Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 41 / 1

An order parameter distinguishes phases through a macroscopic observable, such as the mean spin $m$ of a ferromagnet. Its vanishing can reflect an unbroken symmetry, while choosing one of several nonzero equilibrium values gives spontaneous symmetry breaking. Landau-Ginzburg theory treats it as a slowly varying field and expands the free energy in terms allowed by symmetry:Here $K>0$ penalizes spatial variation, $h$ is the conjugate field, and the highest retained even coefficient must stabilize large $|m|$. For a spin-flip-symmetric system, odd powers are forbidden at $h=0$. Equilibrium in the Landau approximation minimizes this functional; near an ordinary continuous transition $r=r_{t}t$ with $t=(T-T_c)/T_c$ and $r_t>0$.

For $u>0$, the quartic Landau free energy at $h=0$ minimizes at $m=0$ for $r>0$ and at $m=\pm \sqrt{-r/u}$ for $r<0$. The order parameter changes continuously at $r=0$, giving a continuous phase transition and a divergent response. The minimized potential is $0$ above the transition and $-r^2/(4u)$ below it, so its first temperature derivative is continuous while its second derivative jumps.

For $u<0$, retain $v>0$. A first-order phase transition occurs when the $m=0$ minimum and a nonzero minimum have equal free energy. Combining stationarity $r+u{m}^2+v{m}^4=0$ with equality of potential values givesThe order parameter jumps at a positive $r$ before the disordered minimum loses local stability. Generic temperature dependence then also produces a jump in entropy and a latent heat. Equality of global minima locates coexistence; the disappearance of a metastable minimum instead defines a spinodal point.

At fixed average order parameter, a nonconvex local potential need not describe the equilibrium free energy. A mixture of two phases can lower it to the common-tangent convex envelope. If their magnetizations are $m_1,m_2$, the coexistence field satisfiesEquivalently, ${\int }_{m_1}^{m_2}[f^{\prime }(m)-h_{\mathrm{coex}}]dm=0$, the equal-area Maxwell construction. An average $\bar{m}$ between $m_1$ and $m_2$ is obtained by choosing their volume fractions according to the lever rule. The gradient term in Landau-Ginzburg theory gives a finite interfacial energy and smooth domain walls; in the thermodynamic limit the bulk gain dominates the interface cost. Domains are therefore especially natural under a constrained mean magnetization or suitable boundary conditions. Without such a constraint, one pure phase is also an equilibrium state at coexistence.

Critical exponents describe singular behavior near a continuous phase transition. Define $m(t,0)\sim (-t{)}^{\beta }$ below $T_c$, susceptibility $\chi (t,0)=\partial m/\partial h\sim |t{|}^{-\gamma }$, singular heat capacity $C_s\sim |t{|}^{-\alpha }$, and critical isotherm $m(0,h)\sim \mathrm{sgn}(h)|h{|}^{1/\delta }$. The ordinary quartic Landau free energy gives the equation of stateAt zero field its ordered minimum is proportional to $(-t{)}^{1/2}$. Differentiating the equation of state gives ${\chi }^{-1}=r_{t}t+3u{m}^2$, proportional to $|t|$ on either side. At $t=0$, $h=u{m}^3$, and the minimized free energy below the transition is proportional to $-t^2$. ThusHere $\alpha =0$ denotes a finite heat-capacity jump, rather than a power divergence. Including gradients gives the Landau scalar correlation length $\xi \propto |t{|}^{-1/2}$ and correlation-length exponent $\nu =1/2$.

A tricritical point occurs at $r=u=h=0$ with $v>0$: tuning $u$ changes a continuous transition into a first-order one. At $u=0$, the equation of state is $h=rm+v{m}^5$. It gives $m\propto (-r{)}^{1/4}$ at $h=0$, $m\propto h^{1/5}$ at $r=0$, and a minimized free energy proportional to $-|r{|}^{3/2}$ below the transition. The mean-field tricritical exponents areIn the three-dimensional parameter space $(r,u,h)$, the zero-field disordered-to-ordered continuous line $r=0,u>0$ meets the first-order line $r=3u^2/(16v),u<0$ at the tricritical point. The zero-field surface also contains coexistence of positive and negative ordered phases. For negative $u$, tilting with $h$ produces two symmetric tricritical wings of first-order coexistence, each bounded by a line of ordinary critical endpoints. Those wings and their critical edges meet at the tricritical point. The upper critical dimension of tricritical sextic theory is three, so mean-field values can acquire logarithmic corrections there.

The scaling hypothesis for critical phenomena says that the singular equilibrium free-energy density is a generalized homogeneous function of the thermal and field variables, after analytic backgrounds are removed. To derive the ordinary mean-field form, put $r=r_{t}t$, $u>0$ and neglect the higher powers near the transition. Rescale $m=\sqrt{r_t/u}|t{|}^{1/2}\psi$. Minimizing the quartic potential becomesThis is the mean-field scalar free-energy scaling form with $a=r_t^2/u$, $b=\sqrt{u}/r_t^{3/2}$. The two functions correspond to $t>0$ and $t<0$; the latter has two pure-phase branches and a cusp at zero field. The question's $>$ and $<$ subscripts distinguish these temperatures above and below $T_c$.

Differentiating with respect to $h$ gives $m=-{\partial }_h{A}_s\propto |t{|}^{1/2}$ on an ordered pure branch, and differentiating again gives $\chi =-{\partial }_h^2{A}_s\propto |t{|}^{-1}$. Thus the displayed scaling form independently yields $\beta =1/2$ and $\gamma =1$.

To allow anomalous powers, replace it byThen $\beta =2-\alpha -\Delta$ and $\gamma =2\Delta -(2-\alpha )$. Taking $t\to 0$ at fixed small $h$ gives $A_s\propto |h{|}^{(2-\alpha )/\Delta }$, so $1/\delta =(2-\alpha -\Delta )/\Delta =\beta /\Delta$. Eliminating $\Delta$ proves the Rushbrooke scaling relation and Widom scaling relation:These power-law relations assume the simple scaling form; marginal corrections or dangerously irrelevant couplings can require additional scaling variables.