The specific heat usually diverges at a phase transition - typically as a power-law, giving a critical exponent $\alpha > 0$. (Although in 2D, sometimes the divergence is only logarithmic, as with the Ising model, and sometimes the specific heat remains smooth but non-analytic so there is no divergence at all, as with the Kosterlitz-Thouless transition. In both cases we formally have $\alpha = 0$.) This makes sense intuitively: the heat capacity roughly counts the number of "actively fluctuating" internal degrees of freedom through which a temperature increase can raise the internal energy, and at a second-order phase transition you have huge domains that are actively fluctuating together. (Or at a first-order transition, any additional thermal energy you add to the system goes into changing chunks of the low-temperature phase into the high-temperature phase, so it doesn't raise the temperature at all.)
But this paper claims that for 3D locally-coupled spin glasses (of either Ising, XY, or Heisenberg spins), $\alpha \approx -2$ is strongly negative at the spin-glass transition, so the singular part of the specific heat goes as $\sim |\tau|^{2 + \epsilon}$ for some $|\epsilon| \ll 1$. This seems to imply that the specific heat reaches a local minimum at the spin-glass transition temperature, which seems very counterintuitive to me. What is the physical interpretation of this strange temperature dependence of the specific heat?