Why doesn't Coleman-Mermin-Wanger theorem contradict phase transitions in systems with 1D reaction coordinate?

Question

How does both hold:

• Coleman-Mermin-Wanger theorem

  continuous symmetries cannot be spontaneously broken at finite temperature in systems with sufficiently short-range interactions in dimensions $d\le2$

• There are observable systems with an effective 1D reaction coordinate that go under phase transitions, like proteins or Spin-Glass.

---

For example below there is a 1D system with 2 phases state:

Here we have the energy landscape of protein folding which can be study by 1 reaction coordinate and has a few different phases:

---

How comes there are systems with $d \le 2$ that have phase transitions and why doesn't it contradict the Mermin-Wagner theorem? How is it related to Noeather's theorem?

Here are more examples of 1D systems which have phase transitions: Kittel’s Model and 1D Ising Model.

Answer 1

Short answer

While there are commonly no phase transitions in one dimension, they can occur in special circumstances:

• when there are long range interactions, or
• when each local degree of freedom has an unbounded (local) state space, or
• when there are constraints (configurations with infinite energy),

or in other more specialized situations. All of the examples in the question fall into one of these categories.

Details

As Yvan Velenik commented, several of the examples presented in the question have discrete rather than continuous symmetries, and thus the Mermin-Wagner theorem does not apply. The question is still relevant, though, as there is another often quoted "law" of equilibrium statistical mechanics which does hold for systems with discrete symmetries, and which states that "there are no phase transitions in one-dimensional systems with short range interactions". This law, often referred to as "Landau's argument", is indeed correct but with some important caveats. The ultimate reference (to the best of my knowledge) which discusses many fine details related to phase transitions in 1d systems is Cuesta and Sanchez, J Stat Phys 2004. Even this excellent paper does not claim to classify all possibilities of 1d phase transitions, and indeed this is an ongoing area of research (see, e.g., this very recent paper by Saryal et al).

Going over the particular examples in the question:

1. I am not sure I understand the first graph. Presumably G is the Gibbs free energy and x is the order parameter. If this interpretation is correct, the content of the theorem implies that the Gibbs free energy of a 1d system with short range interactions (and some further fine print) cannot be of the form presented in the graph.

2. A protein is a 1 dimensional molecule but importantly it lives in 3 dimensional space. If you'd like, you can think of it as a 1 dimensional system with long range interactions (as distant parts of the protein may come in contact). As far as I understand, the 1 dimensional funnel picture is a caricature. To the extent that this caricature can be made precise, the situation here is like in example 1: the free energy cannot have such a form which allows phase transitions when the underlying microscopic model is 1d with short range interactions (and some more fine print).

3. The nearest-neighbors Ising model and Edwards-Anderson spin glass model do not have a phase transition in one dimension. When the interactions are long-ranged phase transitions can occur (as discussed in the Wikipedia page linked in the question). The mean-field versions of these models also have a phase transition, but these are essentially models with infinite range interactions (every spin interacts with every other spin).

4. Kittel's zipper model is discussed in the paper of Cuesta and Sanchez. This is an example where a phase transition can occur because the state space has constraints: all bonds on one side of the "zipper" must be closed, and all bonds on the other side must be open (in other words, configurations with alternating closed and open segments have infinite energy).

Answer 2

The question seems very confused. The Mermin-Wagner theorem deals with spatial dimensions, because it considers how continuous symmetry breaking is affected by spatial fluctuations.

It has absolutely nothing to do with the dimension of the configuration space for a single particle. For example, a spin system in three spatial dimensions can have a symmetry breaking phase transition. This holds even if the state of a single spin is described by a vector with $82$ components, a single real number, or even a discrete $0$ or $1$.

Similarly, whether or not the Mermin-Wagner theorem applies to protein folding depends on the number of spatial dimensions the proteins exist in. It doesn't have anything to do with the number of coordinates you need to describe the state of one protein.

Additionally, the question seems to be claiming that a single protein can undergo a phase transition. That's simply incorrect. Materials can undergo phase transitions, single molecules can't. A trough in an potential is not a phase.