In 1979–80, my last year at Princeton (although I didn’t know it at the time!), I gave a course on rigorous results in the statistical mechanics of discrete lattice models. In preparing the course, I realized that a twenty-year period of explosive growth in our knowledge seemed to be drawing to a close as the most approachable problems were solved. So it seemed like a good time to think about a book summarizing the state of our knowledge.

It was clear such a book would need to be comprehensive, even encyclopedic, since it was describing a mature subject. Little...

Lattice models are caricatures invented to illuminate various aspects of elementary statistical mechanics, especially the phenomena of phase transitions and spontaneously broken symmetry. The simplest of all models is the Ising (or Lenz-Ising) model. This model was suggested to Ising by his thesis adviser, Lenz. Ising [1925] solved the one-dimensional model, an easy task (we will solve it three times: once in this section, once using transfer matrices in Section II.5, and once using high-temperature expansions in Section V.6), and on the basis of the fact that the one-dimensional model had no phase transition, he asserted there was no phase...

In this section, we will present the basic abstract formalism for lattice gases. We will always take our underlying lattice to be ℤ^ν, the set of ν-tuples (n₁,…,n_ν) of integers. We remark that any subsetL⊂ ℝ^νobeying: (i)x, y∈Limpliesx+yand −xlie inL; (ii)Lis discrete; (iii)Lis contained in no proper subspace of ℝ^ν; is isomorphic to ℤ^νin the sense that there exist vectorsx₁,…,x_ν∈ ℝ^νso that$L=\left\{ \sum\limits_{i=1}^{\nu}{{{n}_{i}}{{x}_{i}}\left| n\in {{\mathbb{Z}}^{\nu}} \right.} \right\}$. This means that we are not losing any models by taking the lattice to be...

In order to motivate the eventual definition we shall take for a state, consider the intuitive notion of states for an Ising model. Given any finite Λ, we would like nonnegative numbersw_Λ(s) for eachs∈ {−1, 1}^Λ, so that\[\sum\limits_{s\, \in\, {{\Omega }_{\Lambda }}}{{{w}_{\Lambda }}(s)=1.}\caption {(\text {III.1.1})}\]

w_Λ(s) should represent the probability that the spins in Λ have the values. Clearly, this notion requires the consistency condition that for Λ ⊂ Λ′\[\sum\limits_{t\,\in\, {{\Omega }^{{\Lambda }'\backslash \Lambda }}}{{{w}_{{{\Lambda }'}}}(s\times t)={{w}_{\Lambda }}(s).}\caption {(\text {III.1.2})}\]

A state for an Ising model should be equivalent to a choice of nonnegative numbers,w_Λ(s), obeying (III.1.1), (III.1.2). A set of weightsw_Λ(s) obeying (II.1.1) defines a measure on Ω^Λ...

Given the notion of state in the classical case (Section III.1), the analogy between measures and states on aC^*-algebra (Section 1.7) and our discussion of local and quasilocal observables in quantum lattice systems (the end of Section II.1), it is clear what we want to mean by a state (in the physical sense) of a quantum lattice system: namely, a state (in the mathematical sense) on theC^*-algebra,$\script a$, of quasilocal observables.

There is an analog of Prop. III.1.1. Since$\underset{\Lambda\, \in\, {{\script P}_{f}}}{\cup }\, {{\script a}_{\Lambda }}$is dense in$\script a$and states are automatically continuous (the argument preceding Prop. 1.7.4), a state ρ...

In this section, we will present a method of treating systems at “high temperature” (where the interaction is small in some norm) or “low densities” (where the a priori measuredμ₀is concentrated near a single point). Among the theorems we will prove are the following pair:

Theorem V.1.1: Consider a general classical system. Let Φ ∈ ℬ₁be an interaction with\[{{\left\| \Phi \right\|}_{2}}\equiv \sum\limits_{0\,\in\, X}{(\left| X \right|-1){{\left\| \Phi (X) \right\|}_{\infty }}<1.}\caption {(\text {V.1.1})}\]

Then, there is exactly one equilibrium state for the interaction Φ.

Theorem V.1.2: Let Ω = {−1, 1} with σ_α= ±1. Suppose that Φ ∈ ℬ₁has the form Φ(X) = −J_Xσ^Xwith${{\sigma }^{X}}=\prod\limits_{\alpha\, \in\, X}{{{\sigma }_{\alpha }}}$with\[\sum\limits_{0\,\in\, X}{(\left| X \right|-1)\tanh (\left| {{J}_{X}} \right|)>1.}\caption {(\text {V.1.2})}\]...