XY model reduces to an annealed {site-diluted } %\textcolor{green}{ \textit{plane rotator} %} model. With minor modifications we can also treat generalized (3-component Heisenberg interactions for n-vector spins with $n \geq 4$) or annealed dilute Heisenberg models. In that case we expect that in $d=2$ there will be a transition between two phases with exponentially decaying correlations, similarly to what is expected for the nonlinear models of \cite{ES1,ES2}. \smallskip %\bl{What kind of details do you like that I add ? Maybe say that by %adding narrow-well or narrow-ditch terms the first-order transition %is strengthened?; or do you just mean that these models also have %been proven to have first-order transitions by similar proofs? } %\\ %\rd{ I mean the following:} \\ %\rd{In the lattice case dilution is achieved by multiplication of spin %variables $S_j$ %by random occupation number $n_j = \{0,1\}$, i.e, one gets as new variables %$\tilde{S}_j:=\{n_j S_j\}_{j\in \Lambda}$. %Density of occupation numbers is governed %by the free lattice gas Hamiltonian %$L^\Lambda: = - \mu \sum_{j\in \Lambda} n_j$ via chemical potential $\mu$. %After transition to %new variables \cite{CSZ}, or redefinition of "bad"-"good" %configurations \cite{GZ} (Ch.2.4), %one profit the our (standard energy-entropy) arguments to prove %the coexistence of a dense (large $\mu$) %(magnetic \cite{CSZ} or eventual BKT \cite{GZ} (Ch.2.4)) %phase with a diluted (small $\mu$) %non-magnetic phase.}\\ %\rd{In the continuum magnetic systems \cite{GTZ} one has first map it % (via space discritization) %onto a diluted lattice model and then apply the strategy above, %or as it is in the case \cite{GTZ} %to apply the Wells inequality to map this diluted model further onto %a non-diluted lattice system. %Again for large $\mu$ we have BKT phase, while for small $\mu$ not.}\\ %\rd{If we inject the spin-spin interaction defined by generalized X-Y %interaction \cite{RZ}, into \cite{GZ} (Ch.2.4), %\cite{CSZ}, \cite{GTZ}, the arguments of our present remark seems %go though verbatim showing that %by narrowing ditch (or by increasing $p$) the first-order transition %is strengthened.}\\ %\textbf{But may be it will deviate us too far from our main message.} \smallskip \section{Model, proofs and results} For general background on the theory of Gibbs measures on lattice systems we refer to \cite{EFS, Geo, GeoHagMae, Sin, Sim}. The method of reflection positivity and chessboard estimates \cite{RP} is reviewed in \cite{Shl1} and in the last 4 chapters of \cite{Geo}. The fact that our models satisfy the property of reflection positivity follows immediately from the conditions described there. \smallskip Our systems are as follows. On each site of the lattice $\bbZ^d$ we have a three-component unit spin, described by spherical coordinates $\phi$ and $\theta$; our models are then described by {a nearest neighbour generalized} XY interaction, {i.e. the plane rotator interaction in the $\phi$-variables combined with a product of $p$-powers of} single-site terms in the $\theta$-variables \cite{RZ}. For a finite $\Lambda \subset \bbZ^d$ the (dimensionless) Hamiltonian reads: %\textcolor{green}{ \begin{equation}\label{Ham} H^{\Lambda}(\phi, \theta): = - \sum_{\langle i,j \rangle \subset \Lambda} [\sin (\theta_i) \sin (\theta_j)]^{p} \cos (\phi_i - \phi_j) \end{equation} %} The integer exponent $p > 0$ is a parameter in our model, and a large value of $p$ means that spins can only interact noticeably when they are in a narrow \textit{ditch} around the equator $\theta = {\pi}/{2}$, whose width is of order {$O({1}/{\sqrt p})$}. This narrow and deep ditch plays a similar role as the narrow-well potentials of \cite{ES1,ES2,BiKo}. We present our proof for the \textit{two-dimensional} model where the ditch has a \textit{square}, instead of a polynomial, shape. Extensions to polynomial shapes then can be done as in \cite{ES1, ES2,BiKo}. Indeed, our proof can be seen to be almost a corollary of these papers, to which we refer for further details. So, we consider the \textit{square-ditch} {approximation of the generalized} XY model (\ref{Ham}) with Hamiltonians \begin{equation}\label{ditch-Ham} H^{\Lambda}_{\varepsilon}(\phi, \theta) := - \sum_{\langle i,j \rangle \subset \Lambda} \ n(\theta_i) n(\theta_j) \ \cos (\phi_i - \phi_j) \ , \ \ n(\theta):= 1_{\varepsilon} (\theta) \ , \end{equation} where $1_{\varepsilon}(x)$ denotes the characteristic function of the interval $[\pi/2 -\varepsilon, \pi/2 +\varepsilon]$. {The square-ditch approximation (\ref{ditch-Ham}) implies that two spins can interact only when they both are in the ditch; in other words, one can interpret (\ref{ditch-Ham}) as the Hamiltonian of annealed site-diluted plane rotator model with lattice-gas occupation numbers $n = {0, 1}$, as discussed in \cite{CKS,CSZ}, and as is also suggested by the notation. Notice that the (\textit{a priori}) one-site distribution of those numbers is induced by the uniform probability $\theta$-measure on the interval $[0,\pi]$. Therefore, $\varepsilon$ is related to the chemical potential $\nu$ that governs the lattice-gas overall particle density. By the standard definition of the lattice-gas chemical potential we obtain: \begin{equation}\label{chem-poten} \nu = \beta^{-1}\ln \frac{2 \varepsilon}{\pi - 2\varepsilon} \ \ \ , \end{equation} where $\beta^{-1}=\Theta$ denotes the (dimensionless) temperature of the system. Hence at fixed temperature the chemical potential becomes negative and large in magnitude when $\varepsilon$ is close to zero, which corresponds to a small lattice-gas density; whereas for $\varepsilon \rightarrow \pi/2$, i.e. for $\nu \rightarrow +\infty$, one obtains a %\textcolor{green}{ non-diluted plane rotator %} model (\ref{ditch-Ham}) with $n(\theta)= 1$ for all $\theta$. } {For the proof} we consider in a two-dimensional lattice a square $\Lambda$, of a linear size $N$ which is a multiple of $4$, with \textit{periodic} boundary conditions. Associated to Hamiltonians $H^{\Lambda}_{\varepsilon}(\phi, \theta)$ are Gibbs measures \begin{equation*} \mu^{\Lambda}(d \phi, d \theta)= {\frac{1}{Z^{\Lambda}}} \exp [-\beta H^{\Lambda}_{\varepsilon} (\phi, \theta)]\mu_{0}^{\Lambda}(d \phi, d \theta) \ , \end{equation*} {which are \textit{reflection positive} (RP).} Here $\mu_{0}$ denotes the rotation-invariant product measure, and $\beta$ is the dimensionless inverse temperature. {RP is the key property for the \textit{chessboard estimates}. They allow us then, following e.g. \cite{ES2}, to obtain \textit{contour} estimates. First we can establish the estimate on the partition function} \begin{equation*} Z^{\Lambda} \geq 1 \end{equation*} and furthermore, by integrating over intervals $|\theta| \leq \varepsilon$ and $|\phi| \leq {\pi}/{20}$, we see that also \begin{equation*} Z^{\Lambda} \geq (C_{1} \ \varepsilon \exp (2 C_{2} \beta))^{|\Lambda|} \end{equation*} with constants $C_{1}$ and $C_{2}= \cos ({\pi}/{20})$ (which is close to $1$) which are independent of $\varepsilon$. On the other hand, let us call a site \textit{ordered}, if the spin on that site, as well as all its neighbours, are in the ditch, and \textit{disordered}, if it is not in the ditch, and consider the same universal contour as in \cite{Shl1,ES2}, consisting of alternating diagonals at distance $2$, which, in turn, consist of ordered and disordered sites (separated by sites which are neither); thus we find that the restricted partition function obtained by integrating all configurations compatible with the universal contour satisfies \begin{equation*} Z_{univcont}^{\Lambda} \leq ((2 \varepsilon)^{{3}/{4}} \exp (\beta))^{|\Lambda|}. \end{equation*} Then, just as in the proof of Theorem 1 of \cite{ES1} we obtain \begin{equation*} \frac{Z_{univcont}^{\Lambda}}{Z^{\Lambda}} \leq {\varepsilon}^{|\Lambda|/(4+C_{3})} \end{equation*} with $C_{3}$ some constant determined by the choice of $C_{1}$ and $C_{2}$. This implies by standard arguments that, when $\varepsilon$ is chosen small enough, contours separating ordered and disordered sites are suppressed, uniformly on a temperature $\Theta$ interval; since at low temperatures most sites are ordered and at high temperatures most sites are disordered, there will be a temperature, where disordered and ordered phase(s) (or infinite-volume Gibbs measures) \textit{coexist}. We notice that by the Mermin-Wagner theorem \cite{MerWag,DS2,IofShlVel, Pfi}, in two dimensions all Gibbs measures are rotation-invariant, so that the spontaneous magnetisation is necessarily zero. {Since the results of \cite{RZ} imply that the generalized XY models (\ref{Ham}) at low temperatures display a BKT phase, our present statement says that for these models with high-exponent-$p$-potentials the transition between this phase and the high-temperature one is first-order.} In the three-dimensional version of the model (\ref{Ham}), however, the low-temperature phase is magnetized, but again the transition in temperature to the high-temperature regime is first-order. Notice also that in general proofs involving contour arguments do not provide very sharp estimates about the optimal parameter values for which a first-order transition appears, therefore we will not pursue this road. Below we discuss what we expect to be the situation based on numerical data, that is, which is the value of the parameter $p$ above which one might expect the first-order transition to appear. Before ending this section, let us go back in some more detail to the lattice-gas interpretation of the square-ditch approximation (\ref{ditch-Ham}). The theorem just proven implies that, for $d=2$, the XY lattice-gas model possesses a first-order transition on a suitable curve in %\textcolor{green}{ the %} $(\Theta, \varepsilon)$-plane. %======================================================================= This answers an open question from \cite{CSZ} about the first-order phase transition in $d=2$ diluted plane rotator model (\ref{ditch-Ham}). On the whole our results complement those of \cite{CSZ,GTZ} and \cite{RZ} for this model, since they concern different parts of the phase diagram in the $(\Theta,\nu)$- or in the $(\Theta, \varepsilon)$-plane, as well as the mechanism of the phase transition. For example, in \cite{GTZ} the existence of the low-temperature BKT phase in the diluted plane rotator model is proven for a relatively large \textit{positive} $\nu$, but without any conclusion on the mechanism connecting low-temperature and high-temperature behaviour. On the other hand, the Ginibre and the Wells inequalities applied to this model (in the same way as it was done for the generalized XY model in \cite{RZ}) ensure the existence of a low-temperature BKT phase for \textit{any bounded} $\nu$, but again without conclusion on the mechanism of the transition. %\textcolor{green}{THE IMPORTANT POINT FOLLOWS} %\textcolor{green}{ Moreover, the same analysis leads also to the existence of a low-temperature BKT phase for generalized XY models %in their lattice-gas version and for bounded with annealed site-dilution, for all $\nu$, %i.e. similarly to what happens for the plane rotator model. %counterpart.} In \cite{CSZ} the first-order phase transition with simultaneous jumps of magnetization and particle density was established in the $d>2$ diluted plane rotator model in some domain of both types of $\nu$ (\textit{positive} and \textit{negative}) at very low temperatures. In the present note we find (as a byproduct of our generalized-XY-model analysis) a first order-phase transition in the diluted plane rotator model (\ref{ditch-Ham}) for \textit{moderate negative} $\nu$, since we consider in (\ref{chem-poten}) sufficiently small $\varepsilon$ as well as sufficiently small $\Theta$. As mentioned before, this answers positively the question from \cite{CSZ} about the first-order phase transition in model (\ref{ditch-Ham}), at least for those $\nu$. Based on \cite{CKS}, one knows that staggered states may be involved in the mechanism of a first order transition at positive chemical potentials, at intermediate temperatures; this will not happen in the regime of negative chemical potentials which is covered by our results. %connecting the low-temperature and high-temperature behaviour. But %as in \cite{CSZ} we leave this as a plausible hypothesis. %in any case, this plausible hypothesis can not explain the hole %line of the first order transitions, since the staggered states are %probably sandwiched between not too small and not too large %densities, see \cite{CKS} for exapmle of diluted Potts model. The %second conjecture bolstering the first-order phase transition for %law temperatures and for not very large $\nu$ comes from the %observation that for the large $\nu$ the system is too dense and the %energy always wins making transition smooth. The proof of existence %of the critical point on the first-order transition line in %$(\Theta,\nu)$ - plane is an open question while this is confirmed %by the mean-field models \cite{GZ2}. %=============================================================================== %This answers an open question from \cite{CSZ}. For further results about %annealed dilute $n$-vector models, in different parts of the phase diagram, %we refer to \cite{CSK,CSZ,GTZ}. Note that although we expect %a BKT slow decay at low temperatures, we have no rigorous proof for this, %as it neither follows from the results of the above papers, which prove it %for higher chemical potential, nor from the equivalent statement %of \cite{RZ} about the generalized XY models with polynomial ditches. %Moreover, by the arguments %of \cite{GTZ} concerning the lattice approximation (which coincides %with (\ref{ditch-Ham}) for the chemical potential calculated in %(\ref{chem-poten})) we know that the low-temperature region is a BKT %phase, if the chemical potential $\nu$ is large enough. The %counterpart of this result for $d \ge 3$ was proven in Ref. %\cite{CSZ}, and in this case, the low-temperature phase possesses %magnetic order.} %\newpage %\parindent 0cm \section{Related results and extensions} Extensions of the above theorem to other pair interaction potentials are also possible. For example, the models studied in \cite{ES1,ES2,ES3,BiKo} involve $n-$component spins for any ($n \ge 2$) and possess an $O(n)$ symmetry. Indeed, their interactions are functions of the \textit{scalar product} between the two interacting spins, having the shape of a narrow well. {The spin interaction in \cite{GZ}, Sect. 2.4, is of the same type and may be viewed as a diluted version of the Patrascioiu-Seiler model with a \textit{narrow-ditch interaction}. For discussion of its low-temperature phase see e.g. \cite{Aiz}.} The proof scheme indicated above works for various \textit{combinations} of \textit{single-site} terms and \textit{nearest neighbour} interaction terms, of which at least some need to have a narrow shape. The spins can be $n$-component spins, and the symmetry can be \textit{green}{either O(n) or O(2) or some symmetry in between}. {For example, we might have narrow-ditch single-site potentials (as mentioned above), or narrow-well single-site potentials as in \cite{DS1}, as well as narrow-well interactions (\cite{ES1,ES2,ES3}) or narrow-ditch interactions (see \cite{GZ}, Sect. 2.4, and \cite{BiChStar})}. The interactions can have \textit{one} well (or ditch) for ferromagnets, \textit{two} (for liquid crystal models, possessing $RP^{(n-1)}$ symmetry), or more and could also include diagonal nearest neigbour terms (as in \cite{MeNa}, inspired by the model \cite{Shl0}). The narrowness of such terms then either creates or reinforces the first-order behaviour. {Another kind of possible extensions is related to \textit{quantum} versions of our models. This observation is inspired by the recent paper \cite{BiChStar}, which studies, in particular, a non-linear quantum XY model. The main ingredient for their arguments is the \textit{quantum} RP property, which is a quite subtle matter, but the \textit{ferromagnetic} quantum XY model does verify it. Since in (\ref{ditch-Ham}) the interaction terms are multiplied by simple single-site classical random variables (the scalar occupation numbers), the square-ditch Hamiltonian also verifies the \textit{quantum} RP property. Then according to \cite{BiChStar} we can claim the existence of the first order phase transition in this quantum model, since we proved it for the classical model (\ref{ditch-Ham}) and we know that its quantum analogue verifies the quantum RP property.} %\newpage %\parindent 0cm \section{Numerical estimates of transition orders and temperatures} ~~~ When $d=3$, a Mean Field (MF) study of the ordering transition is at least qualitatively correct, and relatively feasible in computational terms; moreover, this treatment can be refined by using various %\textcolor{green}{ cluster-variational %} techniques; we used here a Two-Site Cluster (TSC) approach, and both treatments follow Ref. \cite{RZ}. Calculations were carried out for $5 \le p \le 12$, and then $p=12,16,20$. \\ In both cases we found that, upon increasing $p$, the transition changes from second to first order; the two treatments exhibited different thresholds, i.e. a threshold of $p$ between $5$ and $6$ for MF, and a threshold of $p$ between $10$ and $11$ for TSC; results of both treatments are presented and compared in Tables (\ref{t01}) and (\ref{t02}), where %here transitional properties for the first-order first-order transitional properties, such as the energy jump, $\Delta U^*$, and the order parameter at the transition, $\overline{M}$, are shown for $p \ge 11$, where both treatments predict a first-order transition. As a side remark, we also notice that the results of \cite{BiCh} imply the existence of a first-order transition for any $p \ge 6$ in sufficently high dimension. 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Theory of Phase Transitions: Rigorous Results, Pergamon Press, Budapest, 1982. \end{thebibliography} \newpage \parindent 0cm \begin{table}[ht!] \caption[]{ MF results for transitional properties of the generalized XY models in three dimensions, obtained with different values of the exponent p. \\~~~} \label{t01} \begin{tabular}{rlccc} \hline \\ p & $\Theta_{MF}$ & type & $\Delta U^*$ & $\overline{M}$ \\ \hline \\ 5 & ~1.1082~ & II & ~ & ~ \\ 6 & ~1.0287~ & I & ~ & ~ \\ 7 & ~0.9741~ & I & ~ & ~ \\ 8 & ~0.9336~ & I & ~ & ~ \\ 9 & ~0.9019~ & I & ~ & ~ \\ 10 & ~0.8762~ & I & ~ & ~ \\ 11 & ~0.8548~ & I & ~1.2336~ & ~0.7506 \\ 12 & ~0.8366~ & I& ~1.3140~ & ~0.7687 \\ 16 & ~0.7836~ & I & ~1.5355~ & ~0.7836 \\ 20 & ~0.7486~ & I & ~1.6712~ & ~0.8387 \\ ~ & ~ & ~ & ~ & ~ \\ \hline \end{tabular} \end{table} \begin{table}[h!] \caption[]{ TSC results for transitional properties of the generalized XY models in three dimensions, obtained with different values of the exponent p. \\~~~} \label{t02} \begin{tabular}{rlccc} \hline \\ p & $\Theta_{TSC}$ & type & $\Delta U^*$ & $\overline{M}$ \\ \hline \\ 5 & ~1.1011~ & II & ~ & ~ \\ 6 & ~1.0416~ & II & ~ & \\ 7 & ~0.9935~ & II & ~ & ~ \\ 8 & ~0.9537~ & II & ~ & ~ \\ 9 & ~0.9199~ & II & ~ & ~ \\ 10 & ~0.8907~ & II & ~ & ~ \\ 11 & ~0.8659~ & I & ~0.3242~ & ~0.3994~ \\ 12 & ~0.8461~ & I & ~0.5437~ & ~0.5098~ \\ 16 & ~0.7906~ & I & ~1.0097~ & ~0.6721~ \\ 20 & ~0.7549~ & I & ~1.2578~ & ~0.7374~ \\ ~ & ~ & ~ & ~ & ~ \\ \hline \end{tabular} \end{table} \end{document} ---------------0604260716620--