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The Ising Model 3 The Ising Model In this chapter, we study the Ising model on Zd , which was introduced informally in Section 1.4.2. We provide both precise definitions of the concepts involved and a detailed analysis of the conditions ensuring the existence or absence of a phase transition in this model, therefore providing full rigorous justification to the discus- sion in Section 1.4.3. Namely, • In Section 3.1, the Ising model on Zd is defined, together with various types of boundary conditions. • In Section 3.2, several concepts of fundamental importance are introduced, including: the thermodynamic limit, the pressure and the magnetization. The latter two quantities are then computed explicitly in the case of the one- dimensional model (Section 3.3). • The notion of infinite-volume Gibbs state is given a precise meaning in Sec- tion 3.4. In Section 3.6, we discuss correlation inequalities, which play a cen- tral role in the analysis of ferromagnetic systems like the Ising model. • In Section 3.7, the phase diagram of the model is analyzed in detail. In par- ticular, several criteria for the presence of first-order phase transitions, based on the magnetization and the pressure of the model, are introduced in Sec- tion 3.7.1. The latter are used to prove the existence of a phase transition when h 0 (Sections 3.7.2 and 3.7.3) and the absence of a phase transition = when h 0 (Section 3.7.4). A summary with a link to the discussion in the 6= Introduction is given in Section 3.7.5. • Finally, in Section 3.10, the reader can find a series of complements to this chapter, in which a number of interesting topics, related to the core of the chapter but usually more advanced or specific, are discussed in a somewhat less precise manner. We emphasize that some of the ideas and concepts introduced in this chapter are not only useful for the Ising model, but are also of central importance for sta- tistical mechanics in general. They are thus fundamental for the understanding of other parts of the book. 79 Revised version, August 22 2017 To be published by Cambridge University Press (2017) © S. Friedli and Y. Velenik www.unige.ch/math/folks/velenik/smbook 80 Chapter 3. The Ising Model 3.1 Finite-volume Gibbs distributions In this section, the Ising model on Zd is defined precisely and some of its basic properties are established. As a careful reader might notice, some of the definitions in this chapter differ slightly from those of Chapter 1. This is done for later conve- nience. I Finite volumes with free boundary condition. The configurations of the Ising d model in a finite volume Λ b Z with free boundary condition are the elements of the set def Ω { 1,1}Λ . Λ = − A configuration ω ΩΛ is thus of the form ω (ωi )i Λ. The basic random variable ∈ = ∈ associated to the model is the spin at a vertex i Zd , which is the random variable def ∈ σ : Ω { 1,1} defined by σ (ω) ω . i Λ → − i = i We will often identify a finite set Λ with the graph that contains all edges formed by nearest-neighbor pairs of vertices of Λ. We denote the latter set of edges by def E {i, j} Λ : i j . Λ = ⊂ ∼ To each configuration ω Ω , we associate© its energyª , given by the Hamiltonian ∈ Λ def H ∅ (ω) β σ (ω)σ (ω) h σ (ω), Λ;β,h = − i j − i {i,j} E i Λ X∈ Λ X∈ where β R 0 is the inverse temperature and h R is the magnetic field. The su- ∈ ≥ ∈ perscript ∅ indicates that this model has free boundary condition: spins in Λ do not interact with other spins located outside of Λ. Definition 3.1. The Gibbs distribution of the Ising model in Λ with free boundary condition, at parameters β and h, is the distribution on ΩΛ defined by def 1 µ∅ (ω) exp H ∅ (ω) . Λ;β,h = ∅ − Λ;β,h ZΛ;β,h ¡ ¢ The normalization constant def Z∅ exp H ∅ (ω) Λ;β,h = − Λ;β,h ω ΩΛ X∈ ¡ ¢ is called the partition function in Λ with free boundary condition. I Finite volumes with periodic boundary condition. We now consider the Ising model on the torus Tn, defined as follows. Its set of vertices is given by def V {0,...,n 1}d , n = − and there is an edge between each pair of vertices i (i1,...,id ), j (j1,..., jd ) such d = = that r 1 (ir jr ) mod n 1; see Figure 3.1 for illustrations in dimensions 1 and = | − per | = 2. We denote by E the set of edges of Tn. P Vn Configurations of the model are now the elements of { 1,1}Vn and have an en- − ergy given by per def HV ;β,h(ω) β σi (ω)σj (ω) h σi (ω). n = − per − {i,j} E i Vn X∈ Vn X∈ Revised version, August 22 2017 To be published by Cambridge University Press (2017) © S. Friedli and Y. Velenik www.unige.ch/math/folks/velenik/smbook 3.1. Finite-volume Gibbs distributions 81 Figure 3.1: Left: the one-dimensional torus T12. Right: the two-dimensional torus T16. Definition 3.2. The Gibbs distribution of the Ising model in Vn with periodic boundary condition, at parameters β and h, is the probability distribution on { 1,1}Vn defined by − per def 1 per µ (ω) per exp H (ω) . Vn ;β,h = − Vn ;β,h ZV ;β,h n ¡ ¢ The normalization constant def Zper exp H per (ω) Vn ;β,h = − Vn ;β,h ω ΩV ∈X n ³ ´ is called the partition function in Vn with periodic boundary condition. I Finite volumes with configurations as boundary condition. It will turn out to be useful to consider the Ising model on the full lattice Zd , but with configurations which are frozen outside a finite set. Let us thus consider configurations of the Ising model on the infinite lattice Zd , that is, elements of def d Ω { 1,1}Z . = − Fixing a finite set Λ Zd and a configuration η Ω, we define a configuration of b ∈ the Ising model in Λ with boundary condition η as an element of the finite set η def Ω ω Ω : ω η , i Λ . Λ = ∈ i = i ∀ 6∈ © η ª The energy of a configuration ω Ω is defined by ∈ Λ def H (ω) β σ (ω)σ (ω) h σ (ω), (3.1) Λ;β,h = − i j − i {i,j} E b i Λ X∈ Λ X∈ where we have introduced b def d E {i, j} Z :{i, j} Λ ∅, i j . (3.2) Λ = ⊂ ∩ 6= ∼ b © ª Note that EΛ differs from EΛ by the addition of all the edges connecting vertices inside Λ to their neighbors outside Λ. Revised version, August 22 2017 To be published by Cambridge University Press (2017) © S. Friedli and Y. Velenik www.unige.ch/math/folks/velenik/smbook 82 Chapter 3. The Ising Model Figure 3.2: The model in a box Λ (shaded) with boundary condition. + Definition 3.3. The Gibbs distribution of the Ising model in Λ with boundary con- η dition η, at parameters β and h, is the probability distribution on ΩΛ defined by η def 1 µ (ω) exp H (ω) . Λ;β,h = η − Λ;β,h ZΛ;β,h ¡ ¢ The normalization constant η def ZΛ;β,h exp HΛ;β,h(ω) = η − ω Ω X∈ Λ ¡ ¢ is called the partition function with η-boundary condition. η It will be seen later (in particular in Chapter 6) why defining µΛ;β,h on configura- tions in infinite volume is convenient (here, we could as well have defined it on ΩΛ and included the effect of the boundary condition in the Hamiltonian). Two boundary conditions play a particularly important role in the analysis of def the Ising model: the boundary condition η+, for which η+i 1 for all i (see + = + def Figure 3.2), and the boundary condition η−, similarly defined by η− 1 for − i = − all i. The corresponding Gibbs distributions will be simply denoted by µΛ+;β,h and µΛ−;β,h; similarly, we will write ΩΛ+,ΩΛ− for the corresponding sets of configurations. On the notations used below. In the following, we will use the symbol # to denote # ∅ per a generic type of boundary condition. For instance, ZΛ;β,h can denote ZΛ;β,h, ZΛ;β,h η or ZΛ;β,h. In the case of periodic boundary condition, Λ will always implicitly be assumed to be a cube (see below). Following the custom in statistical physics, expectation of a function f with re- spect to a probability distribution µ will be denoted by a bracket: f . When the 〈 〉µ distribution is identified by indices, we will apply the same indices to the bracket. Revised version, August 22 2017 To be published by Cambridge University Press (2017) © S. Friedli and Y. Velenik www.unige.ch/math/folks/velenik/smbook 3.2. Thermodynamic limit, pressure and magnetization 83 # For example, expectation of a function f under µΛ;β,h will be denoted by def f # f (ω)µ# (ω). 〈 〉Λ;β,h = Λ;β,h ω Ω# X∈ Λ We will often use # and µ# ( ) interchangeably. 〈·〉Λ;β,h Λ;β,h · 3.2 Thermodynamic limit, pressure and magnetization 3.2.1 Convergence of subsets It is well known that various statements in probability theory, such as the strong law of large numbers or the ergodic theorem, take on a much cleaner form when considering infinite samples.
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