I Spontaneous symmetry breaking from a QM perspective

[Vanadium 50]

This is not what I asked for. It is not the Hamiltonian for the 1D situation. It is not the Hamiltonian for the 2D situation. It is not even a Hamiltonian. There is no wavefunction.

Here's what I think. I think you are in way over your head here and just don't have the background to understand what is going on with SSB, since you are not answering even basic QM questions about the system. In lieu of being able to calculate, you are forced to rely on stream of words that explain the outcome of the calculation you are not able to do.

You have a choice - you can go back to the QM and solidfy your background, or you can hope that someone will hit on the right arrangement of words.

 

[atyy]

One more thing. I read this article and on page 2 it states: "the state of a system, if it is to be stationary, must always have the same symmetry as the laws of motion which govern it". But if I understand it right from your reply: "the ground states are not symmetric under the symmetry transformation. They are nevertheless eigenstates of the Hamiltonian and thus stationary states." you are stating exactly the opposite. And now I am really confused. Do stationary states have, or not, the same symmetry as the Hamiltonian of the system?

The ferromagnet cited by @vanhees71 is a classic example of spontaneous symmetry breaking. However the "More is Different" article is by Anderson, who atypically does not consider the ferromagnet to be an example of spontaneous symmetry breaking. https://condensedconcepts.blogspot.com/2011/06/ferromagnetism-is-not-spontaneously.html

For the ferromagnet, even a finite system will demonstrate spontaneous symmetry breaking in the sense that @vanhees71 mentioned, because the finite system has ground states that are exactly degenerate - and stable - they don't tunnel into each other.

However, in most examples of quantum spontaneous symmetry breaking, there is only approximate ground state degeneracy that becomes more and more exact as the system becomes larger, ie. the thermodynamic limit is required for spontaneous symmetry breaking.

With the ammonia molecule, there is a high tunneling rate between the approximately degenerate states, which is why the ammonia molecule is usually not considered an example of spontaneous symmetry breaking. In contrast, with the best examples of quantum spontaneous symmetry breaking, tunneling between approximately degenerate low-energy states is increasingly suppressed as the system gets larger.

You can see more discussion on why the ferromagnet is atypical as an example of quantum spontaneous symmetry breaking in these papers:
Spontaneous Symmetry Breaking in Quantum Mechanics by Jasper van Wezel, Jeroen van den Brink (see footnote 11)
An Introduction to Spontaneous Symmetry Breaking by Aron J. Beekman, Louk Rademaker, Jasper van Wezel (see Exercise 2.7 on p56)

 

[vanhees71]

There are few topics which are confusing for students because of sloppy jargon of physicists than spontaneous symmetry breaking. The reason may be that it took some time to get a clear understanding about the different cases.

The ammonia molecule cannot be an example for spontaneous symmetry breaking, because by definition if there is spontaneous symmetry breaking, the ground state must be degenerate. If the broken (global!) symmetry is a Lie symmetry, then you have a Goldstone boson, i.e., a massless excitation. The ammonia molecule is rather an example for explicit symmetry breaking, and indeed there's no Goldstone boson.

What was not clear for a long time also is that if the symmetry is a local gauge symmetry, there cannot be spontaneous symmetry breaking, though almost all textbook describe it as such. I'd rather talk about the "Anderson-Higgs mechanism". It has been discovered by Anderson in a famous paper on superconductivity and was fully understood by Higgs as not only a mechanism to describe massive gauge (non-Abelian) gauge bosons without breaking the local gauge symmetry but also that there must be (in the non-Abelian case) necessarily also a massive scalar boson, the Higgs boson. The formalism used to write down a Anderson-Higgsed gauge model indeed at the first glance looks a lot like the way you write down a model with a spontaneously broken global symmetry. The crucial discovery by Anderson however was that in fact here one has no spontaneously broken symmetry and also no degenerate ground state, because the apparent degenerate ground states are all connected by a local (and that's the crucial difference to the case of a spontaneously broken global symmetry!) gauge transformation and thus to be identified as the same state. The immediate implication is the also crucial point that the field degrees of freedom which would describe massless Goldstone modes in the global-symmetry case do not occur in the physical spectrum of the Anderson-Higgsed gauge theory but (in the socalled unitary gauge) are completely absorbed into the gauge-boson fields providing the additional degree of freedom of a massive vector boson as compared to a massless one. For more general gauges (particularly the important 't Hooft $R_{\xi}$ gauges) you get apparent would-be Goldstone field degrees of freedom, which however in such gauges are "ghosts" in the same sense as you have to introduce Fadeev-Popov ghosts, i.e., to cancel out the contributions from unphysical field degrees of freedom, leading to gauge invariant description of observable quantities like cross sections.

 

[atyy]

Another note on the ammonia molecule. As stated above, the ammonia molecule is usually not considered an example of spontaneous symmetry breaking. Spontaneous symmetry breaking is most commonly demonstrated with a thermodynamic limit. However, there is a way to start from the ammonia molecule and modify the potential to get a an example of spontaneous symmetry breaking - like the thermodynamic limit, the modification involves a strong non-analyticity, but it is a simpler modification than the thermodynamic limit.

The ammonia molecule can be modeled as a double-trough potential. If one now replaces the troughs with rectangular wells, and makes the barrier between the two wells infinitely high, then one can get spontaneous symmetry breaking. This is discussed qualitatively in Anderson's More is Different, and mentioned in Weinberg's Quantum Mechanics (p179-181): "These considerations point up a general feature of spontaneous symmetry breaking: It is always associated with systems that in some sense are very large. It is only the very large barrier in molecules like proteins and sugars that allow these molecules to have a definite handedness. In quantum field theory, it is the infinite volume of the vacuum state that allows other symmetries to be spontaneously broken."

Similar discussions:
https://arxiv.org/abs/1205.4773
https://thiscondensedlife.wordpress.com/2016/05/12/broken-symmetry-and-degeneracy/.

 

[vanhees71]

And to make the list complete, here a very nice paper on the case of local gauge theories (here for superconductivity)

https://arxiv.org/abs/cond-mat/0503400