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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.

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  • $\begingroup$ I am not sure I understand your analogy: in the 1d Ising model, $T=0$ is the only temperature at which the symmetry is broken. In any case, this is of course extremely general: the same will be true (at the classical level) for any one-dimensional model with compact spins, and periodic interactions. This, of course, includes models with continuous symmetry (the one-dimensional $O(N)$-models, for example). $\endgroup$ – Yvan Velenik Mar 6 '15 at 10:46
  • $\begingroup$ @YvanVelenik, sorry for the confusion. What I meant is this: in the 1D Ising model any fluctuation (in this case thermal ones) break the symmetry. In the SUSY case the supersymmetry prevents vacuum fluctuations, but if you add thermal ones the supersymmetry is broken. Is there another quantum field theory where for some reason the vacuum fluctuations do not break the symmetry but any thermal ones do or is the $T=0$ phase transition only possible in classical systems and the SUSY case is dependent on supersymmetry to ward off vacuum fluctuations. I want to understand how much SUSY is important $\endgroup$ – cesaruliana Mar 6 '15 at 17:38
  • $\begingroup$ I still don't see what you mean. In the 1d Ising model, the thermal fluctuations restore the symmetry which is broken at $T=0$ (the model is symmetric whenever $T\neq 0$, but is not symmetric when $T=0$). This is the opposite of what you seem to say. $\endgroup$ – Yvan Velenik Mar 7 '15 at 9:15
  • $\begingroup$ @YvanVelenik, I see what you say. I was trying to put SUSY breaking by finite temperature in the context of phase transition, but you're comment convinced me that the analogy is fundamentally flawed, since the temperature destroys rather than restores the symmetry. In fact now I think that the analogy I tried to make in the question cannot be possibly correct by this argument and SUSY breaking is physically different from spontaneous symmetry breaking, thank you for your time. Since I don't think the question makes sense I'll wait a couple more days, but if nothing else will delete it. $\endgroup$ – cesaruliana Mar 7 '15 at 23:25
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One such example is the 2D Heisenberg antiferromagnet. The ground state breaks spin rotation symmetry, but the Mermin-Wagner theorem tells us that the symmetry is unbroken at any finite temperature.

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  • $\begingroup$ In 2D it would be a finite-temperature phase transition. The OP seems to ask for "$T=0$" transition (although I do not really know what that means, honestly speaking). $\endgroup$ – Meng Cheng Mar 7 '15 at 6:16
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    $\begingroup$ I think the OP asks exactly for systems which have some kind of order at $T=0$ which immediately disappears at $T>0$, and I would say the Heisenberg model is exactly such an example. Whether this should be called a phase transition, I don't know. $\endgroup$ – Norbert Schuch Mar 8 '15 at 2:13

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