Chapter 5: Landau theoryPart 1: order parameters and the Landau free energy

5.1Order parameters

In Chapters 3 and 4 we obtained classical critical exponents by expanding the equation of state near the critical point. The expansion parameters had the same temperature dependence and are called order parameters: the spontaneous magnetization \(M\) of a ferromagnet, and the half-width of the coexistence curve of a fluid, \(\phi\equiv\nu-1 = (V-V_c)/V_c\). An order parameter can be a scalar, a vector, a tensor, a pseudo-scalar, or a group element of a symmetry group such as \(SU(N)\). It is not unique for a given system (\(M\) or \(M^3\) would do): any thermodynamic variable that is zero in the disordered phase and nonzero in an adjacent, usually ordered, phase qualifies.

The Heisenberg and XY models

In the Heisenberg model \(\mathcal H_\Omega = -J\sum_{\langle ij\rangle}\vec S_i\cdot\vec S_j - \vec H\cdot\sum_i\vec S_i\), each \(\vec S_i\) is a vector of fixed magnitude (chosen to be 1), \(|\vec S_i|^2 = 1\), with variable direction. When the system orders below \(T_c\) in more than two dimensions,

a vector in three dimensions: the \(O(3)\) symmetry of the zero-field Hamiltonian, \(\mathcal H_\Omega\{\vec S_i\} = \mathcal H_\Omega\{R\vec S_i\}\) for any rotation \(R\) acting on all spins, is spontaneously broken. With \(\vec H\ne0\), \(\mathcal H_\Omega\) is still invariant under rotations of the spins in the plane perpendicular to \(\vec H\), an \(O(2)\) symmetry. The XY model is the two-component version, \(\vec S_i = (S_{ix},S_{iy})\) and \(\vec M = (M_x,M_y)\), in any spatial dimension \(d = 1,2,3,\dots\). In \(d=2\) it has the Kosterlitz–Thouless transition (Chapter 11), recognized by the 2016 Nobel Prize in Physics to Thouless, Haldane, and Kosterlitz.

A more exotic example: the order parameter of the superfluid transition in \(^3\)He is an anomalous pairing amplitude \(\Psi\), built from quantum field operators, describing Cooper pairs of \(^3\)He atoms in a \(p\)-wave state. It has 18 components and transforms as a representation of \(SO(3)\times SO(3)\times O(2)\): roughly, three orbital angular-momentum values, three spin values, and a complex phase.

5.2Common features of mean-field theories

Van der Waals. In reduced variables \(\pi = 8\tau/(3\nu-1) - 3/\nu^2\), with \(\nu = v/v_c = \rho_c/\rho\) and \(t = \tau-1\). For later convenience take the order parameter to be \(\eta\equiv1/\nu-1 = \rho/\rho_c-1\), so \(\phi = \nu-1 = 1/(1+\eta)-1\approx-\eta\) for \(|\eta|\ll1\). The equation of state of Chapter 4 becomes

which can be derived from a Gibbs-like free energy, regarding \(\eta\) as an independent parameter whose value minimizes

Weiss. With \(\tau = T_c/T\) and \(t = (T-T_c)/T_c\), so \(\tau = 1/(1+t)\) and \(1-\tau\approx t\), the order parameter \(\eta\equiv M\) satisfies (Chapter 3)

(Whether the cubic coefficient is written \(\tfrac13\) or 1 is immaterial: a rescaling of \(\eta\) and \(H\) by constants moves it around without changing any exponent.) The similarity of the two free energies is why the critical exponents, obtained by differentiating the thermodynamic potentials, are the same; in the van der Waals case the role of the magnetic field is played by \(\pi-1-4t\). (Whether these are "Gibbs" or "Helmholtz" free energies depends on the constraints, and does not matter here: the expanded quantity is strictly neither.)

5.3Phenomenological Landau theory

In retrospect, mean-field exponents were inevitable. The Taylor expansions in \(\eta\) and \(t\) contained the most general lowest-order terms, and the absence of a quadratic term in the equation of state (a cubic one in the free energy) was guaranteed by the potential being even in \(\eta\) at zero field. Landau suggested applying similar considerations, at least in spirit, to all phase transitions. The resulting theory, based on general considerations of symmetry and analyticity, is phenomenological Landau theory. (It can also be motivated by a calculation from the microscopic Hamiltonian, with limited use, as we will see.)

Landau theory writes down a function \(L\), the Landau free energy (or functional), depending on the couplings \(\{K_i\}\) and the order parameter \(\eta\). The state of the system is the global minimum of \(L\) with respect to \(\eta\). \(L\) has dimensions of energy and is related to, but not identical with, the Gibbs free energy; thermodynamic functions follow by differentiating \(L\) as if it were (the precise interpretation is discussed in §5.6). The constraints on \(L\) are:

1. \(L\) is consistent with the symmetries of the system.
2. Near \(T_c\), \(L\) is an analytic function of \(\eta\) and \([K]\), and can be expanded in a power series. For a spatially uniform system of volume \(V\), the Landau free energy density is \(\mathcal L\equiv L/V = \sum_{n=0}^\infty a_n([K])\eta^n\).
3. In an inhomogeneous system with order-parameter profile \(\eta(\vec r)\), \(\mathcal L\) is local: it depends only on \(\eta(\vec r)\) and a finite number of its derivatives.
4. In the disordered phase \(\eta = 0\); in the ordered phase near the transition \(\eta\) is small and nonzero (a continuous transition). So \(\eta=0\) solves the minimization for \(T>T_c\), and \(\eta\ne0\) does for \(T\lt T_c\).

We therefore write \(\mathcal L = \sum_{n=0}^{4}a_n([K],T)\eta^n + O(\eta^5)\), in the expectation that \(\eta\) is small and that all the essential physics near \(T_c\) appears at this order, as it did for the van der Waals fluid and the Weiss magnet. Whether the truncation is valid turns out to depend on the dimensionality and on the codimension of the singular point.

Construction. \(\partial\mathcal L/\partial\eta = a_1 + 2a_2\eta + 3a_3\eta^2 + 4a_4\eta^3 = 0\) must be solved by \(\eta=0\) for \(T>T_c\), so \(a_1 = 0\). For an Ising ferromagnet at \(H=0\), with \(\eta = M\), the system is symmetric under \(\eta\to-\eta\): the probability distribution of \(\eta\) in a finite system is even, \(P(\eta) = P(-\eta)\). Assuming \(P\sim e^{-\beta L}\) (the precise connection comes later), \(\mathcal L(\eta) = \mathcal L(-\eta)\), so \(a_3 = a_5 = \dots = 0\):

Analyticity excludes terms like \(|\eta|\). We need \(a_4>0\), otherwise \(\mathcal L\) is minimized by \(|\eta|\to\infty\), whereas we want \(\eta\) to rise from zero to a finite value as the couplings pass through the transition. \(a_0\) is the value of \(\mathcal L\) in the high-temperature phase. It describes the degrees of freedom not captured by the order parameter, a smooth background on which the singular behavior is superimposed; we usually set it to zero. Expanding near \(T_c\), \(a_4 = a_4^0 + (T-T_c)a_4^1+\cdots\), and the temperature dependence will not affect the leading behavior, so \(a_4\) can be taken to be a positive constant. Similarly \(a_2 = a_2^0 + a_2^1(T-T_c)/T_c + O[(T-T_c)^2]\). Minimizing, \(2a_2\eta + 4a_4\eta^3 = 0\) gives \(\eta=0\) or \(\eta = \sqrt{-a_2(T)/2a_4}\); since \(\eta=0\) at \(T=T_c\), \(a_2^0 = 0\). With \(t = (T-T_c)/T_c\) and the usual names \(a\equiv a_2^1\), \(b/2\equiv a_4^0\), and adding the field (whose energy is \(-H\sum_iS_i = -HNM\)),

where \(a\) and \(b\) are phenomenological parameters, obtainable from an appropriate microscopic theory. A term \(\propto H\eta^3\) is also allowed by symmetry (\(H\to-H\), \(\eta\to-\eta\)) but is not a leading term near the critical point. We have considered only the Ising universality class; in general one writes down all scalar terms built from powers and products of the order-parameter components that are consistent with the symmetry of the system (Landau and Lifshitz, Statistical Physics, §145).

Continuous transitions

The figure below is the interactive version of the book's Figure 5.1. At \(H=0\), as \(T\) is lowered through \(T_c\), the single minimum at \(\eta = 0\) splits into two degenerate minima at \(\eta = \pm\eta_s(T)\): a continuous transition. For \(T\lt T_c\), varying \(H\) from negative to positive makes the global minimum jump from one well to the other: the first-order transition of the magnet.

Critical exponents

At \(H=0\), for \(t\lt 0\), \(\eta = \sqrt{-a_2(T)/2a_4} = \sqrt{-at/b} = \sqrt{a(T_c-T)/bT_c}\), so \(\beta = \tfrac12\). For \(t>0\), \(\mathcal L = 0\); for \(t\lt 0\), substituting \(\eta^2 = -at/b\),

Since \(\mathcal L\) plays the role of the free energy at fixed volume, \(C_V = T(\partial S/\partial T)_V = -T\,\partial^2\mathcal L/\partial T^2\) (the thermodynamic square gives \(S = -\partial F/\partial T|_V\)), and

With \(H\ne0\), \(\partial\mathcal L/\partial\eta = 2at\eta + 2b\eta^3 - H = 0\) (the derivative is with respect to \(\eta\)), so \(at\eta + b\eta^3 = \tfrac12H\). On the critical isotherm \(t=0\), \(H\propto\eta^3\): \(\delta = 3\). The isothermal susceptibility is

which for \(t>0\) (\(\eta=0\)) is \(1/2at\propto(T-T_c)^{-1}\) and for \(t\lt 0\) (\(\eta^2 = -at/b\)) is \(1/2(at-3at) = -1/4at\propto(T_c-T)^{-1}\). So \(\gamma = \gamma' = 1\), with the familiar amplitude ratio 2. We have found all the mean-field thermodynamic exponents, in agreement with Weiss and van der Waals. The exponents describing spatial correlations require an extension of Landau theory to inhomogeneous systems (§5.4).

Aside: Landau theory on scale-free networks

For the Ising model on a scale-free network with degree distribution \(p(k)\propto k^{-\lambda}\) (Goltsev, Dorogovtsev, and Mendes, Phys. Rev. E 67, 026123 (2003); see also Rev. Mod. Phys. 80, 1275 (2008)), the heterogeneity of the network adds a non-analytic term to the Landau free energy,

the free-energy counterpart of the singular term in the self-consistency equation of the Chapter 3 appendix. Its sign is fixed by that microscopic theory: \(C\lt 0\) for \(2\lt\lambda\lt3\) and \(C>0\) for \(3\lt\lambda\lt5\).

1. \(2\lt\lambda\lt3\): the lowest-order term is \(C\eta^{\lambda-1}\) with \(1\lt\lambda-1\lt2\), and with \(C\lt 0\) it always favors \(\eta\ne0\). There is no transition: the system is always magnetized.
2. \(3\lt\lambda\lt5\): \(\eta^{\lambda-1}\) lies between \(\eta^2\) and \(\eta^4\), and the \(\eta^4\) term is higher order. Then \(\partial\mathcal L/\partial\eta\approx2at\eta + (\lambda-1)C\eta^{\lambda-2} = 0\) gives \(\eta\propto(-t)^{1/(\lambda-3)}\), so \(\beta = 1/(\lambda-3)\), recovering the result of the Chapter 3 appendix.
3. \(\lambda>5\): \(\eta^{\lambda-1}\) is higher order than \(\eta^4\), so \(2at\eta + 2b\eta^3\approx0\) and \(\beta = \tfrac12\), the ordinary mean-field result.

First-order transitions

Landau theory assumes \(\eta(t)\to0\) as \(t\to0\). Without the symmetry argument, the general \(\partial\mathcal L/\partial\eta = a_1 + 2a_2\eta + 3a_3\eta^2 + 4a_4\eta^3\) still has \(a_1 = 0\) (so that \(\eta=0\) for \(T>T_c\)), but \(a_3\) can be nonzero:

For \(H=0\), \(\eta(2at + 2b\eta^2 + 3C\eta) = 0\), so \(\eta = 0\) or

A real nonzero solution requires \(c^2 - at/b>0\), i.e. \(t\lt t^*\equiv bc^2/a>0\): it appears at a temperature above \(T_c\) (the temperature where \(t\) vanishes), whereas for a continuous transition \(\eta\ne0\) only for \(t\lt 0\). What is going on? Take \(C\lt 0\) without loss of generality. For \(t\lt t^*\) a secondary minimum and a maximum develop in addition to the minimum at \(\eta=0\). At \(t = t_1\) the secondary minimum reaches \(\mathcal L = 0\), and for \(t\lt t_1\) it becomes the global minimum, so the minimizing \(\eta\) jumps discontinuously from \(0\) to a nonzero value: a first-order transition. Solving \(\mathcal L = 0\) and \(\partial\mathcal L/\partial\eta = 0\) together gives

At the first-order transition \(\eta(t_1)\) is not arbitrarily small, so Landau theory, an expansion in small \(\eta\), is not strictly valid there. Unless a symmetry forces \(C=0\), a cubic term will in general cause a first-order transition.

When the order parameter has several components, \(\mathcal L\) is built from combinations of the components that are invariant under the symmetries of the problem. A sufficient but not necessary condition for a continuous transition is that there are no cubic invariants. That it is not necessary is shown by the replica theory of spin glasses and rubber: the \(n\times n\) order parameter \(Q^{\alpha\beta}\) becomes, as \(n\to0\), a non-negative function \(q(x)\), and the Landau free energy, a functional of \(q(x)\), has a cubic term. But its sign is such that the secondary minimum that would preempt the continuous transition occurs only for negative, unphysical values of \(q\), so the transition is continuous despite the cubic term.

The Landau criterion is, of course, a statement made within mean-field theory, and fluctuations can change the order of a transition. Two examples: (a) the three-state Potts model in two dimensions, where mean-field theory predicts a first-order transition but the actual transition is continuous; (b) the type I superconductor–normal metal transition in three dimensions, where mean-field theory predicts a continuous transition but fluctuations of the electromagnetic field make it first order.

5.4Toward inhomogeneous systems

An order parameter \(\eta(\vec r)\) varying in space can result from an inhomogeneous field \(H(\vec r)\), and the spatial correlations grow as \(T\to T_c\): the correlation length \(\xi(T)\) is the linear dimension of patches of spins in which the magnetization is approximately constant. Describing them requires coarse graining, the subject of Part 2.

Chapter 5 continues in Part 2.

Further reading

• N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group (Addison-Wesley, 1992), Chapter 5.
• L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1, 3rd ed. (Pergamon, 1980), §145.
• A. V. Goltsev, S. N. Dorogovtsev, and J. F. F. Mendes, "Critical phenomena in networks," Phys. Rev. E 67, 026123 (2003).
• S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, "Critical phenomena in complex networks," Rev. Mod. Phys. 80, 1275 (2008).
• P. M. Goldbart and N. Goldenfeld, "Microscopic theory for cross-linked macromolecules," Phys. Rev. A 39, 1402 (1989).