As is well known the Ising model exhibits a phase transition, except the one dimensional case in which the phase transition occurs strictly at $T=0$. Now I have always thought that this makes the case uninteresting. Until I started learning supersymmetry.
As is also well known supersymmetry is spontaneously broken at any finite temperature. Intuitively one can argue that since Fermi-Dirac and Bose-Einstein distributions are very different it is impossible to maintain a boson-fermion symmetry at finite temperature. Per usual arguments relating SSB and phase transitions one could think that any model of SUSY has a phase transition at $T=0$.
In order to better understand this analogy I was wondering: what kind of models, like the 1D Ising, have phase transition exactly at $T=0$? Are there any one with continuous global symmetry (and thus a Goldstone mode)? Is there a model in quantum field theory?
Just to clarify, I do not intend here to ask for the so called Quantum Phase Transitions that occur at $T=0$ under variation of a external parameter. I'm concerned with phases that exist only at absolute zero.
EDIT: I was going to delete the answer bu it ocurred to me that maybe it will help someone with the same misunderstanding that I had. The key is in the comment which clarified that one cannot compare SUSY breaking at finite temperature with usual phase transitions because in phase transition the High temperaure phase has the symmetry restored while in SUSY the high temperature case is the one with symmetry broken. Therefore I do not regard the question here as meaningful.