Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 303 / 1

In the Landau theory approximation, choose the global minimum of the Landau free energy as a function of the magnetization. Since ${\mathcal{A}}_6>0$, the polynomial is a coercive function and a global minimum exists. The candidates are stationary points satisfyingCheck their values of the free-energy density; a stationary point or a metastable local minimum alone need not be the equilibrium state. Positive curvature ${\mathcal{A}}_2+3{\mathcal{A}}_4{m}^2+5{\mathcal{A}}_6{m}^4$ ensures local stability; when it vanishes, higher terms must be checked. The selected global minimum is the equilibrium magnetization within this approximation.

At zero magnetic field, equal opposite minima describe distinct phases exhibiting spontaneous symmetry breaking. A spontaneous magnetization is obtained by taking the thermodynamic limit before $h\to 0^{+}$ or $h\to 0^{-}$. A symmetric finite-volume average can instead have zero expectation even when the minima are nonzero. Thus one must specify the phase when interpreting the minimizer as the equilibrium expectation.

Discuss the zero-field phase boundaries, taking $h=0$ and writing $r={\mathcal{A}}_2$, $q={\mathcal{A}}_4$, $s={\mathcal{A}}_6>0$. This qualification matters because a conjugate field generally rounds the zero-field continuous transition. Nonzero stationary points satisfy $r+q{m}^2+s{m}^4=0$.

For $q>0$, the continuous phase transition occurs atOn the $r>0$ side the only stable phase is $m=0$; on the $r<0$ side the equilibrium global minima are $\pm m_0$ with $m_0^2\sim -r/q$. The magnetization therefore grows continuously from zero as $r$ changes sign.

For $q<0$, the first-order phase transition instead occurs while $r$ is positive. At phase coexistence, $m=0$ and $m=\pm m_0$ are equally deep global minimum points. With $a=m_0^2>0$, stationarity and equality of free energies giveEliminating $r$ gives $-q{a}^2/4-s{a}^3/3=0$, henceAt this coexistence value, the sextic even Landau potential factors as $f(m)=s{m}^2(m^2+3q/(4s){)}^2/6\ge 0$. This verifies that the three equal stationary values are global minima. Above this coexistence value of $r$ the disordered phase wins; below it the ordered phase wins, with a finite jump in the phase-selected magnetization. The values $r=0$ and $r=q^2/(4s)$ are the disordered and ordered spinodal points, respectively, not the coexistence boundary: the latter follows from the repeated root of $r+qa+s{a}^2=0$.

The two transition loci meet at the tricritical point,Two independent control parameters such as $T$ and $g$ can tune these two conditions. In a generic local phase diagram their coefficient map has a Jacobian matrix of rank two at the intersection; unspecified functions ${\mathcal{A}}_2(T,g)$ and ${\mathcal{A}}_4(T,g)$ do not by themselves determine the geometric orientation of the boundaries. The sextic term stabilizes this intersection and the negative-quartic first-order side.

These are mean-field critical exponents at a tricritical point. Hold the other control at its tricritical value and take the coefficients to be real analytic functions of temperature along a thermal approach with ${\mathcal{A}}_2=a(T-T_c)+o(T-T_c)$, $a>0$, ${\mathcal{A}}_6\to s>0$, and ${\mathcal{A}}_4=0$ or ${\mathcal{A}}_4=o(|{\mathcal{A}}_2{|}^{1/2})$. At fixed tricritical $g$, a generic real analytic temperature path with nonzero thermal slope has ${\mathcal{A}}_4=O(T-T_c)$ and meets the last condition. Paths tangent to a scaling direction can have different apparent temperature powers.

At zero field on the ordered side with ${\mathcal{A}}_4=0$, stationarity gives $m^4=-{\mathcal{A}}_2/s$; under the more general approach ${\mathcal{A}}_4=o(|{\mathcal{A}}_2{|}^{1/2})$, this relation holds to leading order. Thus the order-parameter critical exponent isFor the zero-quartic representative, putting this minimum into the Landau free energy gives the singular order-parameter contribution (and the leading asymptotic contribution under the stated negligible-quartic approach)Using $C_{\mathrm{s}}=-T{\partial }_T^2{f}_{\mathrm{s}}$ at fixed other control yields $C_{\mathrm{s}}\sim (T_c-T{)}^{-1/2}$, specifically $T{a}^{3/2}/(4\sqrt{s(T_c-T)})$ to leading order. HenceThis heat-capacity critical exponent refers to the nonzero ordered-side singular amplitude; in this mean-field approximation to Landau theory, the disordered global minimum contributes zero singular free energy, while any regular background remains. Fluctuation corrections lie outside the present approximation.

In the disordered phase, differentiating the equation of state at $m=h=0$ gives the magnetic susceptibility $\chi =1/{\mathcal{A}}_2$. On the ordered branch its curvature is ${\mathcal{A}}_2+5s{m}^4=4|{\mathcal{A}}_2|$, so $\chi =1/(4|{\mathcal{A}}_2|)$. Both give the magnetic-susceptibility critical exponentAt the tricritical temperature with both lower coefficients zero, $h=s{m}^5$, so $m=\mathrm{sgn}(h)(|h|/s{)}^{1/5}$ and the critical-isotherm exponent isThe Widom scaling relation $\gamma =\beta (\delta -1)$ and Rushbrooke scaling relation $\alpha +2\beta +\gamma =2$ both check these powers. Their validity here is a mean-field conclusion, not a claim that fluctuations in every spatial dimension preserve them.

For ${\mathcal{A}}_2\ne 0$, rescale the order parameter by $m=(|{\mathcal{A}}_2|/{\mathcal{A}}_6{)}^{1/4}\psi$. Write $\sigma =\mathrm{sgn}{\mathcal{A}}_2$ and introduce dimensionless scaling variablesEvery term then has the common energy-density factor $|{\mathcal{A}}_2{|}^{3/2}/{\mathcal{A}}_6^{1/2}$:Minimizing over $\psi$ defines the tricritical crossover scaling functionsFor the minimized potential, following the paper's notation $\mathcal{F}$, this provesThe exponent names $x,y$ here are unrelated to the couplings in question 2. The expression concerns the specified order-parameter polynomial; a general material can also have a smooth, field-independent background free-energy density. Such a background should be separated before writing a homogeneous singular scaling form. A potential at prescribed magnetization, instead of prescribed field, is a different Legendre transform and is not obtained by this minimization.

There is a needed qualification to the printed single $\Phi$ and nonzero-value condition. At $(X,Y)=(0,0)$, the ordered branch has minimizers $\psi =\pm 1$ and ${\Phi }_{-}(0,0)=-1/3$, whereas the disordered branch has minimizer zero and ${\Phi }_{+}(0,0)=0$. ThusFor example, ${\mathcal{A}}_2>0$, ${\mathcal{A}}_4=h=0$ gives an identically zero minimized polynomial, contradicting a nonzero $\Phi (0,0)$ on that side. The valid scaling statement uses separate sign branches, or restricts the asserted nonzero value to approach from the ordered side. At ${\mathcal{A}}_2=0$ the displayed coordinates are singular; the critical isotherm is obtained as a limit or directly from the equation of state.

At an ordinary zero-field continuous phase transition, the positive quartic coefficient remains nonzero while ${\mathcal{A}}_2\to 0$. Approaching from the ordered side therefore sends the variable $X$ in the tricritical crossover scaling form to $+\infty$, rather than to zero. With $Y=0$ the dimensionless minimizer obeys ${\psi }^4+X{\psi }^2-1=0$, soAt that stationary point, substitution gives ${\Phi }_{-}(X,0)=-{\psi }^2/4-{\psi }^6/12$. Its large-quartic asymptotic of tricritical scaling is consequently ${\Phi }_{-}(X,0)\sim -1/(4X)$, andIf ${\mathcal{A}}_2$ crosses zero linearly with temperature, this is quadratic in $T-T_c$, giving a finite heat capacity jump rather than a power divergence. ThereforeFor locally constant ${\mathcal{A}}_4>0$ and slope $a={\partial }_T{\mathcal{A}}_2{|}_{T_c}$, the leading singular heat capacity jump from this term is $\Delta C=T_c{a}^2/(2{\mathcal{A}}_4)$. Treating $\Phi$ as a nonzero constant while $X\to \infty$ would incorrectly assign the tricritical exponent to this ordinary transition.

On a tricritical thermal path, the quartic scaling variable $X$ tends to zero, and at zero field $Y=0$. On the ordered side the tricritical crossover scaling function tends to ${\Phi }_{-}(0,0)=-1/3$. Consequentlyprovided the quadratic coefficient has a nonzero thermal slope and the sextic coefficient has a finite positive limit. Since the heat-capacity critical exponent satisfies $f_{\mathrm{s}}\sim |T-T_c{|}^{2-\alpha }$,Equivalently, twice differentiating this ordered-side free-energy density produces the $(T_c-T{)}^{-1/2}$ singularity. The scaling path and branch qualifications are the same as for the direct Landau theory calculation: a nonzero fixed positive quartic coefficient would instead send $X\to \infty$ and yield the ordinary transition of part (e).