Since φ is a real scalar, if η is a complex number, vector, or tensor of odd rank, we must have for all odd j because any term containing odd powers of η can never be a real scalar. In which case, eq(3.63a) simplifies to
where, to ensure (formal) global stability, we must have
[ implies global minima of φ at ]
For fixed Y and T, each equilibrium state (phase) is a minimum of φ so that
There are 2 solutions to eq(3.65a), namely,
Obviously, (3.65b) and (3.65c) correspond to the disordered ( ) and ordered ( ) states, respectively. Thus, the transition, or critical, point satisfies
For a given Y, let be the solution of eq(3.65d), i.e.,
Thus, describes a coexistence curve in the Y-T plane. The conditions (3.65b,c,e) are guaranteed if we write
where is expected to be slowly varying in the neighborhood of the transition point. Hence, eq(3.65c) becomes,
Putting everything into (3.64) gives
Since φ is the Gibbs free energy, the molar heat capacity is given by,
Thus, there is a discontinuity at of magnitude
which gives the vs T plot a shape of a λ (see Fig.3.21) and hence the reason for calling the critical point a λ-point.
The normal- superfluid transition of is continuous.
The order parameter is the macroscopic superfluid (complex) wave function so that
with . Similar to (3.69), we have
where the phase θ is a real number that can be set to zero when current flow is absent.
In the presence of an external force f that couples to the order parameter η, the relevant free energy is [see (3.64)]
which implies the Legendre transform (for equilibrium states),
where . Typical plots of can be found in Fig.3.22.
For fixed T,Y, and f, the equilibrium phases are minima of ψ so that
See Fig.3.22 for a few typical solutions. Note that (3.75) is simply the assertion that for equilibrium states, f is indeed the force conjugate to η, i.e., .
More interesting is the susceptibility
with the implicit assumption that all quantities take on their equilibrium values. Taking the partial derivative of eq(3.75) gives
For , eq(3.65b,c) gives
so that with , we have
which exhibits a divergence at .
In a para- to ferro- magnetic transition, critical values are called Curie values.
The order parameter is the magnetization vector M.
The symmetry that is broken is the rotational symmetry. Thus,
For , the analog of (3.76a) is
where is a unit vector.
The heat capacity exhibits the λ shape as shown in Fig.3.23.