Chapter 3: How phase transitions occur in practicetransfer matrices, low-temperature expansions, and mean-field theory

3.1Ad hoc solution methods

The nearest-neighbor Ising model in one dimension, with \(J>0\) (ferromagnetic), is

where the range of the bond sum depends on the boundary conditions.

Free boundary conditions, \(H=0\)

With free ends there are \(N-1\) bonds; spin \(N\) has no right-hand neighbor. Define bond variables \(\eta_i = S_iS_{i+1}\), \(i=1,\dots,N-1\), equal to \(+1\) if the neighbors are parallel and \(-1\) otherwise. Then \(\{S_1,\eta_1,\dots,\eta_{N-1}\}\) specifies a state (\(S_2 = S_1\eta_1\), and so on), and the sum factorizes:

We can calculate everything from this.

Periodic boundary conditions, \(H=0\)

With \(S_{N+1}\equiv S_1\) there is an extra bond, \(KS_NS_1\). The state is still specified by \(\{S_1,\eta_1,\dots,\eta_{N-1}\}\), and since \(S_N = S_1\eta_1\cdots\eta_{N-1}\), the extra bond is \(S_NS_1 = \eta_1\eta_2\cdots\eta_{N-1}\), which couples all the \(\eta\)'s. Expanding \(e^{K\eta_1\cdots\eta_{N-1}} = \sum_\alpha (K\eta_1\cdots\eta_{N-1})^\alpha/\alpha!\) restores the factorization:

separating even \(\alpha\) (which sum to \(\cosh K\)) from odd \(\alpha\) (which sum to \(\sinh K\)). Hence

Recursion

Another method for \(H=0\) with free boundaries: adding spin \(N+1\) to a chain of \(N\),

because \(\sum_{S_{N+1}}e^{KS_NS_{N+1}} = 2\cosh(KS_N) = 2\cosh K\) regardless of \(S_N\) (cosh is even). Iterating from \(Z(1)=2\) gives \(Z(N) = 2(2\cosh K)^{N-1}\), as before.

Effect of boundary conditions

For free boundaries, \(F_\Omega = -Nk_BT\big[\log2 + \tfrac{N-1}{N}\log\cosh K\big] \to -k_BTN\big[\log(2\cosh K) + O(1/N)\big]\). For periodic boundaries, \(F_\Omega = -k_BT\log\big[(2\cosh K)^N\big(1+\tanh^NK\big)\big] \to -k_BTN\log(2\cosh K)\), since \(0\le\tanh K\lt 1\). The difference between boundary conditions becomes negligible as the system grows.

3.2The transfer matrix

The ad hoc methods are hard to apply when \(H\ne0\). With periodic boundary conditions, split the field term symmetrically between neighboring bonds:

treating each factor as an element of a matrix

By the matrix multiplication rule \((\mathbf{BC})_{ij} = \sum_kB_{ik}C_{kj}\), each sum over an intermediate spin is a matrix product, so

where \(\lambda_{1,2}\) are the eigenvalues of \(\mathbf T\). To see the last step, diagonalize: since \(\mathbf T\) is real and symmetric, there is a matrix \(\mathbf S\) of eigenvectors with \(\mathbf S^{T} = \mathbf S^{-1}\), the eigenvalues are real, and \(\mathbf T' = \mathbf S^{-1}\mathbf T\mathbf S = \operatorname{diag}(\lambda_1,\lambda_2)\). The cyclic property of the trace, \(\operatorname{Tr}(\mathbf{ABC}) = \operatorname{Tr}(\mathbf{BCA})\), gives \(\operatorname{Tr}\mathbf T^N = \operatorname{Tr}\mathbf T'^N\). With \(\lambda_1>\lambda_2\),

The eigenvalues follow from \(\det(\mathbf T-\lambda) = \lambda^2 - 2e^K\cosh h\,\lambda + e^{2K}-e^{-2K} = 0\):

3.3Phase transitions

For \(T>0\) the square root in \(f = -J - k_BT\log\big[\cosh h + \sqrt{\sinh^2h+e^{-4K}}\big]\) is strictly positive for real \(h\) and \(K\), so \(f\) is manifestly analytic for \(T>0\). This agrees with the heuristic argument of Chapter 2: there is no phase transition for \(T>0\) in one dimension. The only possibilities for a transition are \(T=0\) or \(h=0\) with \(e^{-4K}\to0\); there the two eigenvalues become degenerate, both tending to \(e^K\), so that \(\lambda_2/\lambda_1\to1\).

The same conclusion follows from Perron's theorem: for an \(N\times N\) matrix (\(N\lt\infty\)) with all \(A_{ij}>0\), the eigenvalue of largest magnitude is (a) real and positive, (b) non-degenerate, and (c) an analytic function of the \(A_{ij}\). In one dimension, the transfer matrix for finite-ranged interactions satisfies these requirements, so \(\lambda_1\ne0\), \(\lambda_1\ne\lambda_2\), and \(\lambda_1\) is analytic: there is no transition for \(T>0\). Why can't the theorem be applied in higher dimensions? In one dimension the Ising transfer matrix is \(2\times2\); in two and higher dimensions it is \(\infty\times\infty\) in the thermodynamic limit, and Perron's theorem does not apply.

What happens at \(T=0\) (\(K\to\infty\))? Then \(\lambda_1 = e^K\big[\cosh h + |\sinh h|\,(1+O(e^{-4K}))\big]\) and \(\cosh h + |\sinh h| = e^{|h|}\), so \(\lambda_1 = e^{K+|h|}\) and

even for finite \(N\), as expected from Chapter 2. The non-analytic behavior comes from \(\sqrt{h^2} = |h|\); for \(T>0\), the smoothed version \(\sqrt{h^2+\epsilon^2}\) is analytic at \(h=0\) as long as \(\epsilon\ne0\).

3.4Thermodynamic properties

At \(h=0\), \(\lambda_1 = e^K(1+e^{-2K}) = 2\cosh K\) and \(Z = (2\cosh K)^N\) as \(N\to\infty\), confirming the ad hoc result. Then

At low temperature the free energy is dominated by the energy, and at high temperature by the entropy.

The heat capacity has no singularity, only a broad peak: \(x^2\operatorname{sech}^2x\) is maximal at \(x\approx1.2\), i.e. \(k_BT\approx0.83J\). This is a Schottky anomaly, typical of a system with a gap in its excitation spectrum. For general \(H\), with \(w^2\equiv e^{-4K}\) the relative Boltzmann weight of two configurations differing by a single spin flip (\(\uparrow\uparrow\uparrow\) vs \(\uparrow\downarrow\uparrow\), energy difference \(4J\)),

At \(K=\infty\), \(w=0\) and \(M=\pm1\) depending on the sign of \(h\); for \(T>0\), \(w\) smooths this step. The isothermal susceptibility \(\chi_T\equiv\partial M/\partial H\) follows for small fields from \(\sinh h\approx h\), \(M\approx h/w = he^{2K}\):

3.5Spatial correlations

In zero field and for \(T>0\), \(\langle S_i\rangle = 0\) in one dimension. The two-point correlation function

measures correlations in the fluctuations of spins at different sites. It is also related to the probability that spins \(i\) and \(j\) have the same value: \(P_{ij}\equiv\langle\delta_{S_iS_j}\rangle = \langle\tfrac12(1+S_iS_j)\rangle = \tfrac12 + \tfrac12\big(G(i,j)+\langle S_i\rangle\langle S_j\rangle\big)\).

Ad hoc calculation. With free boundaries, give each bond its own coupling \(K_i\). Then \(Z_N\{K_i\} = 2^N\prod_{i=1}^{N-1}\cosh K_i\), and differentiating brings down spins:

(With all \(K_i = K\), this gives \(E = -J\sum_i\langle S_iS_{i+1}\rangle = -NJ\tanh\beta J\), as before.) For larger separations, use \((S_{i+1})^2 = 1\):

and by induction \(G(i,i+j) = \tanh K_i\tanh K_{i+1}\cdots\tanh K_{i+j-1}\). With uniform couplings,

The result depends only on \(j\): it is translationally invariant, as it must be when \(K_i\) does not depend on \(i\) (in a continuous system, \(G(\vec r,\vec r') = G(\vec r-\vec r')\)), even in a finite system away from the boundaries. At \(T=0\), \(\tanh K = 1\) and \(G=1\) for all \(j\): a perfectly correlated state with long-range order. The correlation length \(\xi\) measures the distance over which spins are correlated with probability of order one. As \(K\to0\), \(\coth K\to\infty\) and \(\xi\to0\); as \(K\to\infty\), \(\coth K\approx1+2e^{-2K}\) and

an essential singularity: \(\xi\) diverges exponentially fast. Contrast this with the generic behavior near a continuous transition at \(T_c>0\), \(\xi(T)\sim(T-T_c)^{-\nu}\), with \(\nu\) a critical exponent.

Correlations from the transfer matrix

For \(H\ne0\), write the spin at site \(i\) inside the chain of matrix products. Summing over \(S_i\) gives a matrix element \(A_{ab} = \sum_{S_i}T_{aS_i}S_iT_{S_ib}\), i.e. \(\mathbf A = \mathbf T\sigma_z\mathbf T\) with the Pauli matrix \(\sigma_z = \operatorname{diag}(1,-1)\), and by the cyclic property

with \(e,f,g,k\) functions of \(h\) and \(K\) computed from the eigenvectors (Exercise 3-1). Then \(\langle S_i\rangle = (e\lambda_1^N + k\lambda_2^N)/(\lambda_1^N+\lambda_2^N)\to e\), and similarly

Again the correlations decay exponentially for \(T>0\), because \(\lambda_1>\lambda_2\). At \(h=0\), \(\lambda_1 = 2\cosh K\), \(\lambda_2 = 2\sinh K\), and \(\xi = 1/\log\coth K\), as before. Notes: (1) \(\xi\) cannot diverge unless \(\lambda_1=\lambda_2\); there can be no phase transition unless the largest eigenvalue of the transfer matrix becomes degenerate, a general result. (2) For \(h\ne0\), \(\lambda_1>\lambda_2\), so there is no transition. (3) For \(h=0\), the eigenvalues become degenerate when \(K=\infty\), so a transition can occur at \(T=0\), where \(\xi\to\infty\).

3.6Low-temperature expansion

Start at \(T=0\) and raise the temperature slightly. Consider \(\mathcal H_\Omega\{S\} = -J\sum_{\langle ij\rangle}S_iS_j\) in zero field on a \(d\)-dimensional hypercubic lattice with coordination number \(z=2d\). At \(T=0\) all spins are aligned; choose the ground state \(\uparrow\). For \(T>0\), clusters of \(1, 2, \dots\) flipped spins appear, each with an energy and a degeneracy:

At low temperature few spins are flipped. With \(w^2 = e^{-4\beta J}\),

with \(B_1 = N(w^2)^d\) and \(B_2 = dN(w^2)^{2d-1} + \tfrac12N(N-1-2d)(w^2)^{2d}\). For \(d>1\), \(B_k\ll B_{k-1}\), and expanding \(\log(1+\epsilon)\approx\epsilon-\epsilon^2/2\) to second order gives \(B_1 + B_2 - \tfrac12B_1^2\). The terms of order \(N^2\) cancel, leaving an extensive free energy:

For \(d=1\), the exact transfer-matrix result is \(f = -k_BT\log(2\cosh K) = -J - k_BT\log(1+e^{-2K}) \approx -J - k_BTe^{-2J/k_BT}\). The factor \(e^{-2J/k_BT}\) disagrees with the low-temperature expansion, which starts at \((w^2)^d = e^{-4J/k_BT}\). The problem is the non-commutativity of the limits \(T\to0\) and \(N\to\infty\). At finite \(N\),

so \(f = -J - k_BTNe^{-4K}+\cdots\), which in the thermodynamic limit does not reduce to the expansion and is unbounded below: the ground state \(E/N=-J\) is not stable for \(T>0\). The expansion is valid only for \(d>1\), where the \(B_k\) decrease with \(k\). In \(d=1\), two adjacent down spins contribute \(d(w^2)^{2d-1} = (w^2)^1\), the same order as a single flip \((w^2)^d\), and so on for larger blocks. Blocks of adjacent reversed spins, which cost only the two walls at their ends, are the dangerous thermal fluctuations that destroy the ordered state in \(d=1\). This makes concrete the domain-wall argument of §2.8.

3.7Mean-field theory

Mean-field theory is the simplest treatment of an interacting statistical system, short of ignoring the interactions altogether. Consider Weiss's theory for the nearest-neighbor Ising model in \(d\) dimensions,

Suppose first that the spins are independent (\(J=0\)). Then

a paramagnet. When \(J\ne0\), each spin feels an effective field from the magnetic moments of all the others, which is itself unknown at the outset. Write \(\mathcal H_\Omega = -\sum_iS_iH_i\) with

the external field, the mean field, and the fluctuation. Ignoring the fluctuation term, \(H_i = H + 2dJM\) (\(z=2d\) neighbors, each with \(\langle S_j\rangle = M\)), and the paramagnet result becomes

the self-consistency equation. Graphically, for \(T>T_c\) the tanh curve lies below \(y=M\) for \(M>0\), and the only intersection is \(M=0\). For \(T\lt T_c\) there are also intersections at \(\pm M_s(T)\). Non-analyticity arises from how the solution changes with \(T\), and the mean-field critical temperature is

Critical behavior

Let \(\tau = T_c/T\), so that \(1/k_BT = \tau/2dJ\) and \(M = \tanh(H/k_BT + M\tau)\). Using \(\tanh(x+y) = (\tanh x+\tanh y)/(1+\tanh x\tanh y)\),

for small \(H\) and \(M\). This one expansion yields all the mean-field exponents.

• Order parameter (\(H=0\), \(T\to T_c^-\)): \(M^2\big(\tau(1-\tau)+\tau^3/3\big) = -(1-\tau)\), so \(M^2\approx3(T_c-T)/T_c\) and \(M\propto(T_c-T)^{\beta}\) with \(\beta = \tfrac12\).
• Critical isotherm (\(\tau=1\)): \(H/k_BT\approx M^3/3\), so \(H\sim M^{\delta}\) with \(\delta = 3\).
• Susceptibility: differentiating, \(1/k_BT = \chi_T(1-\tau) + 3M^2\chi_T(\tau-\tau^2+\tfrac13\tau^3)\). For \(T>T_c\), \(M=0\) and \(\chi_T = 1/k_B(T-T_c)+\cdots\). For \(T\lt T_c\), \(M^2\approx3(T_c-T)/T\) gives \(\chi_T = 1/2k_B(T_c-T)+\cdots\). With \(\chi_T\sim|T-T_c|^{-\gamma}\), \(\gamma = \gamma' = 1\), and the amplitude below \(T_c\) is half that above.
• Heat capacity: \(\bar E = \langle\mathcal H\rangle_{\rm MF} = -J\sum_{\langle ij\rangle}\langle S_i\rangle\langle S_j\rangle = -NJdM^2\). For \(T>T_c\), \(\bar E=0\) and \(C=0\). For \(T\to T_c^-\), \(M^2\approx3(T_c-T)/T_c\) gives \(\bar E\approx-3dNJ(1-T/T_c)\) and \(C/N = 3dJ/T_c = \tfrac32k_B\). The heat capacity jumps discontinuously at \(T_c\): \(\alpha = 0\) (discontinuity), with \(C\sim|T-T_c|^{-\alpha}\).

Spatial correlations and the susceptibility sum rule

Mean-field theory and spatial correlations might seem antithetical: how can there be interesting correlations when fluctuations have been averaged away? The answer is that the correlation length governs not only the spatial extent of order-parameter fluctuations but also how the order parameter varies in space in response to an inhomogeneous external field. A self-consistent solution can still be built for such a perturbation in the spirit of mean-field theory, as we will see in Landau theory.

The connection is the static susceptibility sum rule. With \(Z_\Omega = \operatorname{Tr}\exp\big[\beta J\sum_{\langle ij\rangle}S_iS_j + \beta H\sum_iS_i\big]\),

with \(a\) the lattice spacing. A divergent \(\chi_T\) must therefore be reflected in \(G\).

We will see later that for \(|\vec r|\gg\xi\),

whose integral converges when \(\xi\) is finite. As \(T\to T_c\), \(\chi_T\) diverges, so \(\xi\) must diverge too (as we saw in \(d=1\) as \(T\to0\)). Assuming isotropy and substituting \(z = r/\xi\),

and the \(z\) integral converges, so \(\xi\sim\big((T-T_c)/T_c\big)^{-\nu}\) with \(\nu = \tfrac12\). Finally, the exponent \(\eta\) describes how \(G\) behaves at large distances exactly at the critical point, \(G(r)\sim r^{-(d-2+\eta)}\); the same mean-field calculation gives \(\eta = 0\), though in principle \(\eta\) can be nonzero.

How good is mean-field theory?

The table compares mean-field exponents with experiments on fluids (which, by the lattice-gas mapping, should be in the Ising universality class) and with the Ising model in two and three dimensions. The values are illustrative, as quoted by Goldenfeld from the literature of the time; \(\nu\) and \(\delta\) are not independent of the others, being related by scaling laws.

The mean-field exponents are in reasonable agreement with experiment and the three-dimensional Ising model, with clear systematic differences. They do not depend on dimension, whereas the exact exponents do. Mean-field theory can give dimension-dependent exponents: for the size of a self-avoiding walk, \(R\sim N^\nu\) with \(\nu_{\rm MFT} = 3/(2+d)\) (Flory theory), which gives \(1\), \(3/4\), \(3/5\), and \(1/2\) in \(d=1\) to \(4\), compared with exact or best values \(1\), \(3/4\), \(0.588\), and \(1/2\). The dimension dependence is quantitatively off, and the values are slightly but definitely wrong. Still, mean-field theory does exhibit universality, a feature that emerges from Landau theory, and it predicted the correct phase diagram and an expression for \(T_c\). Its predictions are not always reliable, but they are always a good first step toward a more complete theory.

Mean-field \(T_c = 2dJ/k_B\) also overestimates the transition temperature, because it ignores the fluctuations that disorder the system: \(2J\), \(4J\), and \(6J\) in \(d=1,2,3\), against the exact or numerical values \(0\), \(2.269J\), and \(4.511J\).

Appendix: the mean-field Ising model on networks

This appendix, following S. H. Lee, M. Ha, H. Jeong, J. D. Noh, and H. Park, Phys. Rev. E 80, 051127 (2009), applies the annealed approximation and heterogeneous mean-field theory (HMFT) to an Ising model on a network with degree distribution \(p(k)\). Take \(N\) spins connected by \(L\) edges, with adjacency matrix \(a_{ij} = 1\) if \(i\) and \(j\) are connected and \(0\) otherwise (\(\sum_{ij}a_{ij} = 2L\)), and

The annealed approximation replaces \(a_{ij}\) by its average over networks with the same degrees, \(a_{ij}\approx k_ik_j/2L\), where the degree \(k_i = \sum_ja_{ij}\) is the number of neighbors of spin \(i\), \(\sum_ik_i = 2L\), and \(\langle k\rangle = 2L/N\). Ignoring the diagonal terms \(i=j\),

The Hubbard–Stratonovich transformation linearizes the square,

after which the spins decouple:

For \(L\to\infty\) the integral is dominated by the saddle point \(\partial X/\partial y|_{y_0} = 0\) (the method of steepest descents), so \(Z\propto e^{-\beta LX(y_0)}\) with

Since \(\langle S_i\rangle = \beta^{-1}\partial\ln Z/\partial h_i = \tanh[\beta(Jy_0k_i+h_i)]\), this is \(y_0 = \langle kS(k)\rangle/\langle k\rangle\), assuming \(\langle S_i\rangle\) depends only on \(k_i\): a degree-weighted average magnetization, \(-1\le y_0\le1\), which serves as the order parameter. Writing \(m\equiv y_0\), in zero field and replacing the sum over nodes by a sum over the degree distribution,

Expanding about \(m=0\), \(m = \beta J\frac{\langle k^2\rangle}{\langle k\rangle}m - \frac{(\beta J)^3}{3}\frac{\langle k^4\rangle}{\langle k\rangle}m^3+\cdots\). A nonzero solution requires the linear coefficient to exceed 1:

and if all the moments are finite, \(T_c/T - 1 = \frac{\langle k^4\rangle\langle k\rangle^2}{3\langle k^2\rangle^3}\big(\frac{T_c}{T}\big)^3m^2\), so \(m\propto(T_c-T)^{1/2}\), the ordinary mean-field \(\beta = \tfrac12\). For a regular lattice (\(k = z\) for every node) this reduces to Weiss's \(T_c = zJ/k_B\).

For scale-free networks, \(p(k) = ck^{-\gamma}\) (here \(\gamma\) is the degree exponent, not the susceptibility exponent), the moments need not be finite:

• \(2\lt\gamma\lt3\): \(\langle k\rangle\) is finite but \(\langle k^2\rangle\propto\int^\infty k^{2-\gamma}dk\to\infty\), so \(T_c\to\infty\). The system is magnetized at every finite temperature.
• \(3\lt\gamma\lt5\): \(\langle k^2\rangle\) is finite but \(\langle k^4\rangle\propto\int^\infty k^{4-\gamma}dk\to\infty\), and the expansion is singular. Writing \(m = \frac{\langle k^2\rangle}{\langle k\rangle}\beta Jm + \frac{c}{\langle k\rangle}\int_{k_{\min}}^\infty dk\,k^{1-\gamma}\big[\tanh(\beta Jkm)-\beta Jkm\big]\) and substituting \(x = \beta Jkm\), the integral scales as \(m^{\gamma-2}\). Defining the constant \(-A\) by \(\frac{c}{\langle k\rangle}\int(\cdots) = -Am^{\gamma-2}\), we get \(1 = T_c/T - Am^{\gamma-3}\), so

while for \(\gamma>5\), or any distribution with finite \(\langle k^4\rangle\), \(\beta = \tfrac12\).

Further reading

• N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group (Addison-Wesley, 1992), Chapter 3.
• H. A. Kramers and G. H. Wannier, "Statistics of the two-dimensional ferromagnet," Phys. Rev. 60, 252 (1941).
• C. Domb, "On the theory of cooperative phenomena in crystals," Adv. Phys. 9, 149 (1960): low-temperature series.
• J. C. Le Guillou and J. Zinn-Justin, "Critical exponents from field theory," Phys. Rev. B 21, 3976 (1980).
• S. H. Lee, M. Ha, H. Jeong, J. D. Noh, and H. Park, "Critical behavior of the Ising model in annealed scale-free networks," Phys. Rev. E 80, 051127 (2009).
• S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, "Critical phenomena in complex networks," Rev. Mod. Phys. 80, 1275 (2008).