There are many so-called “theorems” in physics. The most famously quoted in the field of condensed matter are the ones associated with the names of Goldstone, Mermin-Wagner, and McMillan.

If you aren’t familiar with these often (mis)quoted theorems, then let me (mis)quote them for you:

1) Goldstone: For each continuous symmetry a phase of matter breaks, there is an associated collective excitation that is gapless for long wavelengths, usually referred to as a Nambu-Goldstone mode.

2) Mermin-Wagner: Continuous symmetries cannot be spontaneously broken at finite temperature in systems with sufficiently short-range interactions in dimensions *d* ≤ 2. (From Wikipedia)

3) McMillan (PDF link!): Electron-phonon induced superconductivity cannot have a higher Tc than approximately 40K.

All these three theorems in condensed matter physics have been violated to a certain extent. My gut feeling, though, is that these theorems can have the adverse consequence of limiting one’s imagination. As an experimental physicist, I can see the value in such theorems, but I don’t think that it is constructive to believe them outright. The number of times that nature has proven that she is much more creative and elusive than our human minds should tell us that we should use these theorems as guidance but to always be wary of such ideas.

For instance, had one believed the Mermin-Wagner theorem outright, would someone have thought the existence of graphene possible? In a solid, which breaks translational symmetry in three directions and rotational symmetry in three directions, why are there only three acoustic phonons? McMillan’s formula still holds true for electron-phonon coupled superconductors (marginal case being MgB2 which has a Tc~40K), though a startling discovery recently may even shatter this claim. However, placed in its historical context (it was stated before the discovery of high-temperature superconductors), one wonders whether McMillan’s formula disheartened some experimentalists from pursuing the goal of a higher transition temperature superconductor.

My message: One may use the theorems as guidance, but they are really there to be broken.

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You should really add Luttinger’s theorem to this list – it is kind of doing the same thing you claim for the others. I am happy to elaborate at length.

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That’s an interesting one as well and very pertinent (or so I hear) in the context of the cuprates. Please do elaborate; I am much more familiar with the previous three, which is why I failed to mention Luttinger’s.

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The problem does not stem from the statement that these are theorems (which for the two first at least, they are), but from the fact that physicists tend to forget that theorems come with assumptions, and are not valid in general (that is, for an arbitrary physical system).

For example, the Mermin-Wagner theorem is valid only for short range interactions. And we know that membrane-like fluctuations of graphene have long-range interactions, which implies that MW theorem does not apply.

Same thing for Goldstone theorem. The often quoted result was in fact derived for a relativistic theory and a continuous internal symmetry. To apply it to a non-relativistic theory, and/or space-time symmetry, you have to rederive another version of it, which of course implies only three phonons, and not six, for solids.

I don’t know for the last one, it had not hear about it before reading this post.

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Hi Adam,

Thanks for your comment — in some sense the point you are making is the point I was trying to address. In every one of these theorems there are assumptions, either implicit or explicit. What I was trying to say is that sometimes these rules can be broken because we don’t know the implicit assumptions. Sometimes experiments surprise us because we only realize afterward the implicit assumptions in these theorems. To my mind, therefore, it is better to keep theorems that say something cannot exist, such as the Mermin-Wagner theorem, at arm’s length. Condensed matter is too rich to say something cannot exist.

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