Past exam of the thermal and statistical physics course of the University of Cambridge / 2015 / Paper 1 / 4

Landau description. Introduce a macroscopic order parameter and expand an appropriate free energy analytically in that parameter, retaining terms permitted by the physical symmetries. Stable equilibrium minimizes this free energy globally; a local minimum with larger free energy is metastable. Coefficient changes with temperature can exchange the preferred minima. A continuous emergence from zero can produce a continuous transition, while competing separated minima can produce a first-order transition. Stability requires a positive highest retained even-power term. This local mean-field description neglects spatial fluctuations and nucleation dynamics unless gradient and dynamical terms are added; near a critical point those omissions can matter.

For this temperature-dependent cubic-quartic Landau free energy, abbreviate $B=a/(12\pi )>0$ and $A(T)=c(T^2-T_0^2)$, so $F=A{\varphi }^2/2-BT{\varphi }^3+\lambda {\varphi }^4/4$ and $\gamma =2B^2/(\lambda c)$.

Local behavior at the origin. To determine constrained order-parameter boundary stability, note that $F^{\prime }(0)=0$ and $F^{\prime \prime }(0)=c(T^2-T_0^2)$, the origin is a local minimum for $T>T_0$ and a local maximum for $0\le T<T_0$. At $T=T_0$, the quadratic term vanishes, but $F=-B{T}_0{\varphi }^3+O({\varphi }^4)<0$ for small positive $\varphi$. ThereforeThe equality case uses the stipulated domain $\varphi \ge 0$. On an unrestricted real line, the nonzero cubic term would instead make it a stationary inflection point.

Global stability in both inline cases. For $\varphi >0$, complete the square in $F/{\varphi }^2$:The quartic coefficient guarantees a bounded global minimum. A negative value exists precisely when the minimum of this quadratic factor is negative. Hence case a, $\gamma >1$, has a nonzero globally stable phase with $F<0$ for every physical temperature. Case b, $0<\gamma <1$, has such a phase exactly forAt $T=0$, the square's centre is at the boundary, but its negative constant still gives $F<0$ for small positive $\varphi$, so this endpoint is included.

At $T=T^{\ast }$,The two global minima $\varphi =0,{\varphi }^{\ast }$ have equal free energy, and $F^{\prime \prime }({\varphi }^{\ast })=\lambda ({\varphi }^{\ast }{)}^2/2>0$. The coexistence barrier of a cubic-quartic Landau transition has its maximum at ${\varphi }^{\ast }/2$, with barrier height $\lambda ({\varphi }^{\ast }{)}^4/64$.

For locating the extrema away from coexistence,Retain only real positive roots; when both are positive, ${\varphi }_{-}$ is a local maximum and ${\varphi }_{+}$ a local minimum. For $T<T_0$, only ${\varphi }_{+}$ is positive. If $\gamma <8/9$, the pair first appears on cooling at $T_{\mathrm{sp}}=T_0/\sqrt{1-9\gamma /8}>T^{\ast }$; if $8/9\le \gamma <1$, a higher-free-energy nonzero minimum already exists at arbitrarily high temperature. These local-stability limits must not be mistaken for the global coexistence condition. The boundary $\gamma =1$ also favors a nonzero minimum at every finite temperature, but has no finite $T^{\ast }$.

Cooling and transition temperature. For $\gamma <1$, equilibrium starts at the zero minimum at high temperature and switches to the nonzero minimum at $T^{\ast }$. The jump is finite, $0\to {\varphi }^{\ast }>0$, and there is a barrier between the coexisting states. This is a first-order transition. The latent heat of a temperature-dependent cubic-quartic Landau transition follows from differentiating the minimized free energy: the heat released on cooling isIf $F$ is an energy density rather than total energy, $L$ is likewise a density. An actual system can supercool below $T^{\ast }$ before nucleation, so $T^{\ast }$ is the equilibrium transition temperature.

For thermal symmetry nonrestoration in a cubic-quartic model, with $\gamma >1$, the equilibrium global minimum is nonzero already at arbitrarily high temperature. Following it does not force a transition. A system initially prepared in the metastable zero minimum can still leave it by nucleation while $T>T_0$; that minimum loses local stability at $T_0$. This is the sense in which a change is possible rather than inevitable. There is no equilibrium coexistence temperature for $\gamma >1$ in this model. A kinetic transition can depend on preparation and nucleation dynamics; $T_0$ is the limiting loss of local stability of the zero state, not a second coexistence temperature. Continuing $T^{\ast }$ to an imaginary value does not define a physical transition temperature.

Mass window in the specified electroweak phase transition model. For the mass window in a cubic electroweak transition model, put $x=m_H^2/m_0^2>0$. The supplied ratios implyThe two strict conditions therefore requireSince $x(A+x)$ is increasing for $x>0$, the finite window isUsing the specified $A$ and $m_0$ givesAt $m_H=134$ u, $x=0.3581801$, $\gamma \simeq 0.0030033<1$ but ${\varphi }^{\ast }/T^{\ast }\simeq 0.1481<1$. Thus 134 u lies outside the required window, failing the order-parameter-strength condition. This does not remove the polynomial model's finite first-order jump; it fails the additional requirement ${\varphi }^{\ast }>T^{\ast }$. The result is a bound within the specified simplified model, not a model-independent Higgs-mass bound.