Past exam of the theoretical physics I course of the University of Cambridge / 2019 / 4

A Landau free energy is a coarse-grained thermodynamic potential expressed as a function of a proposed macroscopic order parameter. Its coefficients include microscopic and thermal effects already averaged over. At fixed external thermodynamic conditions, equilibrium selects its global minimum; other local minima can represent metastable states. It is not the internal energy of a single microscopic magnetic configuration. Here the real scalar $m$ represents a selected component of magnetization: zero describes the disordered state, while nonzero values distinguish two oppositely ordered states.

Every power is even, so the internal discrete symmetry isFor a magnetic order parameter, this represents reversal of the magnetization, as under time reversal in zero applied magnetic field. A nonzero field would generally add an odd term and break this symmetry explicitly. This scalar model has no continuous internal rotation symmetry; physical rotational symmetry of a vector magnet is not implied by treating one real component. The condition $c>0$ makes the polynomial bounded below at large $|m|$.

For the requested ordered phase on cooling, take the usual implicit assumption $a>0$, and write $r=a(T-T_c)$. If $a<0$ the temperature directions reverse; if $a=0$ this expansion has no temperature-driven transition. The source explicitly states only $c>0$.

The stationary points of the Landau free energy obeyFor $b>0$, $r\ge 0$ makes every term of $f$ nonnegative, so $m=0$ is the unique global minimum. For $r<0$, its curvature $f^{\prime \prime }(0)=2r$ is negative. Setting $x=m^2$ gives $c{x}^2+bx+r=0$, whose unique positive root isAt a nonzero stationary point,Thus both $m=\pm \sqrt{x_{+}}$ are stable and have lower free energy than $m=0$. The equilibrium magnetization isFor fixed $b>0$ and $T\to T_c^{-}$, expand the square root:It tends continuously to zero. Hence the continuous phase transition has the order-parameter critical exponentThis is the mean-field critical exponent; fluctuation corrections beyond Landau theory are not included.

For $b<0$ at $T=T_c$, the quartic term lowers the free energy near $m=0$, while the positive sextic term stabilizes it at larger $|m|$. The nonzero stationary points haveThey are global minima. The stationary point $m=0$ has zero quadratic curvature but is a flat local maximum because its leading nonzero term is negative. Thus the system is already ordered at $T_c$, with a finite magnetization.

More explicitly, a continuous onset from $m=0$ could occur only where $f^{\prime \prime }(0)=2r$ vanishes, namely at $r=0$. But the global minima there are the finite values above. For sufficiently large positive $r$, only $m=0$ is stable. Therefore the equilibrium minimum must change at some positive $r$ by crossing between a zero and a finite nonzero order parameter. This is a first-order phase transition. Its actual temperature lies above $T_c$ and is calculated next; $T_c$ itself is the disordered spinodal point, not the coexistence temperature.

At phase coexistence, the finite ordered global minimum has the same free energy as $m=0$. With $x=m^2>0$ and $f(0)=0$, stationarity and equal values giveSubtracting eliminates $r$:Substitute back to get $r_{\ast }=3b^2/(16c)$, soThe three degenerate equilibrium values at this temperature are $0$ and $\pm m_{\mathrm{jump}}$. A useful global check, avoiding confusion with a spinodal point, is the exact factorizationAll three zeros are therefore global minima. Their curvatures are positive: $f^{\prime \prime }(0)=3b^2/(8c)$ and $f^{\prime \prime }(\pm m_{\mathrm{jump}})=3b^2/(2c)$.

The ordered local minimum first appears on cooling at $r=b^2/(4c)$, where the nonzero stationary roots merge. It becomes globally stable only at $r_{\ast }$; the disordered minimum remains metastable down to $r=0$. These different temperatures distinguish phase coexistence from loss of local stability.

Let $\Delta T=T-T_c$, with $a,c>0$ fixed. The equilibrium phase diagram has boundariesFor either branch, larger $\Delta T$ is disordered and smaller $\Delta T$ is ordered. Both signs of the ordered magnetization are symmetry-related global minima throughout the ordered region, rather than separate regions selected by temperature and $b$.

The branches meet at the tricritical point $(\Delta T,b)=(0,0)$. Directly on $b=0$, minimizing $r{m}^2+c{m}^6/3$ givesThe transition there remains continuous, with a different mean-field critical exponent. For $b<0$, the metastability boundaries are $\Delta T=0$ for the disordered state and $\Delta T=b^2/(4ac)$ for the ordered state. The first-order phase transition curve lies strictly between them, as shown in the figure. These spinodal curves describe possible hysteretic continuation of local minima, not equilibrium phase boundaries.