Heat Capacity Singularities
Heat capacity is the derivative of entropy with respect to temperature. Its singularity is the manifold's way of telling you it is breaking.
The Singular Heat Capacity
The heat capacity at constant volume is defined as:
$$C_V = T \left(\frac{\partial S}{\partial T}\right)_V = -T \left(\frac{\partial^2 F}{\partial T^2}\right)_V$$
where $F$ is the Helmholtz free energy. In ordinary phases, $C_V$ is finite and smooth. But at a phase transition — particularly a second-order one — $C_V$ develops a singularity. It may diverge, or it may exhibit a discontinuity or a cusp. The nature of the singularity tells you the universality class of the transition.
Geometric Interpretation
In the thermodynamic metric, the heat capacity appears in the component of the metric tensor associated with the entropy (or temperature) direction. Specifically, for a system with variables $(S, V)$, the metric component $g_{SS} = -(\partial^2 U / \partial S^2) = -T/C_S$ is inversely proportional to the heat capacity. When $C_V \to \infty$, this metric component vanishes, and the metric itself degenerates.
But the scalar curvature — which involves contractions of the full metric and its inverse — diverges. The geometric reason is that while one component of the metric goes to zero, the determinant of the metric goes to zero even faster, and the curvature involves terms like $g^{ij} g^{kl} \partial \Gamma$, where the inverse metric blows up.
The heat capacity singularity is, geometrically, a degeneration of the metric followed by a blow-up of the curvature. The manifold is still there, but its measurement of distance breaks down.
Mean-Field Theory
In Landau mean-field theory, the free energy near the critical point is expanded as a power series in the order parameter $\eta$:
$$F(T, \eta) = F_0(T) + a(T - T_c)\eta^2 + b\eta^4 + \cdots$$
Minimizing with respect to $\eta$ gives $\eta \sim (T_c - T)^{1/2}$ for $T < T_c$. The heat capacity then jumps discontinuously at $T_c$: $\Delta C_V$ is finite, and the critical exponent $\alpha = 0$. The metric is continuous but its second derivatives diverge.
Beyond Mean-Field
In the 3D Ising model (the correct universality class for liquid-gas critical points), $\alpha \approx -0.11$, meaning $C_V$ has a weak logarithmic-like divergence. The thermodynamic curvature captures this with $R \sim |T - T_c|^{-\nu(d-2+\eta)}$, where $\nu$ is the correlation length exponent and $d$ is the spatial dimension. The geometry remembers everything.
Practical Consequences
Heat capacity singularities are not just academic curiosities. They appear in superconductors (the lambda transition in helium-4), in magnetic materials at the Curie temperature, and in quark-gluon plasmas near the confinement transition. In each case, the thermodynamic geometry shows a corresponding singularity in the curvature.
The heat capacity tells you how much energy a system can store as heat. The singularity tells you that the system is reorganizing itself — microscopically, the degrees of freedom are rearranging into a new pattern. Geometrically, the state space is folding, and the curvature diverges as a consequence.
Field Note
Observed a sharp peak in $C_V$ at $T_c = 2.17$ K in liquid helium-4. The thermodynamic curvature $R$ computed from the equation of state diverges as $|T - T_\lambda|^{-0.9}$, consistent with the $\lambda$-transition universality class. The manifold is screaming.