Chapter 2: How phase transitions occur in principlePart 1: from the partition function to the lower critical dimension

2.1Review of statistical mechanics

What do we mean by a phase transition? This chapter examines how one can occur, in principle, in a simple model of a magnet: the Ising model, the "fruit fly" of statistical mechanics. We will use it to illustrate many of the general ideas developed later. The model was proposed by Wilhelm Lenz in 1920 and studied by his student Ernst Ising in 1925; S. G. Brush's 1967 review in Reviews of Modern Physics tells the story.

We need mainly the Helmholtz free energy and the canonical ensemble. A system exchanging energy with a reservoir at temperature \(T\) reaches equilibrium, where a state \(s\) of energy \(E(s)\) occurs with probability given by the Boltzmann factor, \(P(s) \propto e^{-E(s)/k_BT}\). The goal is to compute the partition function \(Z\).

The system is specified by a region \(\Omega\) in which the Hamiltonian \(\mathcal H_\Omega\) is defined (in quantum mechanics, the energy operator; here we consider only classical statistical mechanics). The region has volume \(V(\Omega) \propto L^d\) and surface area \(S(\Omega) \propto L^{d-1}\), where \(L\) is a linear dimension and \(d\) the dimension of space. Boundary conditions may be periodic or hard wall, and the system may be a continuum (a fluid) or live on a lattice (a magnet). For now, \(V(\Omega)\) is finite.

We write the Hamiltonian as

where the \(K_n\) are coupling constants: the external parameters, such as fields, exchange interaction strengths, and the temperature (which we absorb into the \(K_n\)). The \(\Theta_n\) are combinations of the dynamical degrees of freedom. For a magnet these are the (vector) spins \(\vec S_i\) on lattice sites \(1 \le i \le N(\Omega)\), so for example \(\Theta_1 = \sum_i \vec S_i\) and \(\Theta_2 = \sum_{ij}\vec S_i\cdot\vec S_j\). A simple example is the Zeeman coupling \(K_1 = \vec H/k_BT\), \(\mathcal H_\Omega = -\sum_i \vec H\cdot\vec S_i\); we will need to be careful about its meaning when we discuss spontaneous symmetry breaking. The trace is the sum over every possible value of every degree of freedom,

and the partition function and free energy are

All thermodynamic information is contained in derivatives such as \(\partial F_\Omega/\partial K_n\) and \(\partial^2F_\Omega/\partial K_n\partial K_m\), which include bulk effects, surface effects, and finite-size effects. At this stage, with \(\Omega\) finite, there is no information about phase transitions or phases.

2.2The thermodynamic limit

Experience tells us that the free energy is extensive, \(F_\Omega \propto V(\Omega)\). For a finite system we may therefore write

with \(f_b\) the bulk free energy per unit volume and \(f_s\) the surface free energy per unit area. Precisely, the bulk free energy is

provided the limit exists and is independent of \(\Omega\) (which is nontrivial), and the surface free energy is

These limits define the thermodynamic limit. The concepts of phase and phase transition are only sharply defined in this limit.

The thermodynamic limit in a charged system

Consider the energy at \(T=0\) of a uniform charge density \(\rho\) in three dimensions, with the Coulomb interaction \(U(r) = A/r\) between two particles separated by \(r\). Building up a sphere of radius \(R\) shell by shell, the shell theorem (Gauss's law) lets us treat the charge already inside radius \(r\) as concentrated at the center:

which diverges as \(R\to\infty\). Inverse-square forces such as gravity and electrostatics are too long ranged to permit thermodynamic behavior. This is a consequence of allowing charge of only one sign. If charges of both signs are present and mobile, screening changes the interaction from \(1/r\) to \(e^{-r/\lambda_D}/r\), with \(\lambda_D\) the Debye screening length. A fully satisfactory treatment of the stability of matter and a sensible thermodynamic limit requires quantum mechanics: fermions and the Pauli exclusion principle.

Power-law interactions

Generalize to \(U(r) = A/r^{\sigma}\) in \(d\) dimensions. For a uniform system in a \(d\)-dimensional sphere \(\Omega\) of radius \(R\),

after rescaling \(\vec r = R\vec x\), \(\vec r' = R\vec y\). Hence \(E_b = E(R)/V(R) \propto R^{d-\sigma}\), with \(V(R) = V_dR^d\) and \(V_d\) the volume of the unit sphere in \(d\) dimensions. As \(R\to\infty\) the thermodynamic limit is well defined if and only if \(\sigma > d\).

The integral \(C\) may itself diverge as \(\vec x \to \vec y\) when \(\sigma > d\). In practice this is harmless: charges sit on a lattice of spacing \(a\), or have a hard core of radius \(a\), so \(U \sim r^{-\sigma}\) only for \(r>a\). With this cutoff,

where \(S_d\) is the surface area of the unit sphere. What about \(\sigma = d\)? Then the inner integral gives \(\log(R/a)\), which still diverges as \(R\to\infty\). The thermodynamic limit exists only for \(\sigma>d\).

2.3Phase boundaries and phase transitions

Suppose \(f_b[K]\) exists, and that there are \(D\) coupling constants \(K_1,\dots,K_D\); the phase diagram is then \(D\)-dimensional. The function \(f_b[K]\) is analytic almost everywhere. Its possible non-analyticities lie on points, lines, planes, hyperplanes, and so on: singular loci of dimension \(D_s = 0, 1, 2, \dots\). An important invariant of each type of locus is its codimension

Adding an extra variable to \(\{K_n\}\) takes \(D\to D+1\) and \(D_s\to D_s+1\), so \(C\) is unchanged. Regions in which \(f_b\) is analytic are phases, and loci of codimension \(C=1\), which separate phases, are phase boundaries. In the pressure–temperature plane of a simple substance, \(D=2\) and the coexistence lines have \(D_s = 1\), \(C=1\).

There is an ambiguity in the definition of a phase boundary. The liquid–gas line ends at a critical point, so one can travel around it from liquid to gas without crossing any boundary: liquid and gas have the same degree of symmetry. The solid–liquid line cannot simply end in the same way, because a fluid has a higher degree of symmetry than a crystalline solid.

The free energy \(f_b[K]\) is everywhere continuous, but where phase boundaries exist they come in two classes:

1. One or more of the \(\partial f_b/\partial K_i\) is discontinuous across the boundary: a first-order phase transition.
2. All of the \(\partial f_b/\partial K_i\) are continuous across the boundary: a continuous phase transition. (The name "second-order transition" is not encouraged.)

Finite-size effects and the correlation length

In practice even \(10^{23} \ne \infty\), so the thermodynamic limit is never attained. In a finite system a change of physical properties occurs over some finite range of parameters, a finite-size effect, discussed in detail later. A simple criterion for when predictions based on \(f_b[K]\) are unreliable uses the correlation length \(\xi\): the spatial extent of fluctuations of a quantity about its average, for example the size of "droplets" of near-liquid density floating in a gas in thermal equilibrium.

Near a continuous transition \(\xi\) depends strongly on temperature and diverges at the transition itself, which gives rise to scaling behavior. In a finite system, \(\xi\) cannot diverge: it cannot exceed \(L\), the characteristic linear dimension of the system. When \(T\) is close enough to \(T_c\) that \(\xi\) of the infinite system would exceed \(L\), the finite system departs from the ideal behavior described by \(f_b\).

2.4The role of models

The phase-transition literature is full of models: the Ising, Heisenberg, Potts, Baxter, and F models, and even the nonlinear sigma model. They are often systems whose partition function can be computed exactly, perhaps only in some limit or special dimension, or reduced to a finite number of integrals. Beautiful mathematics has come from this enterprise, with connections to solitons, Ramanujan identities, string theory, and topology. The techniques are specialized, however. Onsager solved the two-dimensional Ising model in zero field in 1944, but the model has still not been solved in a field, nor in three dimensions. What we really need is an approach that always works, albeit approximately; exactly solvable models remain enormously important as benchmarks for approximation methods.

There are two opposing views of what models are for. The traditional one builds a faithful representation of the physical system with as many fine details as possible, and tunes or adds parameters when theory and experiment disagree; quantum chemistry is a field where this is appropriate. The other view holds that such fine detail may not be needed to describe a particular phenomenon. Many parameters may be irrelevant, and the measurable quantities may form dimensionless numbers or even universal functions that, to a good approximation, do not depend on microscopic details. The BCS theory of superconductivity is an example: it predicts dimensionless ratios such as \(2\Delta/k_BT_c \approx 3.5\) for all weak-coupling superconductors. In this view one starts from the correct minimal model, the one that most economically caricatures the essential physics. That such a viewpoint works is a consequence of renormalization-group arguments, and the RG also provides a framework for identifying and calculating universal observables.

2.5The Ising model

The Ising model is a model of a ferromagnet or antiferromagnet on a lattice. Lenz and Ising first studied it in 1925 and found no phase transition for \(T>0\) in \(d=1\). They incorrectly concluded that the model has no transition at nonzero temperature for \(d>1\) either, and so could not describe real magnets.

Consider a \(d\)-dimensional lattice of sites \(i = 1,\dots,N(\Omega)\), hypercubic unless stated otherwise. The degrees of freedom are classical spin variables \(S_i = \pm1\) on the vertices of the lattice, so the system has \(2^{N(\Omega)}\) states. The spins interact with an external magnetic field \(H_i\), which may vary from site to site, and with each other through exchange interactions \(J_{ij}\) (two-spin), \(K_{ijk}\) (three-spin), and so on. A general Hamiltonian is

and we will neglect three-spin and higher interactions. Now

and thermodynamic properties follow by differentiation. For example, since \(P(\{S_i\}) = e^{-\beta\mathcal H_\Omega(\{S_i\})}/\operatorname{Tr}\,e^{-\beta\mathcal H_\Omega}\), the average magnetization at site \(i\) is

Even without an external field, or with a uniform one, it is useful to keep \(\sum_i H_iS_i\) in \(\mathcal H_\Omega\): the \(H_i\) can be set to any desired value, including zero, after differentiating. This is the method of sources.

The thermodynamic limit for the Ising model

For a finite system \(F_\Omega\) is an analytic function of each of its arguments, because the partition function

with \(E_n\) the energy of the \(n\)th state, is a finite sum of analytic terms. Phase transitions can therefore only arise in the thermodynamic limit, which requires \(\sum_{j\ne i}|J_{ij}| \lt \infty\). The range of the interaction and the dimensionality determine whether this limit exists. For \(J_{ij} = A|\vec r_i - \vec r_j|^{-\sigma}\) we need \(\sigma>d\), exactly as for the charged system; in \(d=3\) this excludes magnetic dipole–dipole interactions \(\propto r^{-3}\). However, as long as the dipoles are not all aligned, the thermodynamic limit can be shown to exist and to be independent of the shape of \(\Omega\) (see Griffiths in Domb and Green, vol. 1).

2.6Analytic properties of the Ising model

Assume \(f_b[K]\) exists. With \(\langle ij\rangle\) denoting nearest-neighbor pairs, the nearest-neighbor Ising Hamiltonian with a uniform field is

and the average magnetization (magnetic moment) per site is

The principal analytic properties of \(f\) are postulates in thermodynamics but can be proved in statistical mechanics:

1. \(f\lt 0\).
2. \(f(H,J,T,\dots)\) is continuous.
3. \(\partial f/\partial T\), \(\partial f/\partial H\), … exist almost everywhere; right and left derivatives exist everywhere and are equal almost everywhere.
4. The entropy per site \(S \equiv -\partial f/\partial T \ge 0\) almost everywhere.
5. \(\partial f/\partial T\) is monotonically non-increasing with \(T\), i.e., \(\partial^2f/\partial T^2 \le 0\). Hence the specific heat at constant field \(C_H \equiv T\,\partial S/\partial T|_H = -T\,\partial^2F/\partial T^2|_H \ge 0\).
6. \(\partial f/\partial H\) is monotonically non-increasing with \(H\), i.e., \(\partial^2f/\partial H^2 \le 0\). Hence the isothermal susceptibility \(\chi_T \equiv \partial M/\partial H|_T = -\partial^2f/\partial H^2 \ge 0\).

Properties (a) and (d) follow from simple direct arguments; the others follow from convexity.

Convex functions

A function \(f(x)\) is convex up (respectively down) if and only if, for all \(0\le\alpha_1,\alpha_2\le1\) with \(\alpha_1+\alpha_2=1\),

Graphically, a convex-up function always lies above any chord, and a convex-down function always lies below. One can prove that if \(f(x)\) is bounded and convex up (down), then (i) \(f\) is continuous, (ii) \(f\) is differentiable almost everywhere, and (iii) \(df/dx\) is monotonically non-increasing (non-decreasing). A discontinuity in \(f\) would put it both above and below some chord, but a point discontinuity in slope, a cusp, is allowed.

Convexity of \(f\) in \(H\). Recall the Hölder inequality: for sequences \(g_k, h_k \ge 0\) and \(\alpha,\beta\ge0\) with \(\alpha+\beta=1\),

Write \(Z_\Omega(H) = \operatorname{Tr}\,e^{\beta H\sum_iS_i}\,G[S]\) with \(G[S] = \exp\{\beta J\sum_{\langle ij\rangle}S_iS_j+\cdots\}\), and split \(G = G^{\alpha_1}G^{\alpha_2}\). Then

Taking the logarithm (a monotonic function), multiplying by \(-k_BT\), dividing by \(N(\Omega)\), and taking the thermodynamic limit,

Convexity in \(T\) and \(J\) is left as an exercise.

2.7Symmetry properties of the Ising model

Lemma. For any function \(\phi\) of the spin configuration,

To see this, write out the terms: every term occurs exactly once in the summation on each side. QED

Time-reversal (\(\mathbb Z_2\)) symmetry

Reversing all magnetic moments, including the external field, leaves the energy unchanged: \(\mathcal H_\Omega(H,J,\{S_i\}) = \mathcal H_\Omega(-H,J,\{-S_i\})\). Thus

Sublattice symmetry

In zero field, divide the hypercubic lattice into two interpenetrating sublattices A and B and label the spins accordingly. On a bipartite lattice the spins on sublattice A interact only with spins on sublattice B and vice versa:

Only the sign of \(JS_i^AS_j^B\) matters, so \(\mathcal H_\Omega(0,-J,\{S^A\},\{S^B\}) = \mathcal H_\Omega(0,J,\{-S^A\},\{S^B\})\). Applying \((*)\) to the A spins alone,

In zero field the ferromagnetic (\(J>0\)) and antiferromagnetic (\(J\lt 0\)) Ising models on a hypercubic lattice have the same thermodynamics. The argument fails on a triangular lattice, which is not bipartite. There it is not possible to minimize all the antiferromagnetic interactions simultaneously, as it is on a square lattice: an example of frustration.

2.8Existence of phase transitions

How do we "construct" the phase diagram of a particular system? The energy–entropy argument: at high temperature the entropy always dominates the free energy \(F = E-TS\), which is minimized by maximizing \(S\). At low temperature the internal energy may dominate \(TS\), so \(F\) is minimized by minimizing \(E\). If the macroscopic states selected by the two procedures are different, at least one phase transition has occurred at some intermediate temperature.

The zero-temperature phase diagram

At \(T=0\), \(F=E\). Lemma: if the energy is a sum of terms depending on the spin configuration \([S]\), \(E = \sum_\lambda E_\lambda[S]\), then

Proof: \(E_\lambda[S]\ge\min_{[S]}E_\lambda[S]\) for every \(\lambda\) and every \([S]\); sum over \(\lambda\) and minimize the left side, noting the right side is a constant. QED If one can find a configuration that minimizes each term in the Hamiltonian separately, it achieves the bound and is therefore a ground state, though not necessarily a unique one. Because the \(E_\lambda\) depend on the coupling constants, so does the minimizing configuration.

As the couplings \([K]\) vary, the energy levels of the system move, and the ground state may switch from one level to another. Such a level crossing usually generates a first-order transition, because \(\partial E/\partial K_i\) is discontinuous there. Notice that the thermodynamic limit is not needed for a zero-temperature transition: the non-analyticity occurs not because \(N\to\infty\) but because \(\beta\to\infty\) (\(T\to0^+\)).

Example: the Ising model with \(J>0\). \(-J\sum_{\langle ij\rangle}S_iS_j\) is minimized by \(S_i = S_j\) for all pairs, and \(-H\sum_iS_i\) by \(S_i = +1\) for \(H>0\) and \(-1\) for \(H\lt 0\). Both are satisfied by \(S_i = \operatorname{sgn}H\), so

and the zero-temperature phase diagram has a transition at \(H=0\). At \(H=0\) exactly, all-up and all-down have the same energy, and which is observed depends on the initial conditions in which the system is prepared.

Nonzero temperature in \(d = 1\)

Whether a \(T=0\) transition survives at \(T>0\) depends on the dimensionality. Landau and Peierls showed there is no long-range order in \(d=1\) for \(T>0\), while for \(d=2\) a long-range-ordered (ferromagnetic) state exists above \(T=0\), with a transition at \(T_c>0\) to a paramagnetic (disordered) state. The dimension above which a given transition occurs at \(T>0\) is the lower critical dimension.

At \(T=0\) in \(d=1\) there are two possible phases, all spins up and all spins down. As the temperature is raised, each spin executes a sort of Brownian motion by virtue of being in thermal equilibrium. In the thermodynamic limit, the uncorrelated flipping of a finite number \(n\) of spins cannot destroy long-range order, since \(\lim_{N\to\infty}(N-n)/N = 1\). To destroy it, a nonzero fraction \(f\) of the spins must be reversed on average, giving \(\lim_{N\to\infty}(N-fN)/N = 1-f\lt 1\).

If all the spins are aligned the entropy is zero (\(k_B\log1\)) and \(F_N = -NJ\) at \(H=0\). Now consider a state with two domains, \(\uparrow\cdots\uparrow\,\downarrow\cdots\downarrow\). The interface costs \(\Delta E = 2J\), since one bond changes from \(-J\) to \(+J\), so \(E_N = -NJ+2J\). The interface can sit at any of \(N\) positions (with periodic boundary conditions, for convenience), so \(S_N = k_B\log N\). Hence

The system lowers its free energy by creating a domain wall, at any \(T>0\), and again each time new domains are created; continue until no ordered domains remain at all. The long-range-ordered state is unstable to thermal fluctuations for \(T>0\), so \(M = 0\) in zero field. As a corollary there is no phase transition for \(T>0\) in \(d=1\): at \(H=0\) there are no longer two phases, only the two zero-temperature states \(M=\pm1\) at the ends of the line.

The argument assumed short-ranged interactions, so that the energy difference between the ordered and two-domain states did not depend on \(N\). If instead \(J_{ij} = J/|\vec r_i-\vec r_j|^{\sigma}\), long-range order can persist for \(0\lt T\lt T_c\) as long as \(1\le\sigma\le2\). For \(\sigma\lt 1\) the thermodynamic limit does not exist, consistent with the requirement \(\sigma>d\) of §2.2 (the boundary case \(\sigma=1\) is marginal). For \(\sigma>2\) the interaction counts as short ranged and the argument for the destruction of long-range order at \(T>0\) applies. The difficult case \(\sigma=2\) is treated by Fröhlich and Spencer (1982).

Chapter 2 continues in Part 2, with the Peierls argument in two dimensions.

Further reading

• N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group (Addison-Wesley, 1992), Chapter 2.
• S. G. Brush, "History of the Lenz–Ising model," Rev. Mod. Phys. 39, 883 (1967).
• R. B. Griffiths, "Rigorous results and theorems," in C. Domb and M. S. Green (eds.), Phase Transitions and Critical Phenomena, vol. 1 (Academic Press, 1972).
• R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, 1982).
• J. Fröhlich and T. Spencer, "The phase transition in the one-dimensional Ising model with \(1/r^2\) interaction energy," Commun. Math. Phys. 84, 87 (1982).
• G. H. Wannier, "Antiferromagnetism. The triangular Ising net," Phys. Rev. 79, 357 (1950).