Lecture 28–30 — Phase Transitions, Critical Phenomena, and Landau Theory#
Source: NPTEL Classical Physics, Mod-01 Lec-28 (Phase transitions, Part 2), Lec-29 (Part 3), and Lec-30 (Part 4), Prof. V. Balakrishnan.
This page jumps ahead slightly in Balakrishnan’s syllabus, from the canonical ensemble of Lecture 24–25 to a topic covered a little later in the course. The intervening material (oscillations and normal modes, and further probability distributions in Lec-26–27) isn’t transcribed here yet; see Upcoming Topics for the roadmap. What follows is self-contained regardless.
Recap: one critical point, and one that never appears#
Every simple single-component substance has a \(P\)–\(V\)–\(T\) surface with a liquid–gas coexistence line that runs from the triple point up to a critical point \((T_c, P_c)\) and then simply stops. Past that point there is no sharp transition left to cross: you can walk a substance from gas to liquid along a path that skirts around the critical point, staying in equilibrium the whole way, and never once feel a discontinuity. The solid–liquid coexistence line does no such thing — trace it as far as you like and it never terminates in a critical point.
The reason is symmetry, not energetics. A liquid (like a gas) is homogeneous and isotropic: it looks the same after any translation or rotation. A crystalline solid keeps only a discrete subgroup of those symmetries — the space group of its lattice. Order and symmetry are opposites here: the more ordered the phase, the smaller the group of transformations that leaves it unchanged. Melting a crystal means breaking a discrete symmetry down to nothing in one abrupt step — there is no way to interpolate continuously between “has crystalline order” and “doesn’t,” so the solid–liquid line cannot fade out at a critical point; it must run on indefinitely (or end on a physical boundary, or meet another transition line, as at the triple point). Liquid and gas, by contrast, share exactly the same symmetry — both are isotropic and homogeneous — so nothing forbids the distinction between them from shrinking continuously to zero, which is precisely what happens at \((T_c, P_c)\).
That single observation — that critical points require a symmetry that survives unbroken on both sides of the transition — is the seed of everything below.
The fluid–magnet analogy#
To understand what kind of universal behavior shows up near a critical point, it helps to leave fluids behind and look at magnets instead, because the microscopic model is far simpler and yet, remarkably, produces the same mathematics. The dictionary between the two systems:
Fluid |
Magnet |
|---|---|
Pressure \(P\) |
Magnetic field \(H\) |
Volume \(V\) |
Magnetization \(M\) |
Temperature \(T\) |
Temperature \(T\) |
Equation of state \(P(V,T)\) |
Equation of state \(M(H,T)\) |
Work term \(-P\,dV\) |
Work term \(H\,dM\) (Legendre-conjugate to \(M\,dH\)) |
Liquid / gas |
Ferromagnet / paramagnet |
Critical point \((T_c, P_c)\) |
Curie point \((T_c, H_c{=}0)\) |
The payoff of building this analogy carefully is that the magnet is the simplest system with a genuine phase transition, so we can push the calculation all the way through by hand — and every conclusion transfers back to the liquid–gas transition (and, as it turns out, to a huge variety of unrelated systems) essentially unchanged.
A minimal paramagnet, and Curie’s law#
Model a paramagnetic substance as \(N\) independent, non-interacting atomic dipole moments \(\mu\), each sitting in an external field \(H\) and in thermal contact with a heat bath at temperature \(T\). In the simplest version each dipole can only point along the field or exactly against it, so its energy is \(\epsilon = \mp \mu H\). The canonical (Boltzmann) weights for these two states immediately give the average magnetization of the sample:
This is the magnetic equation of state, the direct analogue of \(P(V,T)\). It is manifestly bounded, \(|M|\le N\mu\), and grows in a small linear region before saturating — which immediately kills the naive high-school definition of susceptibility as \(M/H\): that ratio simply drifts to zero as \(H\to\infty\) even though \(M\) stays finite. The physically meaningful quantity is the initial slope,
the isothermal susceptibility — exactly analogous to the isothermal compressibility \(\kappa_T = -\tfrac1V(\partial V/\partial P)_T\). Differentiating the equation of state at \(H=0\) gives
The susceptibility diverges as \(T\to 0\): cool a paramagnet enough and an infinitesimal field is enough to saturate it completely. (Pierre Curie discovered this experimentally; he died young, in a Paris street accident, having also been the first to recognize the deep role symmetry plays in the properties of condensed matter — the very theme running through this lecture.)
Drag the slider toward \(T=0\): the curve’s initial slope — the susceptibility — visibly steepens without bound, exactly as \(\chi_T \propto 1/T\) predicts.
Dimensionality hides in the coefficient#
The up/down model is the crudest possible caricature; real dipoles can point in any direction. Relaxing to three dimensions, the magnetization along \(H\) becomes a solid-angle average weighted by the Boltzmann factor \(e^{\beta\mu H\cos\theta}\), which evaluates to the Langevin function:
Expanding \(L(x)\) for small \(x\) gives \(L(x)\approx x/3\), so \(\chi_T = N\mu^2/(3k_BT)\) — Curie’s law again, but now with a \(\tfrac13\) where the 1D model had a bare \(1\). Restricting the dipole to a plane instead (a 2D model) replaces the solid-angle measure \(\sin\theta\,d\theta\) with a bare \(d\theta\), and the integral no longer closes in elementary functions — it becomes a ratio of modified Bessel functions, \(I_1(x)/I_0(x)\), whose small-\(x\) slope is \(\tfrac12\).
Three different models, three different numbers — \(1\), \(\tfrac12\), \(\tfrac13\) — but they’re not independent facts to memorize. Each is exactly \(1/(\text{number of components the dipole is free to explore})\): it traces back to a \(\cos^2\theta\) average that comes out to \(1\), \(\tfrac12\), or \(\tfrac13\) depending on whether \(\theta\) ranges over a line, a plane, or a sphere. This is the first hint that critical behavior depends on the dimensionality of the space the order parameter lives in — a theme that becomes central once critical exponents enter the picture.
Turning on interactions: the Weiss molecular field#
Curie’s law diverges only at \(T=0\) — but real ferromagnets order at a finite Curie temperature. Something is missing: the dipoles have been treated as completely independent, with no interaction between them at all. The natural culprit, classical dipole–dipole coupling, actually makes things worse: two bar magnets side by side prefer to align anti-parallel (it lowers their mutual energy), so a chain of purely classical dipoles would settle into an alternating up-down-up-down pattern with zero net magnetization — no permanent magnet at all. Real ferromagnetism comes from the quantum-mechanical exchange interaction: short-ranged (it dies off exponentially with distance) but strong enough between nearest neighbors to overwhelm the weak, long-ranged (\(1/r^3\)) classical dipole coupling and force parallel alignment instead.
Rather than solving the full quantum many-body problem, Weiss proposed a phenomenological fix in exactly the spirit of the Van der Waals correction to the ideal gas: replace the true internal field seen by each dipole with an effective field that grows with the sample’s own magnetization,
and simply substitute this into the paramagnet’s equation of state. The result is implicit — \(M\) now appears on both sides —
a transcendental equation with no closed-form solution, but one that can be solved graphically. Setting \(H=0\) and writing \(m \equiv M/(N\mu)\), \(T_c \equiv \mu^2\lambda/k_B\):
\(m=0\) is always a root. The question is whether there’s a nonzero one: draw the straight line \(y=m\) and the curve \(y=\tanh\!\big((T_c/T)m\big)\) and look for where they cross away from the origin. The curve’s initial slope at \(m=0\) is exactly \(T_c/T\). For \(T>T_c\) that slope is less than \(1\), so the curve stays under the line everywhere except at the origin — no other crossing exists. For \(T<T_c\) the slope exceeds \(1\), the curve pokes above the line near the origin before bending over to saturate at \(\pm1\), and two new crossings appear symmetrically at \(\pm m_0(T)\). This is a pitchfork bifurcation: one stable equilibrium (\(m=0\)) splitting into two stable equilibria (\(\pm m_0\)) plus an unstable one (\(m=0\) itself, now a local maximum of the free energy) as \(T\) drops through \(T_c\).
Watch the two off-axis roots peel away from the origin as the temperature slider crosses \(T/T_c = 1\) — that is the onset of spontaneous magnetization: a nonzero \(M\) that persists even after the external field is switched off.
Critical exponents from mean-field theory#
Everything about the behavior near \(T_c\) falls out of expanding \(\tanh\) to cubic order, \(\tanh(\xi)\approx \xi - \xi^3/3\), in the self-consistency equation. Writing \(T\) slightly below \(T_c\) and keeping \(m_0\) small:
The square-root law hugs the exact curve right at the origin and visibly peels away as \(T\) drops further — exactly as expected of a leading-order expansion valid only in the immediate vicinity of \(T_c\).
Two companion exponents fall out of the same equation of state:
On the critical isotherm (\(T=T_c\) exactly, small field \(h=\mu H/k_BT_c\)): the self-consistency equation becomes \(m = \tanh(h+m)\), which to leading nontrivial order gives \(h \approx m^3/3\) — a cubic, not linear, curve through the origin: \(\boxed{h \propto m^{\delta}}\) with \(\delta = 3\).
The susceptibility \(\chi_T = (\partial m/\partial h)_{h=0}\) diverges as \(T\to T_c\) from either side, \(\boxed{\chi_T \propto |T-T_c|^{-\gamma}}\) with \(\gamma=1\) — the Curie–Weiss law, the natural generalization of the bare Curie law now divergent at the finite temperature \(T_c\) rather than only at absolute zero.
What makes \(\beta=\tfrac12\), \(\gamma=1\), \(\delta=3\) worth remembering by name is that the identical exponents appear at the liquid–gas critical point — a system with a totally different microscopic mechanism — because the Van der Waals correction term \(-a/V^2\) (an average attraction proportional to the square of the number density) has exactly the same mathematical structure as the Weiss molecular field \(\propto M\). Both are mean-field theories: they replace a genuinely fluctuating, spatially varying interaction with its average effect. Real 3D magnets measured close to \(T_c\) show \(\beta\approx\tfrac13\) and \(\gamma\approx 1.3\), not the mean-field values — getting to the asymptotic regime demands controlling temperature to milli-kelvin precision, among the hardest quantities in physics to pin down that tightly — but the qualitative agreement between two unrelated systems is the discovery: near a critical point, most microscopic detail becomes irrelevant, and only a few structural features (here, dimensionality) survive.
Why the Ferro-up/Ferro-down line is exactly flat#
The \(H\)–\(T\) phase diagram has the Curie point sitting on a coexistence line at \(H=0\), \(T<T_c\), separating the “Ferro-up” phase from the “Ferro-down” phase — the direct analogue of the liquid–gas coexistence line in the \(P\)–\(T\) plane. But its slope behaves completely differently, and the reason is worth deriving carefully because it recycles a single trick — a Maxwell relation dressed up as a Clausius–Clapeyron equation — across three physically distinct cases.
Starting from \(dQ = T\,dS\) along a coexistence curve and a Maxwell relation,
with \(\Delta\) denoting (final phase) \(-\) (initial phase), unambiguous once you fix which phase is which.
Liquid–gas: the gas is far more disordered, so \(\Delta S = S_{\text{gas}} - S_{\text{liquid}} > 0\) always; likewise \(\Delta V = V_{\text{gas}} - V_{\text{liquid}} > 0\) always. Both numerator and denominator are positive — but \(\Delta V\) is huge (a liquid’s specific volume barely responds to temperature) — so the slope is positive but small: the familiar shallow-sloped boiling curve.
Solid–liquid: the liquid is always more disordered than the crystal, so \(\Delta S = S_{\text{liquid}} - S_{\text{solid}} > 0\) always — but \(\Delta V = V_{\text{liquid}} - V_{\text{solid}}\) has no fixed sign. Most substances contract on freezing (\(\Delta V>0\), positive steep slope); water is a famous exception — its near-icosahedral short-range order freezes into an open crystal structure that is less dense than the liquid, so \(\Delta V<0\) and ice’s melting curve tilts the other way, which is also why ice floats.
Ferro-up / Ferro-down: flip the field’s sign and the entire microstate distribution flips with it — every spin configuration with a given excess of “up” spins has an exactly degenerate mirror configuration with the same excess of “down” spins, of identical energy. So \(\Delta S \equiv 0\) identically, while \(\Delta M \ne 0\). The slope \(dH/dT = \Delta S/\Delta M\) is therefore exactly zero for every \(T<T_c\) — not approximately, not just “small like the liquid–gas slope,” but exactly flat, forced by the up–down symmetry of zero field. It is also the only sane answer physically: nobody heats a permanent magnet at zero field and watches its polarity spontaneously reverse — flipping it requires actually crossing the \(H=0\) line.
(A related curiosity: despite the huge visual asymmetry between the liquid and gas branches of the \(V\)–\(T\) diagram — compare the nearly incompressible liquid branch to the wildly expanding gas branch — an empirical law of rectilinear diameters shows that a simple linear change of variables restores a hidden symmetry, the same symmetry the magnetic diagram displays outright. The \(\beta=\tfrac12\) exponent right at the tip of the curve is untouched by any of this asymmetry.)
Generalizing the order parameter#
Every example so far has used a quantity that vanishes in the disordered phase and grows continuously from zero below \(T_c\): an order parameter. Identifying the right order parameter is the essential first step for any phase transition, and it isn’t always the obvious “extensive quantity that changes”:
Transition |
Order parameter |
Type |
|---|---|---|
Para → Ferro magnet |
Magnetization \(M\) |
vector |
Liquid → Gas |
Density difference \(\rho_{\text{liquid}} - \rho_{\text{gas}}\) |
scalar |
Liquid → Crystal |
Fourier amplitudes of the density, \(\tilde\rho(\vec k) = \int \rho(\vec r)\,e^{i\vec k\cdot\vec r}\,d^3r\) |
set of Fourier components |
Normal liquid He → Superfluid He |
Condensate wavefunction \(\psi(\vec r)\) |
complex scalar |
Entropy changes at every one of these transitions too, but entropy is not the order parameter — it doesn’t satisfy the right technical requirements. For the liquid–crystal transition in particular, the ordinary density itself is nearly useless (liquids and solids have almost the same density), but the density’s Fourier transform is not: it is essentially flat (no preferred direction) for an isotropic liquid and sharply peaked at the crystal’s reciprocal-lattice vectors, uniquely encoding which crystalline order has appeared. For superfluid helium the order parameter is a genuinely quantum object — a macroscopic condensate wavefunction, zero above the transition and a coherent nonzero complex amplitude below it, with both its modulus and its phase carrying physical meaning. Order parameters, in short, can be scalars, vectors, sets of Fourier amplitudes, or complex numbers — whatever the symmetry being broken demands.
Landau’s phenomenological theory#
Landau’s 1937 insight was to stop building microscopic models altogether and instead ask what the free energy must look like as a function of the order parameter, constrained only by symmetry. Near the critical point the order parameter is small, so expand the free energy at \(H=0\) in powers of \(m\):
At zero field, up and down are physically equivalent — the free energy must be invariant under \(m \to -m\) — which forbids every odd power outright:
Minimizing, \(\partial F/\partial m = 2am + 4bm^3 = 0\), gives \(m=0\) always, plus \(m^2 = -a/2b\) whenever \(a<0\). So the shape of \(F(m)\) does all the work:
\(a(T) > 0\): a single well, minimum at \(m=0\) — the disordered phase.
\(a(T) < 0\): a double well, degenerate minima at \(m = \pm\sqrt{-a/2b}\) — the ordered phase, with \(m=0\) demoted to an unstable local maximum.
For \(a(T)\) to switch sign at \(T_c\), the simplest (and generic) choice is a linear zero-crossing, \(a(T) = a_1(T - T_c)\). This single assumption by itself, with no reference to spins or dipoles at all, reproduces \(m_0 \propto (T_c-T)^{1/2}\) — because any smooth, symmetric free energy truncated at quartic order has no other option. That is the real content of “\(\beta = \tfrac12\) is a mean-field exponent”: it is a generic consequence of Taylor-expanding any symmetric free energy, not a coincidence shared by the Weiss and Van der Waals models specifically. (Restoring a small field adds back the symmetry-breaking linear term \(-Hm\), and re-deriving \(\delta=3\) and \(\gamma=1\) from this Landau form recovers exactly the exponents found above — Landau theory doesn’t just explain \(\beta\), it reproduces the whole mean-field universality class from symmetry alone.)
Spontaneous symmetry breaking with a complex order parameter#
Superfluid helium’s order parameter \(\psi\) is complex, not real, so the same expansion becomes \(F(\psi,T) \approx F_0 + a(T)|\psi|^2 + b|\psi|^4\). Below \(T_c\) the minimum is no longer a pair of isolated points but an entire circle, \(|\psi| = \sqrt{-a/2b}\), of degenerate minima in the complex plane — the “Mexican hat” or wine-bottle potential. Moving around the circle (changing the phase of \(\psi\)) costs no energy at all in this model; only radial motion, toward or away from the rim, does.
Drag the surface around: a ball settled anywhere on the circular trough can slide tangentially along the rim for free but must climb radially to leave it. That asymmetry has real consequences. Whenever a continuous symmetry — here, the phase rotation \(\psi \to \psi\,e^{i\phi}\) — breaks spontaneously, the flat tangential direction guarantees a mode of excitation that costs vanishing energy in the long-wavelength limit: a Goldstone mode. Spin waves in a ferromagnet and acoustic phonons in a crystal are physical examples — both cost energy that vanishes as the wavelength goes to infinity. Ordinary ferromagnets don’t have this: flipping \(m\to -m\) is a discrete symmetry (only two rim points, not a continuous circle), so there is no free tangential direction and no Goldstone mode — a genuinely different kind of symmetry breaking from the continuous case.
First-order transitions, and the limits of mean-field theory#
Not every transition grows continuously out of \(m=0\). Allow the quartic coefficient itself to vary — e.g. \(F - F_0 = am^2 + bm^4 + cm^6\) with \(c>0\) for stability — and as \(b(T)\) sweeps through negative values a second, initially higher local minimum can develop away from \(m=0\), become momentarily degenerate with it, and then drop below it. The order parameter then jumps discontinuously from one minimum to the other rather than growing smoothly from zero — the generic Landau mechanism for a first-order (discontinuous) transition, always accompanied by a nonzero latent heat, as opposed to the second-order (continuous) mechanism worked out above.
Landau’s own motivation was actually structural phase transitions in crystals, classified via group theory and crystallography — well beyond this course — but the same free-energy logic carries over. A later refinement by Ginzburg added a gradient-energy penalty \(k(\nabla m)^2\) for spatial variation of the order parameter (needed because near \(T_c\) a magnet doesn’t flip uniformly — islands of “up” shrink while islands of “down” grow), turning this into the Landau–Ginzburg theory that underlies the modern treatment of correlation functions near criticality.
Mean-field theory itself, finally, is only exact above an upper critical dimension (typically \(d=4\) for these models); below it, the fluctuations mean-field theory ignores start to matter and shift the exponents away from \(\beta=\tfrac12\), \(\gamma=1\), \(\delta=3\) toward the experimentally observed values. There is also a lower critical dimension (often \(d=2\)) below which no ordered phase exists at any positive temperature at all. We live in \(d=3\) — strictly between the two — which is exactly why real magnets show non-mean-field exponents while still displaying every qualitative feature derived here: an order parameter, a symmetry that must break, and a critical point where microscopic detail stops mattering. (Quantum phase transitions — driven by quantum rather than thermal fluctuations, occurring at \(T=0\) — extend this picture further still, but that is a story for another course.)