# Susceptibility at a first-order phase transition

I have two questions about first-order phase transitions:

1) is the susceptibility divergent at a first-order phase transition?

2) if yes, does it diverge in a universal way as in continuous phase transitions?

My understanding is as follows, and I would like to know if it is correct. The susceptibility diverges at a first phase transition, but in a non-universal way. To be concrete, let us consider a transition signaled by spontaneous symmetry breaking, where the system is symmetric above the transition and the symmetry is broken across the transition. If the transition is first order, right at the transition symmetry-broken states will be degenerate with the symmetric state. Then if an infinitesimal field that couples to the order parameter is exerted, a symmetry-broken state will have the lowest energy and the system will be pinned to that state. Therefore, the susceptibility diverges. However, at a first-order phase transition the correlation length does not diverge, so short distance details will be important and the divergence of the susceptibility will depend on them, thus will be non-universal.

I appreciate if anyone tells me whether my understanding is right.

• I think that it is mostly a question of definition. The magnetic susceptibility, say, assumes that a change $\delta h$ of the magnetic field results in a change of order $\delta h$ in the magnetization. This does not happen at a first order phase transition, where the response to an infinitesimal magnetic field is of order $1$. In this sense, the susceptibility is undefined there rather than infinite. – Yvan Velenik May 9 '16 at 15:15
• On the other hand, if you take a very large but finite system and observe it "at" the first order phase transition, then you'll observe a huge value of the susceptibility (actually a huge value of the susceptibility is a natural way of locating the transition in such a finite system). In this sense, you might say that, indeed, the susceptibility in the thermodynamic limit, at the transition point, is infinite. But that's really just terminology. What is important is to understand what's going on, and it seems that you do. – Yvan Velenik May 9 '16 at 15:15
• Finally, you're right that the behavior will not be universal. There is nevertheless a different form of universality at first-order phase transitions. Namely, if you look at the behavior in finite systems (again), and define, say, the point at which the transition occurs using, as suggested above, the point at which the susceptibility is maximal, then this point will be shifted compared to its value in the infinite system. The exponent describing how the infinite-volume point is approached is universal. This type of analysis is called finite-size scaling and is very useful. – Yvan Velenik May 9 '16 at 15:19
• @YvanVelenik All that would actually make a nice answer. – udrv May 9 '16 at 15:50