Content-Type: multipart/mixed; boundary="-------------0103022105817" This is a multi-part message in MIME format. ---------------0103022105817 Content-Type: text/plain; name="01-87.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-87.comments" 23 pages; MSC: 82B10, 82B20, 82B26, 82B41, 60K40 ---------------0103022105817 Content-Type: text/plain; name="01-87.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-87.keywords" Quantum lattice models, phase transitions, Bose-Einstein condensation ---------------0103022105817 Content-Type: application/x-tex; name="probo.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="probo.tex" \def\version{March 2, 2001} \documentclass[reqno,11pt]{amsart} %\usepackage{epsf,showkeys} \usepackage{epsf} %%%%%%%%%% simplifications %%%%%%%%%%%%%%%%%%%%%%%%% \def\be{\begin{equation}} \def\ba{\begin{align}} \def\bm{\begin{multline}} \def\bfig{\begin{figure}[htb]} \def\efig{\end{figure}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%% references %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\newcommand{\bibit}[1]{\vspace{1mm} \bibitem[#1]{#1}} \newcommand{\bibit}[1]{\bibitem[#1]{#1}} \newcommand{\paper}[1]{{\it #1}, } \newcommand{\journal}[4]{#1 {\bf #2}, #3 (#4)} \newcommand{\CMP}{Commun. 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%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \begin{quote} \raggedleft {\small \version } \end{quote} \vspace{2mm} \title[Geometric \& probabilistic aspects of boson lattice models]{Geometric and probabilistic aspects \\ of boson lattice models} \author{Daniel Ueltschi} %\iffalse \address{Daniel Ueltschi \hfill\newline Department of Physics\hfill\newline Princeton University\hfill\newline Jadwin Hall\hfill\newline Princeton, NJ 08544\hfill\newline {\small\rm\indent http://www.princeton.edu/$\sim$ueltschi}} \email{ueltschi@princeton.edu} %\fi \maketitle \vspace{-5mm} %\iffalse \begin{centering} {\small\it Department of Physics, Princeton University, New Jersey\\ } \end{centering} \vspace{5mm} %\fi \begin{quote} {\small {\bf Abstract.} This review describes quantum systems of bosonic particles moving on a lattice. These models are relevant in statistical physics, and have natural ties with probability theory. The general setting is recalled and the main questions about phase transitions are addressed. A lattice model with Lennard-Jones potential is studied as an example of a system where first-order phase transitions occur. A major interest of bosonic systems is the possibility of displaying a Bose-Einstein condensation. This is discussed in the light of the main existing rigorous result, namely its occurrence in the hard-core boson model. Finally, we consider another approach that involves the lengths of the cycles formed by the particles in the space-time representation; Bose-Einstein condensation should be related to positive probability of infinite cycles. \vspace{1mm} } % end \small \end{quote} \renewcommand{\thefootnote}{} \footnote{Work partially supported by the US National Science Foundation, grant PHY-98 20650.} \setcounter{footnote}{0} \renewcommand{\thefootnote}{\arabic{footnote}} \vspace{3mm} \section{Introduction} Statistical Physics is the study of macroscopic properties of systems with a large number of microsopic particles. Its relevance stems from the law of large numbers, allowing the state of a system to be specified by the values of a few `macroscopic variables', although the number of microscopic degrees of freedom is enormous. From a probability theory point of view, the Ising model of classical spins is an example of identically distributed, but not independent, random variables; when couplings are small (high temperature, random variables close to independent), magnetization is zero; for large couplings however (strong dependence, or low temperature), the law of large numbers takes a subtler form, with two typical values for the magnetization. This behavior is a manifestation of a phase transition. Connections between statistical physics and probability theory, such as the relation between the physical entropy and the rate function of large deviations, are discussed in detail by Pfister in his excellent lectures \cite{Pfi}. While the original motivation for the Ising model resides in quantum mechanics, it is considered as a classical model, because energy and observables are functions on the space of configurations --- in quantum systems, these are operators on the vector space spanned by the configurations. There are several reasons for devoting some attention to quantum systems. \begin{itemize} \item They are closer to the physical reality, and usually of more interest to physicists than classical ones. \item They have richer properties; new types of phases such as superfluidity or superconductivity may show up that are intrisically quantum phenomena. \item They pose a number of mathematically interesting questions. \end{itemize} There are three classes of quantum lattice systems. The first class consists of spin systems, such as the quantum Heisenberg model, where each site of the lattice hosts a spin that interacts with nearest neighbors. In the second class are fermionic systems, an example of which is the Hubbard model, where the energy of the quantum particles is provided by a discrete Laplacian (`hopping matrix') for the kinetic part, while the potential part is given by an operator that is a function of the position operators; particles are indistinguishable, so that a permutation of the particles results in the same quantum state, up to a sign for odd permutations. The last class consists in bosonic systems that describe particles hopping on a lattice and interacting among themselves, but a permutation does not alter their wavefunction. There are also other models that have spins and particles, particles with spins, or both kinds of particles. This review is focussed on bosonic systems. Their great advantage over fermionic ones is that they involve only positive numbers, hence natural links with probability theory. They also have extremely interesting behavior with various phase transitions, including the Bose-Einstein condensation (hereafter denoted BEC), that should be one of the mechanisms leading to superfluidity and superconductivity. Section \ref{secmathst} introduces the general formalism and defines equilibrium states. This leads to the notion of phase transitions, and of symmetry breaking. These ideas are then illustrated in a simple boson model with Lennard-Jones potential; its low temperature phase diagram is analyzed and shown to display various phase transitions (Section \ref{secexample}). This can be proven by showing the equivalence of this model with a `contour model' that fits the framework of the Pirogov-Sinai theory (Section \ref{seccontrep}). These techniques, however useful, do not allow discussing the occurrence of BEC. We briefly review the main questions in Section \ref{secBEC}, and state the best result so far --- the occurrence of `off-diagonal long-range order' in the hard-core boson lattice model \cite{DLS,KLS}, see Theorem \ref{thmBEChcb}. We conclude by discussing an approach to the BEC that is both geometric and probabilistic, and that involves `cycles' formed by bosonic trajectories in the Feynman-Kac representation. When the temperature decreases, the probability of observing an infinite cycle should vary from 0 to a positive number, and this transition should be related to BEC. These ideas are described in Section \ref{secinfcyc}. \section{Mathematical structure} \label{secmathst} \subsection{Microscopic description} The physical picture is that of a group of bosons on a lattice, with the kinetic energy described by a discrete Laplacian, and interacting with a two-body potential. Let $\Lambda \subset \bbZ^d$ be a finite volume. The space $\bbC^\Lambda$ of `wave functions' on $\Lambda$ is a Hilbert space, and a normalized vector describes the state of a quantum particle. For $\Psi \in \otimes_{n=1}^N \bbC^\Lambda$ we define the symmetrization operator $S_N$ $$ S_N \Psi(x_1, \dots, x_N) = \frac1{N!} \sum_\pi \Psi(x_{\pi(x)}, \dots, x_{\pi(N)}), $$ where the sum is over all permutations of $N$ elements. Then $S_N(\otimes_{n=1}^N \bbC^\Lambda)$ is the Hilbert space for $N$ bosonic particles, and the Fock space that describes a variable number of particles is $\caF_\Lambda = \oplus_{N=0}^\infty S_N(\otimes^N \bbC^\Lambda)$. There is a natural inner product on this space that makes it into a Hilbert space. This formalism is the natural one from a physical point of view, but it is more practical to consider another Hilbert space that is isomorphic to the Fock space above. Thus we start again, this time in the appropriate setting. Standard references are Israel \cite{Isr} and Simon \cite{Sim}. We consider a Hilbert space $\caH_0$; either $\caH_0 \simeq \bbC^\infty$ (more precisely $\caH_0 \simeq \ell^2(\bbC)$), or $\caH_0 \simeq \bbC^N$ for systems with a `hard-core condition', \ie a prescription that sets a maximal number $N$ of bosons at a given site. Then we define local Hilbert spaces $\{\caH_x\}_{x\in\bbZ^d}$ with each $\caH_x \simeq \caH_0$, and for $\Lambda \subset \bbZ^d$ we set $\caH_\Lambda = \otimes_{x \in \Lambda} \caH_x$. A natural basis for $\caH_0$ is $\{ \ket{n_0} \}_{n_0 \in \bbN}$; for $\caH_\Lambda$, an element of this basis is \be \label{defbasis} \ket n = \otimes_{x \in \Lambda} \ket{n_x}, \end{equation} where $n \in \bbN^\Lambda$. This represents a state where the site $x$ has $n_x$ bosons. The main operators are the {\it creation operator} of a boson at site $x$, noted $c_x^\dagger$, its adjoint the {\it annihilation operator} $c_x$, and the {\it operator number of particles} at $x$, $\hat n_x = c_x^\dagger c_x$. Their actions on the above basis are \ba c_x^\dagger \ket n &= \sqrt{n_x+1} \ket{n+\delta_x}, \nn\\ c_x \ket n &= \sqrt{n_x} \ket{n-\delta_x}, \\ \hat n_x \ket n &= n_x \ket n. \nn \end{align} Here, we denoted $\ket{n+\delta_x}$ the vector that is equal to $\otimes_{y\in\Lambda} \ket{n_y+\delta_{xy}}$. Considering a system with hard-core bosons, we demand that $c_x^\dagger \ket n = 0$ if $n_x = N$. Notice that the operators $\hat n_x$ are diagonal in this basis. Without hard-cores, creation and annihilation operators satisfy the commutation relations \be [c_x, c_y^\dagger] = \delta_{xy}. \end{equation} With a hard-core, the relation is \be [c_x,c_y^\dagger] = \delta_{xy} \Bigl\{ 1 - (N+1) \sum_{n: n_x=N} \ket n \bra n \Bigr\}. \end{equation} In order to avoid extra technicalities associated with unbounded operators, we restrict our interest to models with a hard-core condition. The energy of the particles is given by an `interaction', that is, a collection of operators $H = (H_A)_{A \subset \bbZ^d}$ with $H_A : \caH_A \to \caH_A$. We commit an abuse of notation and still denote $H_A$ the operator $H_A \otimes \bbbone_{\Lambda\setminus A}$. We define operations $(H+H')_A = H_A + H_A'$ and $(\lambda H)_A = \lambda H_A$, and introduce the norm \be \label{defnorm} \|H\|_r = \sup_x \sum_{A \ni x} \|H_A\| \e{r \|A\|} \end{equation} for some positive number $r$, where $\|A\|$ is the cardinality of the smallest connected set that contains $A$. An interaction is periodic iff there exists a subgroup $\Lambda' \subset \bbZ^d$ of dimension $d$, such that $H_{\tau_x A} = H_A$ for all $x \in \Lambda'$. Here, $\tau_x$ is the translation operator. The space of {\it periodic} interactions with finite norm \eqref{defnorm} is a Banach space and we denote it $\caB$. \subsection{Free energy and equilibrium states} The {\it free energy}\footnote{Some authors prefer to define the {\it pressure} instead, that is equal to $-\beta$ times the free energy. In thermodynamics, the pressure is the potential depending on temperature, volume, and chemical potential. It would be physically more appropriate for the discussion of boson models below. The free energy is however more convenient for low temperature studies, since $\lim_{\beta\to\infty} f(H)$ exists in typical situations.} for an interaction $H$ and at inverse temperature $\beta$ is \be \label{deffen} f(H) = -\frac1\beta \lim_{\Lambda \nearrow \bbZ^d} \frac1{|\Lambda|} \log \Tr \e{-\beta \sum_{A \subset \Lambda} H_A}, \end{equation} where the limit is taken over a sequence $(\Lambda_n)$ of volumes such that $\lim_n \frac{|\partial_r \Lambda|}{|\Lambda|} = 0$ for all $r$; here, $\partial_r \Lambda = \{x \in \Lambda : \dist(x,\Lambda^\compl) \leq r\}$ is an enlarged boundary of $\Lambda$. It is well-known that the limit \eqref{deffen} is independent of the way the limit is performed, and that it is a concave function of the interactions. An {\it equilibrium state} $\rho_H$ for the interaction $H$ is a linear, normalized, positive functional on the space of interactions, that is tangent to the free energy at $H$, \ie for all $K \in \caB$, \be \label{defstate} \rho_H(K) + f(H) \geq f(H+K). \end{equation} To motivate this definition, let us consider the free energy at finite volume $f_\Lambda(H)$, given by \eqref{deffen} without taking the limit. The corresponding finite-volume state would be $$ \rho^\Lambda_H(K) = \frac\dd{\dd\lambda} f_\Lambda(H+\lambda K) \Big|_{\lambda=0} = \frac{\Tr(\frac1{|\Lambda|} \sum_{A \subset \Lambda} K_A) \e{-\beta \sum_{A \subset \Lambda} H_A}}{\Tr \e{-\beta \sum_{A \subset \Lambda} H_A}}. $$ The definition \eqref{defstate} is therefore more general, and allows to define states directly with the free energy in the limit of infinite volumes. The set of tangent functionals at a given $H$ is convex; extremal points are the `pure states'. Existence of more than one tangent functionals implies a {\it first-order phase transition}. A popular definition of equilibrium states in quantum lattice systems involves `KMS states'. They are actually equivalent to tangent functionals, see \eg \cite{Isr,Sim}. One could restrict our interest to operators that are diagonal with respect to the basis \eqref{defbasis} above. In this case, one would consider the configuration space $\bbN^\Lambda$ and the interactions would be collections of functions on this space. As a result, we have a classical system, whose free energy is still given by \eqref{deffen}. States can also be defined as tangent functionals to the free energy. Hamiltonians (or interactions, in our case) may possess {\it symmetries}: for instance, a translation by a vector of the lattice often does not affect the energy, nor does a rotation or a reflection. In quantum statistical physics, one says that $U: \caB \to \caB$, $H \mapsto H' = U(H)$ is a symmetry if for all volumes $\Lambda$ that appear in the limit in \eqref{deffen} there exists a unitary operator $U_\Lambda$ in $\caH_\Lambda$ such that \be \label{defsym} U_\Lambda \sum_{A \subset \Lambda} H_A \, U_\Lambda^{-1} = \sum_{A\subset\Lambda} H_A'. \end{equation} Clearly, one has $f(H') = f(H)$. Let us illustrate this notion on two examples that will be relevant in the sequel. The first one is the translation by one site in the direction 1; it is defined by $H_A' = H_{A-e_1}$, where $A-e_1 = \{x: (x(1)+1, x(2), \dots, x(d)) \in A\}$. Let us assume that the boxes $\Lambda$ are rectangles with periodic boundary conditions, and $1 \leq x(1) \leq L$. Then one can choose $U_\Lambda$ to be $U_\Lambda \ket{n_\Lambda} = \ket{n_\Lambda'}$, where $n_x' = n_{(x(1)-1, x(2), \dots, x(d))}$ if $x(1) \neq 1$, $n_x' = n_{(L, x(2), \dots, x(d))}$ if $x(1)=1$. The second example is relevant for the Bose-Einstein condensation and is called a `global gauge symmetry'; $U_\Lambda$ takes the form $U_\Lambda = \e{\ii\alpha \sum_{x\in\Lambda} \hat n_x}$, $\alpha \in [0,2\pi)$. Hamiltonians describing real particles always conserve the total number of particles, and hence possess the global gauge symmetry. It can be broken however, yielding states where the number of particles fluctuates more than usual.\footnote{Large deviations of the number of particles in a finite volume are studied in \cite{LLS} in the ideal Bose gas, outside the condensation regime. They are indirectly affected by BEC, if the deviated phase is a condensate.} We discuss this in Section \ref{secBEC}. \section{Example: Hopping particles with two-body interactions} \label{secexample} In this section we introduce a simple lattice model and study it by means of geometric methods. One obtains that the free energy display angles corresponding to first-order phase transitions, see \fig\ref{figgse} below. Let us mention that the existence of a first-order phase transition in a quantum system {\it in the continuum} has been recently established for the (quantum) Widom-Rowlinson model \cite{CP,Iof}. \subsection{The model} The particles have kinetic and potential energy, so that the Hamiltonian is \be \label{defHam} H = T + V. \end{equation} The kinetic energy $T$ of particles on a lattice is described by a discrete Laplacian that can be written using the creation and annihilation operators in the following way: $T = (T_A)$, with \be T_A = \begin{cases} -t (c_x^\dagger c_y + c_y^\dagger c_x) & \text{if } A = \{x,y\} \text{ with } |x-y|=1 \\ 0 & \text{otherwise.} \end{cases} \end{equation} We consider here two-body interactions given by a function $U(\cdot)$ that depends on the Euclidean distance between two particles. \be V_A = \begin{cases} U(|x-y|) \, \hat n_x \hat n_y & \text{if } A = \{x,y\} \text{ and } x \neq y \\ \tfrac12 U(0) \, \hat n_x (\hat n_x-1) & \text{if } A = \{x\} \\ 0 & \text{otherwise.} \end{cases} \end{equation} The on-site operator $\frac12 \hat n_x (\hat n_x-1)$ is the number of pairs of particles at site $x$, and the energy is naturally proportional to it. The model with only on-site interactions was introduced in \cite{FWGF} and is usually called the Bose-Hubbard model. In order for the Hamiltonian $H=T+V$ to have finite norm \eqref{defnorm}, the interaction $U$ must have exponential decay for large distances. The density of the system is controlled by a term involving a chemical potential, $-\mu N$, where $N$ is the `interaction' that corresponds to the number of particles; $N_{\{x\}} = \hat n_x$ and $N_A = 0$ if $|A|\geq2$. Let us now discuss in more details the case of a Lennard-Jones type of potential; the graph of the corresponding $U$ is depicted in \fig\ref{figpot}. We suppose that $U(0)=+\infty$, corresponding to a hard-core condition that prevents multiple occupancy of the sites. We also suppose that the tail \be u_r = \sum_{|y| \geq 2} |U(|y|)| \e{r|y|} \end{equation} does not play an important role; only important values of the potential are $U(1)$ and $U(\sqrt2)$. The results below are valid for $u_r \leq u_0$, the values of $u_0$ and $r$ depending on $U(1)$ and $U(\sqrt2)$. \bfig \epsfxsize=70mm \centerline{\epsffile{figpot.eps}} \figtext{ \writefig 3.2 1.3 {\small $a$} \writefig -3.2 5.3 {\small $U(a)$} \writefig -2.95 1.1 {\tiny $0$} %\writefig 5.07 1.1 {\tiny $1$} \writefig -2.33 1.1 {\tiny $1$} \writefig -1.75 1.1 {\tiny $2$} } \caption{The graph of a Lennard-Jones type of potential.} \label{figpot} \end{figure} We start with an analysis of the ground states of the `classical model' with configuration space $\{0,1\}^\Lambda$ and a Hamiltonian given as a sum over squares $S$ of four nearest-neighbor sites: \be H_\Lambda^{\text{cl}}(n) = \sum_{S \subset \Lambda} \Bigl[ \frac{U(1)}{2(d-1)} \sumtwo{\{x,y\} \subset S}{|x-y|=1} n_x n_y + U(\sqrt2) \sumtwo{\{x,y\} \subset S}{|x-y|=\sqrt2} n_x n_y \Bigl] - \frac14 \sum_{x \in S} \bigl[ \mu \, n_x + h \, (-1)^x n_x \bigr]. \end{equation} We added a staggered interaction $-h (-1)^x n_x$, with $(-1)^x \equiv (-1)^{\|x\|_1}$. This interaction has no physical relevance, but is mathematically useful to uncover the occurrence of phases of the chessboard type that breaks the symmetry of translation invariance. One is of course interested in what happens when $h=0$. Four configurations are important, namely $(\begin{smallmatrix}0&0\\0&0\end{smallmatrix})$, $(\begin{smallmatrix}1&0\\0&1\end{smallmatrix})$, $(\begin{smallmatrix}0&1\\1&0\end{smallmatrix})$,and $(\begin{smallmatrix}1&1\\1&1\end{smallmatrix})$; respective energies are \ba \label{gsen} e^{\mu,h}(\begin{smallmatrix}0&0\\0&0\end{smallmatrix}) &= 0 \nn\\ e^{\mu,h}(\begin{smallmatrix}1&0\\0&1\end{smallmatrix}) &= U(\sqrt2) - \tfrac\mu2 - \tfrac h2 \nn\\ e^{\mu,h}(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}) &= U(\sqrt2) - \tfrac\mu2 + \tfrac h2 \\ e^{\mu,h}(\begin{smallmatrix}1&1\\1&1\end{smallmatrix}) &= 2U(1) + 2U(\sqrt2) - \mu. \nn \end{align} We make the further assumptions on the potential that $U(1)>0$, ensuring a chessboard phase to be present, and $U(\sqrt2)<0$, so that no phases with quarter density show up --- they are more difficult to study, since the classical model has an infinite number of ground states. In many cases one expects that this degeneracy will be lifted as a result of `quantum fluctuations', that is, the effect of a small kinetic energy $T$. A general theory of such effects combined with the Pirogov-Sinai theory can be found in \cite{DFFR,KU}. Notice that $U(1)>U(\sqrt2)$, meaning that at low temperature, the chessboard phase overcomes the phase with alternate rows or columns of 1's and 0's. Energies \eqref{gsen} provide the zero-temperature phase diagram and allow guesses for the low temperature situation. \subsection{The phase diagram} The situation at high temperature ($\beta$ small) is that of bosons with weak interactions and no phase transitions may occur. The natural condition for high temperature is that $\beta \|H\|_r$ is small; one can however prove slightly more by {\it not} requesting that $U(0)$ be small. So we define (compare with \eqref{defnorm}) \be \|H\|_r^* = \sup_x \sumtwo{A \ni x}{|A| \geq 2} \|H_A\| \e{r \|A\|}. \end{equation} \begin{theorem} \label{thmht} There exists $r<\infty$ such that if $\beta \|H\|_r^* < 1$, there is a unique tangent functional at $H$, and the free energy is real analytic in a neighborhood of $H$. \end{theorem} This theorem is proven in Section \ref{subsecproofht} using high temperature expansions. We shall see below that there may be more than one tangent functionals at low temperature, corresponding to equilibrium states that are not translation invariant. This implies that a transition with symmetry breaking takes place when the temperature decreases. Presumably it is second order, like in the Ising model, but there are no rigorous results to support this. The limit $\beta\to\infty$ is easily analyzed and is depicted in \fig\ref{figgse}. \bfig \epsfxsize=80mm \centerline{\epsffile{figgse.eps}} \figtext{ \writefig 3.8 1.1 {\small $\mu$} \writefig -4.1 2.0 {\small $h$} \writefig -0.7 7.4 {\small $e^{\mu,h}$} \writefig -2.6 3.1 {\footnotesize $(\begin{smallmatrix}0&0\\0&0\end{smallmatrix})$} \writefig 1.1 2.5 {\footnotesize $(\begin{smallmatrix}0&1\\1&0\end{smallmatrix})$} \writefig -0.5 2.1 {\footnotesize $(\begin{smallmatrix}1&0\\0&1\end{smallmatrix})$} \writefig 2.7 1.1 {\footnotesize $(\begin{smallmatrix}1&1\\1&1\end{smallmatrix})$} } \caption{The free energy in the limit $\beta\to\infty$. The phase diagram is divided in four domains, corresponding to the empty, chessboards, and full configurations. For large $\beta$ and small $t$, the flat parts bend but the angles remain.} \label{figgse} \end{figure} The graph of the function $e^{\mu,h}$ is a kind of roof with four flat parts. There are angles between each flat part, so that first derivatives have discontinuities there. The two questions that should be asked are: \begin{itemize} \item Does this picture survive when adding the tail of the potential, and the kinetic energy (hopping matrix)? \item Does this picture survive at non-zero temperatures? \end{itemize} The answer to both questions is yes and is provided by the {\it quantum Pirogov-Sinai theory}. It can be viewed as a considerable extension of the Peierls argument for the Ising model. It was proposed by Pirogov and Sinai for classical lattice models \cite{PS,Sin}, and extended to quantum models in \cite{BKU,DFF,DFFR,KU,FRU}. These ideas are discussed for this model in the next section. One is then led to the phase diagram of \fig\ref{figphd}. Multiple phases and occurrences of first order phase transitions are proven when $\beta$ is large and $t$ small, \ie at low temperature and close to the classical limit of vanishing hoppings. It is expected that BEC and superfluidity are present in dimension $d \geq 3$, when the temperature is low and with sufficient hoppings \cite{FWGF}. Actually, the situation $U(0)=\infty$ and $U(a)=0$ for $a \geq 1$ corresponds to the hard-core boson model, when BEC is proven at low temperature \cite{DLS,KLS}; see Section \ref{secBEC}. \bfig \epsfxsize=80mm \centerline{\epsffile{figphd.eps}} \figtext{ \writefig 3.9 0.5 {\small $t$} \writefig -4.3 5.5 {$\frac1\beta$} \writefig -1 4.7 {\footnotesize Unicity} \writefig -3.7 1.1 {\footnotesize LRO} \writefig -0.5 1.5 {\footnotesize BEC expected} } \caption{The phase diagram $(t,\frac1\beta)$ of the boson model with Lennard-Jones potential. There is a unique state (tangent functional) at high temperature, while a domain with two extremal states, and hence long-range order (LRO), is present for low temperature and small hopping (darker zone). Most of the phase diagram is not rigorously understood yet.} \label{figphd} \end{figure} The proof of existence of phase transitions were obtained in \cite{BKU,DFF}; it was realized in \cite{FRU} that tangent functionals naturally fit in the context of the Pirogov-Sinai theory. The zero-temperature energy takes the form (see \fig\ref{figgse}) \be e^{\mu,h} = \min_{(\begin{smallmatrix}\cdot&\cdot\\ \cdot&\cdot\end{smallmatrix})} e^{\mu,h}(\begin{smallmatrix}\cdot&\cdot\\ \cdot&\cdot\end{smallmatrix}) \end{equation} where the minimum is taken over the four configurations $(\begin{smallmatrix}0&0\\0&0\end{smallmatrix})$, $(\begin{smallmatrix}1&0\\0&1\end{smallmatrix})$, $(\begin{smallmatrix}0&1\\1&0\end{smallmatrix})$, and $(\begin{smallmatrix}1&1\\1&1\end{smallmatrix})$. There are angles at the intersections between different energies. It is not clear whether they subsist at finite temperature however --- an example where angles disappear is the one-dimensional Ising model. The main result of the Pirogov-Sinai theory, in this model, is the claim that there exist four $C^1$ functions that are close to the energies \eqref{gsen}, and that play the same role: the free energy is given by the minimum of these four functions, and hence has angles at their intersections. \begin{theorem}[Free energy at low temperature] \label{thmbosmodel} Assume $d \geq 2$. Let $U(0) \to \infty$, $U(1)>0$ and $U(\sqrt2)<0$. There exist $\beta_0, r<\infty$ such that if $\beta \geq \beta_0$ and $t + u_r \leq 1$, there are real functions $f^{\mu,h}_{(\begin{smallmatrix}0&0\\0&0\end{smallmatrix})}$, $f^{\mu,h}_{(\begin{smallmatrix}1&0\\0&1\end{smallmatrix})}$, $f^{\mu,h}_{(\begin{smallmatrix}0&1\\1&0\end{smallmatrix})}$, $f^{\mu,h}_{(\begin{smallmatrix}1&1\\1&1\end{smallmatrix})}$ such that \begin{itemize} \item $$ \limtwo{\beta\to\infty}{T,u_r \to 0} f^{\mu,h}_{(\begin{smallmatrix}\cdot&\cdot\\ \cdot&\cdot\end{smallmatrix})} = e^{\mu,h}(\begin{smallmatrix}\cdot&\cdot\\ \cdot&\cdot\end{smallmatrix}) \quad\quad \text{and} \quad \limtwo{\beta\to\infty}{T,u_r \to 0} \frac\partial{\partial \mu,h} f^{\mu,h}_{(\begin{smallmatrix}\cdot&\cdot\\ \cdot&\cdot\end{smallmatrix})} = \frac\partial{\partial \mu,h} e^{\mu,h}(\begin{smallmatrix}\cdot&\cdot\\ \cdot&\cdot\end{smallmatrix}) $$ uniformly in $\mu,h$. Limits are taken in any order. The limit $u_r\to0$ means that $U(a)\to0$ for all $a\geq2$. \item The free energy \eqref{deffen} is given by $$ f^{\mu,h} = \min_{(\begin{smallmatrix}\cdot&\cdot\\ \cdot&\cdot\end{smallmatrix})} f^{\mu,h}_{(\begin{smallmatrix}\cdot&\cdot\\ \cdot&\cdot\end{smallmatrix})}. $$ \item The functions are $C^1$ in $\mu,h$ with uniformly bounded derivatives. Furthermore, $f^{\mu,h}_{(\begin{smallmatrix}\cdot&\cdot\\ \cdot&\cdot\end{smallmatrix})}$ is real analytic in $\mu,h$ when $f^{\mu,h}_{(\begin{smallmatrix}\cdot&\cdot\\ \cdot&\cdot\end{smallmatrix})}$ is the unique minimum. \end{itemize} \end{theorem} The phase diagram is therefore governed by these four functions; clearly, it is symmetric under the transformation $h \to -h$. Let $\mu_1$ be the coexistence point of $(\begin{smallmatrix}0&0\\0&0\end{smallmatrix})$ and the chessboards, \ie \be f^{\mu_1,0}_{(\begin{smallmatrix}0&0\\0&0\end{smallmatrix})} = f^{\mu_1,0}_{(\begin{smallmatrix}0&1\\1&0\end{smallmatrix})}, \end{equation} and $\mu_2$ be the coexistence between the chessboard and $(\begin{smallmatrix}1&1\\1&1\end{smallmatrix})$. There are exactly two extremal tangent functionals for $\mu_1<\mu<\mu_2$ and $h=0$. Exactly three for $\mu=\mu_1$ and $h=0$, as well as for $\mu=\mu_2$ and $h=0$. There is a unique tangent functional everywhere else. Among the consequences are various first-order phase transitions. For instance, \be \frac\partial{\partial h} f^{\mu,h} \Big|_{h=0-} \neq \frac\partial{\partial h} f^{\mu,h} \Big|_{h=0+} \end{equation} for $\mu_1<\mu<\mu_2$; also, if $h=0$, \be \frac\partial{\partial\mu} f^{\mu,0} \Big|_{\mu=\mu_1-} \neq \frac\partial{\partial\mu} f^{\mu,0} \Big|_{\mu=\mu_1+}, \end{equation} and similarly at $\mu_2$. Construction of the functions (`metastable free energies' in the Pirogov-Sinai terminology) is done in two steps. First, using a space-time representation of the model, one defines an equivalent {\it contour model}. This step is explained in the next section; it gives the opportunity to make the link with a stochastic process of classical particles jumping on the lattice. The second step is to get an expression for the metastable free energies starting from a contour model, and this is achieved using the standard Pirogov-Sinai theory \cite{PS,Sin}. This is only outlined here. Ideas are described \eg in \cite{Kot}; we also mention \cite{Uel} for a self-contained review which includes precise statements on tangent functionals. \subsection{Incompressibility} The space-time contour representation actually allows us to obtain more. The total number of particles is conserved, and as a consequence the ground state of the quantum model has same density as that of the model without hoppings, and hence the compressibility is zero. The following observations were made in \cite{BKU2}. Since a state is a linear functional on the space of interactions, we have to understand what is the {\it density} of the systems. We consider the interaction $N$: \be N_A = \begin{cases} \hat n_x & \text{if } A = \{x\} \\ 0 & \text{ otherwise;} \end{cases} \end{equation} if $\rho$ denotes a state, than the corresponding density is $\rho(N)$. It is a function of the chemical potential $\mu$. One defines the {\it compressibility} $\kappa_{\text T}$, \be \kappa_{\text T} = \frac\partial{\partial\mu} \rho(N) \end{equation} where the derivative is with constant temperature (\ie $\beta$). The theorem below claims incompressibility of the ground state, and also that the low temperature states are close to incompressible. It holds in all dimensions. \begin{theorem} \label{thmincomp} Let $U(0) \to \infty$, $U(1)>0$ and $U(\sqrt2)<0$. There exist $\beta_0,r<\infty$ such that if $\beta \geq \beta_0$ and $t + u_r \leq 1$, one has \ba &\bigl| \rho(N) - \rho_0(N) \bigr| \leq C \e{-\beta r'} ; \nn\\ &|\kappa_{\rm T}| \leq C \e{-\beta r'} \nn \end{align} for some $C<\infty$, $r'>0$. \end{theorem} \section{The space-time representation and the equivalent contour model} \label{seccontrep} \subsection{Equivalence with a stochastic system} We start with the finite-volume expression for the free energy, \be \label{fenfin} f_\Lambda^{\mu,h} = -\frac1{\beta|\Lambda|} \log\Tr \e{-\beta \sum_{A \subset \Lambda} H_A}, \end{equation} with $H=T+V-\mu N$. Notice that the last two interactions are diagonal with respect to the basis \eqref{defbasis}. One can give various probabilistic interpretations for \eqref{fenfin}, see \eg \cite{Toth}. A natural one is a continous-time Markov chain where the collection of random variables $\{ n(t) \}_{t\geq 0}$ take values in $\{0,\dots,N\}^\Lambda$. Let us introduce the set of `neighbors' of a configuration $n$: \be \caN(n) = \{ n': \exists x,y \text{ with } |x-y|=1 \text{ and } n_x' = n_z - \delta_{zx} + \delta_{zy} \text{ for all } z \in \Lambda \}. \end{equation} The generator of this random process is \be G_{nn'} = \begin{cases} 1 & \text{if } n' \in \caN(n) \\ -|\caN(n)| & \text{if } n'=n \\ 0 & \text{otherwise.} \end{cases} \end{equation} The {\it partition function} $Z_\Lambda = \Tr \e{-\beta \sum_{A \subset \Lambda} H_A}$ is the expectation \be \label{partfctsp} Z_\Lambda = \bbE_{[0,\beta]} \biggl( \chi[n(0)=n(\beta)] \exp\Bigl\{ -\int_0^\beta \dd\tau \Bigl[ \sum_{x,y \in \Lambda} U(|x-y|) n_x(\tau) n_y(\tau) - \mu \sum_{x \in \Lambda} n_x(\tau) \Bigr] \Bigr\} \biggr). \end{equation} Another representation that is more appealing for the physical intuition involves continu\-ous-time simple random walks. It was explicited in \cite{CS} and used to obtain a bound on the free energy of the Heisenberg model \cite{CS2,Toth}. Let $\{ x_j(t) \}_{t\geq0}$, $1 \leq j \leq N$, be random walks with generator \be L_{xy} = \begin{cases} 1 & \text{if } |x-y|=1 \\ -2d & \text{if } x=y \\ 0 & \text{otherwise.} \end{cases} \end{equation} Then the partition function takes the form \bm \label{partfctrw} Z_\Lambda = \sum_{N\geq0}^\infty \frac{\e{\beta\mu N}}{N!} \sum_{x_1, \dots,x_N \in \Lambda} \sum_{\pi \in S_N} \bbE\biggl( \chi\bigl[ x_i(\beta) = x_{\pi(i)}, 1\leq i \leq N \bigr] \\ \exp\Bigl\{ -\int_0^\beta \dd\tau \sum_{i0$. Then there is $\beta_0 < \infty$ such that for $\beta>\beta_0$, $$ \lim_{|x-y|\to\infty} \expval{c_x^\dagger c_y} \neq 0. $$ \end{theorem} This theorem implies the existence of a phase transition in the sense that the state $\expval\cdot$ is not clustering. It is established using `reflection positivity', introduced in \cite{FSS} for proving spontaneous magnetization in the classical Heisenberg model; its difficult extension to quantum systems was done in \cite{DLS}. The claims of \cite{DLS,KLS} that are relevant here deal with spontaneous magnetization in the spin $\frac12$ $x$-$y$ model. Let us discuss analogies between spins and hard-core boson systems. For the latter, we take $\caH_0 \simeq \bbC^2$ and define self-adjoint operators $\{S_x^{(1)}, S_x^{(2)}, S_x^{(3)}\}_{x\in \bbZ^d}$, that commute if they are located on different sites, and satisfy $[S_x^{(1)}, S_x^{(2)}] = \ii S_x^{(3)}$ (and permutations of (1,2,3)) at a same site. (These matrices are called {\it Pauli matrices}.) The $x$-$y$ model has interaction $-S_x^{(1)} S_y^{(1)} - S_x^{(2)} S_y^{(2)}$ on nearest-neighbor sites $x,y$, and zero otherwise. The correspondence to boson models is done by setting \ba &c_x^\dagger = S_x^{(1)} + \ii S_x^{(2)} \nn\\ &c_x = S_x^{(1)} - \ii S_x^{(2)} \\ &n_x = S_x^{(3)} + \tfrac12 \nn \end{align} In the case of hard-cores (with $N=1$) the commutation relations are $[c^{\#}_x, c_y^{\#}] = 0$ if $x \neq y$, and $\{ c_x, c_x^\dagger \} = 1$, where $\{\cdot,\cdot\}$ denotes the anticommutator. It is easy to check that these also follow from the commutation relations of spin operators, and from definitions above. The $x$-$y$ model is equivalent to $H'$, \be H_A' = \begin{cases} -\tfrac12 [c^\dagger_x c_y + c^\dagger_y c_x] & \text{if } A = \{x,y\}, \quad |x-y|=1 \\ 0 & \text{otherwise.} \end{cases} \end{equation} Off-diagonal long-range order is then equivalent to spontaneous magnetization in the 1-2 plane. \subsection{BEC \& symmetry breaking} The Bose-Einstein condensation is related to a symmetry breaking, namely `global gauge invariance'. Let us note that the Hamiltonian \eqref{defHam} conserves the total number of particles, \ie \be \bigl[ \sum_{A \subset \Lambda} H_A, \sum_{x \in \Lambda} \hat n_x \bigr] = 0. \end{equation} Therefore one can define the unitary operator $U_\Lambda = \e{\ii \alpha \sum_{x \in \Lambda} \hat n_x}$, which is a symmetry of the Hamiltonian. Its action on creation and annihilation operators is \ba & U_\Lambda c_x^\dagger U_\Lambda^{-1} = \e{\ii\alpha} c_x^\dagger \nn\\ & U_\Lambda c_x U_\Lambda^{-1} = \e{-\ii\alpha} c_x^\dagger. \end{align} This is easily seen from the action of all these operators on elements of the basis \eqref{defbasis}. To study the properties of the free energies as a function of the interactions, one has to proceed similarly as in Section \ref{secexample}. Recall that we added a non translation-invariant (and non-physical) interaction $h P$ and looked at a phase diagram where $h$ is a parameter. This is similar here. First, we need an interaction that does not conserve the total number of particles. The simplest choice with self-adjoint operators is $Q = (Q_A)$, with \be Q_A = \begin{cases} \e{\ii\alpha} c_x^\dagger + \e{-\ii\alpha} c_x & \text{if } A = \{x\} \\ 0 & \text{otherwise.} \end{cases} \end{equation} Supposedly, there is a unique tangent functional to the free energy at $H + h Q$ for all $h\neq0$, but there should be an infinite number of extremal states at $H$, if the temperature is low enough; each of these extremal states is indexed by $\alpha \in [0,2\pi)$. Since there is a unique equilibrium state at high temperature (Theorem \ref{thmht}), we face here the breakdown of a continuous symmetry. It should occur at low temperature and if the dimension of the lattice is greater or equal to 3. There is no rigorous result to support this discussion, besides the weaker --- but important! --- statement of Theorem \ref{thmBEChcb} in the case of the hard-core boson gas. \section{Infinite cycles: context and conjectures} \label{secinfcyc} \subsection{Heuristics} In the last section of this brief review, we discuss an approach to the BEC initiated by Feynman 50 years ago \cite{Fey}, that focusses on the occurrence of infinite cycles in the space-time representation. Its appeal to probabilists should be evident --- it looks at first sight like a percolation phenomenon. However, the one-dimensional nature of cycles makes them harder to study than clusters. Still, some progress should be possible. The partition function for the Hamiltonian \eqref{genHam} can be expanded via Feynman-Kac; setting $\hbar^2/2m=1$, the partition function is given by \bm \label{fpartcontinuum} Z_V = \sum_{N\geq0} \frac{\e{\beta\mu N}}{N!} \int_V \dd x_1 \dots \dd x_N \sum_{\pi \in S_N} \Bigl( \prod_{i=1}^N \inttwo{\bsx_i(0)=x_i}{\bsx_i(\beta)=x_{\pi(i)}} \dd W_{[0,\beta]}(\bsx_i) \Bigr) \\ \exp\Bigl\{ -\int_0^\beta \dd\tau \sum_{i2$ to account for large interactions. One could also simplify the problem and consider \be \label{xinn} \xi(x,y) = \begin{cases} 0 & \text{if } x=y \\ 1/\beta & \text{if } |x-y|=1 \\ \infty & \text{otherwise.} \end{cases} \end{equation} In any case, we restrict the choice of $\xi$ to one that satisfies \be \sum_x \e{-\xi(0,x)} < \infty, \end{equation} ensuring that particles do not jump to infinity in one step. Let us describe carefully these cycles models. The lattice is $\bbZ^d$, and we denote by $\bbB$ the set of bijections $\bbZ^d \to \bbZ^d$. Given $x,y \in \bbZ^d$, let $B_{xy} = \{ \pi \in \bbB : \pi(x)=y \}$; then we define $\caB'$ to be the algebra made out of all such sets and their complements. Next we set $\bbB(\Lambda) = \{ \pi \in \bbB : \pi(x)=x \text{ for all } x \notin \Lambda \}$ the set of permutations that are trivial out of $\Lambda$. Since $\caB'$ is countable, there exists a sequence of boxes $\bsLambda = (\Lambda_n)_{n\geq0}$ such that for all $B \in \caB'$ the following limit exists: \be \label{defprob} \lim_{\Lambda \in \bsLambda} \frac1{Z(\Lambda)} \sum_{\pi \in \bbB(\Lambda)} \indicator{\pi \in B} \prod_{x\in\Lambda} \e{-\xi(x,\pi(x))} \equiv P(B). \end{equation} The normalization $Z(\Lambda)$ is \be Z(\Lambda) = \sum_{\pi \in \bbB(\Lambda)} \prod_{x\in\Lambda} \e{-\xi(x,\pi(x))}. \end{equation} The probability \eqref{defprob} extends to the smallest $\sigma$-algebra generated by $\caB'$, that we denote $\caB$. A {\it cycle} is a sequence $c = (x_1, \dots, x_{|c|})$ of different sites; we identify $(x_2, \dots, x_{|c|}, x_1) = (x_1, \dots, x_{|c|})$. The set of permutations $B_c = \{ \pi \in \bbB : \pi(x_j) = x_{j+1}, 1\leq j\leq n \}$ (with $x_{|c|+1} \equiv x_1$) is an element of $\caB$, and the set of cycles is countable. Therefore, the set \be B_\infty = \bbB \setminus \union_{c \ni 0} B_c \end{equation} is also in the $\sigma$-algebra $\caB$. It represents the event `the origin belongs to an infinite cycle', and is the central object of our attention. \subsection{Few results and important conjectures} There are no infinite cycles at high temperature; the condition of the following theorem is easy to check for small $\beta$. \begin{theorem} If $$ \sum_{c \ni 0} \prod_{j=1}^{|c|} \e{-\xi(x_j,x_{j+1})} < \infty, $$ then $P(B_\infty)=0$. \end{theorem} \begin{proof} Let $B_{>n}$ be the set of permutations where the origin belongs to a cycle of length greater than $n$. One has \be B_{>1} \supset B_{>2} \supset \dots \quad \text{and} \quad B_\infty = \inter_n B_{>n}. \end{equation} Then $P(B_\infty) = \lim_n P(B_{>n})$. Since \be B_{>n} = \union_{x\neq0} \, \uniontwo{w:0\to x}{|w|=n} B_w, \end{equation} with $w=(0,x_1,\dots,x_{n-1},x)$ a self-avoiding walk from 0 to $x$, and $B_w = \cap_{j=1}^n B_{x_{j-1},x_j}$, one can write \ba P(B_{>n}) &= \sum_{x\neq0} \sumtwo{w:0\to x}{|w|=n} \lim_{\Lambda \in \bsLambda} P_\Lambda(B_w) \nn\\ &\leq \lim_{\Lambda \in \bsLambda} \sum_{x\neq0} \sumtwo{w:0\to x}{|w|=n} P_\Lambda(B_w) \nn\\ &= \lim_{\Lambda \in \bsLambda} \sumtwo{c \ni 0}{|c|>n} \prod_{j=1}^{|c|} \e{-\xi(x_{j-1},x_j)} \frac{Z(\Lambda\setminus c)}{Z(\Lambda)} \nn\\ &\leq \sumtwo{c \ni 0}{|c|>n} \prod_{j=1}^{|c|} \e{-\xi(x_{j-1},x_j)}. \nn \end{align} The first inequality is Fatou's lemma. The last term goes to 0 as $n\to\infty$ since the sum over all cycles containing the origin converges. \end{proof} The typical picture at high temperature is that of \fig\ref{figcyc} $(a)$. Most cycles involve a unique site and have length 1. When the temperature decreases, cycles lengths should increase, as depicted in \fig\ref{figcyc} $(b)$. \bfig \centerline{$\begin{matrix} \quad \epsfxsize=70mm \epsffile{figcyc1.eps} \quad & \quad \epsfxsize=70mm \epsffile{figcyc2.eps} \quad \\ (a) & (b) \end{matrix}$} \caption{Expected typical configurations of cycles, $(a)$ at high temperature and $(b)$ at low temperature.} \label{figcyc} \end{figure} The cycles model resemble that of multiple random walks interacting via exclusions. Assume for a moment that $\xi$ is given by \eqref{xinn} with $\beta=\infty$, that is, cycles have nearest-neighbor jumps. One can generate a configuration of cycles by starting at the origin and doing two self-avoiding random walks in different directions. When they eventually met, we close this cycle and start another pair of walks from a free site, that have to avoid the first one. One repeats the procedure until all the sites have been considered. This actually does not give the same probability distribution on the configurations of cycles, but one can expect similar behavior. There is a natural question in this process: Is there a chance that after $n$ steps the two legs have not crossed? If the non-crossing probability remains finite when $n$ goes to infinity, there are infinite cycles. It is actually known that the random walk is recurrent in dimension 2 and transcient in dimension 3 and higher. Considerably extrapolating this argument, one obtains an illustration on the fact that BEC occurs only in dimensions greater or equal to 3. This also suggests the natural conjecture that infinite cycles do occur in this model at low temperature and $d\geq3$. \subsection*{Acknowlegments} I am grateful to R. Moessner and Y. Velenik for a critical reading of the manuscript. \begin{thebibliography}{99} \vspace{-1mm} \bibit{AN} M. Aizenman and B. Nachtergaele, \paper{Geometric aspects of quantum spin states} \journal{\CMP}{164}{17--63}{1994} \bibit{AEMWC} M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman and E. A. Cornell, \journal{Science}{269}{198--202}{1995} \bibit{BKU} C. Borgs, R. Koteck\'y and D. Ueltschi, \paper{Low temperature phase diagrams for quantum perturbations of classical spin systems} \journal{\CMP}{181}{409--446}{1996} \bibit{BKU2} C. Borgs, R. Koteck\'y and D. Ueltschi, \paper{Incompressible phase in lattice systems of interacting bosons} unpublished (1997) \bibit{BZ} A. Bovier and M. Zahradn\'\i k, \paper{A simple inductive approach to the problem of convergence of cluster expansions of polymer models} \journal{\JSP}{100}{765--778}{2000} \bibit{CP} M. Cassandro and P. Picco, \paper{Existence of phase transition in continuous quantum systems} preprint (2000) \bibit{CS} J. Conlon and J. P. Solovej, \paper{Random walk representations of the Heisenberg model} \journal{\JSP}{64}{251--270}{1991} \bibit{CS2} J. Conlon and J. P. Solovej, \paper{Upper bound on the free energy of the spin 1/2 Heisenberg ferromagnet} \journal{\LMP}{23}{223--231}{1991} \bibit{DFF} N. Datta, R. Fern\'andez and J. Fr\"ohlich, \paper{Low-temperature phase diagrams of quantum lattice systems. I. Stability for quantum perturbations of classical systems with finitely-many ground states} \journal{\JSP}{84}{455--534}{1996} \bibit{DFFR} N. Datta, R. Fern\'andez, J. Fr\"ohlich and L. Rey-Bellet, \paper{Low-temperature phase diagrams of quantum lattice systems. II. Convergent perturbation expansions and stability in systems with infinite degeneracy} \journal{\HPA}{69}{752--820}{1996} \bibit{Dob} R. L. Dobrushin, \paper{Estimates of semi-invariants for the Ising model at low temperatures} in Topics of Statistical and Theoretical Physics, Amer. Math. Soc. Transl. Ser. 2, 177, 59--81 (1996) \bibit{DLS} F. J. Dyson, E. H. Lieb and B. Simon, \paper{Phase transitions in quantum spin systems with isotropic and nonisotropic interactions} \journal{\JSP}{18}{335--383}{1978} \bibit{Fey} R. P. Feynman, \paper{Atomic theory of the $\lambda$ transition in Helium} \journal{\PR}{91}{1291--1301}{1953} \bibit{FWGF} M. P. A. Fisher, P. B. Weichman, G. Grinstein and D. Fisher, \paper{Boson localization and the superfluid-insulator transition} \journal{\PRB}{40}{546--570}{1989} \bibit{FRU} J. Fr\"ohlich, L. Rey-Bellet and D. Ueltschi, \paper{Quantum lattice models at intermediate temperatures} preprint, math-ph/0012011 (2000) \bibit{FSS} J. Fr\"ohlich, B. Simon and T. Spencer, \paper{Infrared bounds, phase transitions and continuous symmetry breaking} \journal{\CMP}{50}{79--95}{1976} \bibit{Gin} J. Ginibre, \paper{Some applications of functional integration in Statistical Mechanics} in Statistical Mechanics and Field Theory, C. De Witt and R. Stora eds, Gordon and Breach (1971) \bibit{Iof} D. Ioffe, \paper{A note on the quantum Widom-Rowlinson model} mp{\_}arc 01-32 (2001) \bibit{Isr} R. B. Israel, \paper{Convexity in the Theory of Lattice Gases} Princeton Univ. Press (1979) \bibit{KLS} T. Kennedy, E. H. Lieb and B. S. Shastry, \paper{The $X$-$Y$ model has long-range order for all spins and all dimensions greater than one} \journal{\PRL}{61}{2582--2584}{1988} \bibit{Kot} R. Koteck\'y, \paper{Phase transitions of lattice models} Rennes lectures (1996) \bibit{KP} R. Koteck\'y and D. Preiss, \paper{Cluster expansion for abstract polymer models} \journal{\CMP}{103}{491-498}{1986} \bibit{KU} R. Koteck\'y and D. Ueltschi, \paper{Effective interactions due to quantum fluctuations} \journal{\CMP}{206}{289--335}{1999} \bibit{LLS} J. L. Lebowitz, M. Lenci and H. Spohn, \paper{Large deviations for ideal quantum systems} \journal{\JMP}{41}{1224--1243}{2000} \bibit{Lieb} E. H. Lieb, \paper{The Bose fluid} in Lectures in Theoretical Physics, Vol.\ VII C, W. E. Brittin ed., Univ.\ of Colorado Press, 175--224 (1965) \bibit{Lieb2} E. H. Lieb, \paper{The Bose gas: a subtle many-body problem} in Proceedings of the XIII Internat. Congress on Math. Physics, London, Internat. Press (2001) \bibit{LY} E. H. Lieb and J. Yngvason, \paper{Ground state energy of the low density Bose gas} \journal{\PRL}{80}{2504--2507}{1998} \bibit{Pfi} Ch.-E. Pfister, \paper{Thermodynamical aspects of classical lattice systems} in this volume (2001) \bibit{PS} S. A. Pirogov and Ya. G. Sinai, \paper{Phase diagrams of classical lattice systems} \journal{Theoretical and Mathematical Physics}{25}{1185--1192}{1975}; \journal{}{26}{39--49}{1976} \bibit{PO} O. Penrose and L. Onsager, \paper{Bose-Einstein condensation and liquid Helium} \journal{\PR}{104}{576--584}{1956} \bibit{Sim} B. Simon, \paper{The Statistical Mechanics of Lattice Gases} Princeton Univ. Press (1993) \bibit{Sin} Ya. G. Sinai, {\it Theory of Phase Transitions: Rigorous Results}, Pergamon Press (1982) \bibitem[S\"ut\H o]{Suto} A. S\"ut\H o, \paper{Percolation transition in the Bose gas} \journal{\JPA}{26}{4689--4710}{1993} \bibitem[T\'oth]{Toth} B. T\'oth, \paper{Improved lower bound on the thermodynamic pressure of the spin 1/2 Heisenberg ferromagnet} \journal{\LMP}{28}{75--84}{1993} \bibit{Uel} D. Ueltschi, \paper{A review of the Pirogov-Sinai theory of phase transitions} in preparation \bibit{ZB} V. Zagrebnov and J.-B. Bru, \paper{The Bogoliubov model of weakly imperfect Bose gas} preprint (2000) \end{thebibliography} \end{document} ---------------0103022105817 Content-Type: application/postscript; name="figcont.eps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="figcont.eps" %!PS-Adobe-2.0 EPSF-2.0 %%Title: figcont.eps %%Creator: fig2dev Version 3.2.3 Patchlevel %%CreationDate: Mon Jan 29 11:46:58 2001 %%For: ueltschi@pc337.princeton.edu (Daniel Ueltschi) %%BoundingBox: 0 0 598 418 %%Magnification: 1.0000 %%EndComments /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 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