Today, in our Mathematical Quantum Mechanics lecture, I put my foot in my mouth. I claimed that under very general circumstances there cannot be spontaneous symmetry breaking in quantum mechanics. Unfortunately, there is an easy counter example:
Take a nucleus with charge Z and add five electrons. Assume for simplicity that there is no Coulomb interaction between the electrons, only between the electrons and the nucleus (this is not essential, you can instead take the large Z limit as explained in this paper by Frieseke. The only way the electrons see each other is via the Pauli exclusion principle. The Hamiltonian for this system has an obvious SO(3) rotational symmetry. The ground state, however is what chemists would call 1S^2 2S^2 2P^1. That is, there is one electron in a P-orbital and in fact this state is six-fold degenerate (including spin). Of course, there is a symmetric linear combination but in that six dimensional eigen-space of the Hamiltonian there are also linear combinations that are not rotationally invariant. Thus, the SO(3) symmetry here is in general spontaneously broken.
This is in stark contrast to the folk theorem that for spontaneous symmetry breaking you need at least 2+1 non-compact dimensions. This for example is discussed by Witten in lecture 1 of the IAS lectures or here and is even stated in the Wikipedia.
Witten argues using the Stone-von Neumann theorem on the uniqueness of the representation of the Weyl group (the argument is too short for me) and explicitly only discusses the particle on the real line with a potential (the famous double well potential where the statement is true). In 1+1 dimensions, there is the argument due to Coleman, that in the case of symmetry breaking you would have a Goldstone boson. But free bosons do not exist in 1+1d since the 2-point function would be a log which is in conflict with positivity.
I talked to a number of people and they all agreed that they thought that "in low dimensions (QM being 0+1d QFT), quantum fluctuations are so strong they destroy any symmetry breaking". Unfortunately, I could not get hold of Prof. Wagner (of the Mermin-Wagner theorem) but maybe you my dear reader have some insight what the true theorem is?