Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 49 / 1

In Landau-Ginzburg theory, a scalar order parameter $M$ distinguishes macroscopic phases: for an Ising model it is the magnetization per site, while near a liquid–gas endpoint it can be a shifted density. An ordered pure thermodynamic phase has a nonzero value even when the conjugate field is removed. For a symmetry-related pair, take the thermodynamic limit before $h\to 0^{\pm }$; a finite-volume symmetric ensemble instead has zero average.

Coarse-grain to a slowly varying real field and use a local, symmetry-constrained free-energy density. A useful convention isFor spin inversion symmetry, odd powers other than the externally imposed source are absent. Take $v>0$ when the quartic coefficient can be negative; for an ordinary transition $u>0$ already stabilizes the quartic truncation. The gradient energy penalizes spatial variations. In the Landau approximation, minimize this functional, usually first over constant fields. The corresponding statistical field theory also integrates fluctuations with weight $e^{-\mathcal{F}/(k_{B}T)}$. The order parameter identifies the phase; its spatial field describes both equilibrium order and its fluctuations.

A first-order phase transition exchanges two distinct global minima of the Landau free energy. The equilibrium free energy remains continuous, but a first derivative such as the magnetization or entropy jumps. A temperature-driven transition generically has latent heat; crossing $h=0$ between symmetry-related ordered phases instead has a magnetization jump without requiring latent heat. A continuous phase transition has no such order-parameter jump and usually has a divergent correlation length and nonanalytic higher responses.

For $u>0$, $h=0$ and $r$ crossing zero, minimizing $V=r{M}^2/2+u{M}^4/4$ gives $M=0$ for $r>0$ and $M=\pm \sqrt{-r/u}$ for $r<0$. The minima merge continuously at zero. The curvature vanishes there, so the long-distance response becomes singular.

For $u<0<v$, put $z=M^2$. A nonzero stationary minimum satisfies $r+uz+v{z}^2=0$. Equality with the $M=0$ free energy givesThe finite separation demonstrates the first-order phase transition. Metastability occurs where a nonglobal local minimum persists; its disappearance at a spinodal point is not the equilibrium coexistence criterion.

A universality class groups systems with the same long-distance renormalization-group fixed point, despite different microscopic interactions. Spatial dimension, order-parameter symmetry, interaction range and additional soft modes matter; microscopic lattice spacing or a change in a short-range coupling usually does not change the critical exponents. Irrelevant microscopic distinctions decay under renormalization-group flow.

For example, a three-dimensional short-range Ising model and a simple-fluid liquid–gas critical endpoint share the scalar Ising universality class. The fluid density deviation plays the role of the order parameter, with appropriately mixed thermal and field-like control variables. Their critical temperatures and dimensional amplitudes differ, but critical powers and suitable dimensionless amplitude ratios agree. Landau theory predicts a common set of mean-field critical exponents; fluctuations can replace these values below the upper critical dimension. Universality concerns the long-distance singular behavior, not equality of microscopic parameters.

A critical exponent specifies a leading singular power in a response or order parameter as the thermodynamic critical point is approached. Analytic backgrounds and the selected thermodynamic branch must be distinguished from that singular part. Write $r=at$, $a>0$, and first take $u>0$. The equation of state is $h=rM+u{M}^3$.

At zero field on the ordered branch, $M=(-r/u{)}^{1/2}$, giving the order-parameter critical exponent $\beta =1/2$. Differentiating the equation of state gives $\chi =(r+3u{M}^2{)}^{-1}$: above the transition $\chi =1/r$, below it $\chi =1/(2|r|)$, so the magnetic-susceptibility critical exponent is $\gamma =1$. At $r=0$, $M=\mathrm{sgn}(h)(|h|/u{)}^{1/3}$, giving the critical-isotherm exponent $\delta =3$.

The minimized singular free-energy density is $f_s=0$ above and $f_s=-r^2/(4u)$ below. Two temperature derivatives produce a finite specific-heat jump, corresponding to the heat-capacity critical exponent $\alpha =0$. The inverse quadratic fluctuation kernel is $K{p}^2+r+3u{M}^2$, so $\xi =\sqrt{K/(r+3u{M}^2)}$ gives $\nu =1/2$; the critical propagator proportional to $p^{-2}$ gives $\eta =0$. ThusThese mean-field critical exponents follow from minimization and linear response. More generally, linearize the renormalization group and differentiate its singular free-energy scaling form to obtain the actual fluctuation-controlled exponents.

In a smooth scalar Landau free energy, a line of coexistence between two distinct minima can end by those minima and the intervening maximum coalescing. At a generic endpoint with no competing third phase,The first condition is stationarity, the second removes the inverse susceptibility, and the third is needed because nonzero cubic curvature at a zero-quadratic point would prevent it being a stable minimum. After shifting $M=M_c+\psi$ and mixing two control parameters, the local potential is $V_c+R{\psi }^2/2+U{\psi }^4/4-H\psi +\cdots$, $U>0$. Its two-phase coexistence line $H=0,R<0$ reaches $R=H=0$, with $\Delta M=2\sqrt{-R/U}\to 0$ and diverging magnetic susceptibility. This is the familiar critical endpoint of a quartic Landau free energy.

The word “must” needs the two-phase endpoint assumptions. A first-order line can instead end at a triple point or a boundary of the available control parameters. There is an explicit scalar counterexample in the sextic potential: fix $u<0<v$ and vary $(r,h)$. The $+M/-M$ coexistence line at $h=0$, $r<3u^2/(16v)$ ends when the $M=0$ phase also becomes globally stable. At that triple point the nonzero minima remain $\pm \sqrt{-3u/(4v)}$; their jump does not vanish. Thus the literal unrestricted assertion is false. The continuous-endpoint argument above establishes the intended case and distinguishes it from this counterexample.

Consider the $(u,r)$ plane at $h=0$ for $V=r{M}^2/2+u{M}^4/4+v{M}^6/6$, with fixed $v>0$. For $u>0$, the continuous boundary is $r=0$: above it the global minimum is zero, and below it there are two symmetry-related ordered pure thermodynamic phases. For $u<0$, the equilibrium boundary is the tricritical three-phase lineAbove this curve the disordered minimum wins; below it the two ordered minima win. On the curve,Its three zeros are exactly $M=0$ and $M=\pm \sqrt{-3u/(4v)}$, proving equality of three global minimum values. Spin inversion symmetry enforces the equality of the two ordered phases automatically, so tuning a single remaining coefficient can produce three-phase coexistence on a line in this symmetry plane. This does not contradict the generic phase-rule counting without symmetry restrictions.

The tricritical point is $(u,r)=(0,0)$, where the discontinuity vanishes, the first-order boundary meets the continuous boundary, and the positive sextic term becomes the leading stabilizer. The local metastability limits $r=0$ and $r=u^2/(4v)$, $u<0$, bracket the first-order curve but are not phase boundaries. In the full $(r,u,h)$ parameter space, the zero-field line is attached to tricritical wings; a nonzero field splits the equality of the two ordered phases.

At the tricritical point, tune $u=0$, hold $v>0$, and let $r=at$ cross zero. The equation of state becomes $h=rM+v{M}^5$. At $h=0,r<0$, $M^4=-r/v$, while the inverse susceptibility on the ordered branch is $r+5v{M}^4=4|r|$. On the disordered side it is $r$. At $r=0$, $M=\mathrm{sgn}(h)(|h|/v{)}^{1/5}$. Therefore $\beta =1/4$, $\gamma =1$ and $\delta =5$.

The minimized singular free-energy density isTwo temperature derivatives give a singular heat capacity proportional to $|t{|}^{-1/2}$ on the ordered side. Consequently the mean-field tricritical exponent calculation givesThe displayed scaling relations for critical exponents hold numerically:These are Landau theory values with both relevant coefficients tuned: along a generic path with fixed $u>0$, the asymptotic transition is ordinary rather than tricritical. The sextic upper critical dimension is three, so fluctuations can change the powers below three dimensions, and logarithmic corrections are possible at three. The vanishing disordered mean-field singular amplitude does not invalidate the ordered-side exponent.