Question

Consider a classical lattice model on Zᵈ and suppose that the system undergoes a phase transition as you lower the temperature, i.e., increase β

. The most general definition of a first-order (discontinuous) transition (as far as I know) is that it is discontinuous if (1) the Gibbs measure (satisfying the DLR condition) is non-unique at criticality βc
. Conversely, it is continuous if the Gibbs state is unique at criticality.

I'm curious why this is equivalent to other definitions of continuous/discontinuous phase transitions, such as (2) the correlation length diverges/finite and (3) the magnetization is zero/nonzero.

Indeed, I know that for the q-states Potts model on Z² , it was proven (I think by Hugo Duminil-Cupin) that (2) and (3) are equivalent. I also think there's a counter example for (3) by Aernout van enter, who shown that for a family of 2D XY spin model (not the standard XY model), the transition is discontinuous, even though by Mermin-Wagner, the magnetization must be zero. However, the model seemed relatively "unphysical".

With that in mind, my question is, are there theorems that show equivalence between definitions (1) and (2) or (1) and (3) for relatively general spin systems?

Answer 1

Theorems can prove the equivalence between definitions (1) and (2) or (1) and (3) for relatively general spin systems.

Step-by-step explanation:

Equivalence between definitions (1) and (2) or (1) and (3) for relatively general spin systems can be proven by certain theorems.

For example, for the q-states Potts model on Z², it has been proven that the divergence or finiteness of the correlation length and the existence or nonexistence of magnetization are equivalent.

Similarly, for a family of 2D XY spin model, the transition is discontinuous even though the magnetization must be zero according to Mermin-Wagner theorem.

However, this model may be considered relatively 'unphysical'.