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How does both hold:


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

enter image description here

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

enter image description here


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.

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    $\begingroup$ The Mermin-Wagner theorem states that a continuous symmetry cannot be broken in two or less dimensions (when the interaction decay fast enough). It does not prevent breaking of discrete symmetries, nor phase transitions without no broken symmetries. In this respect, I don't see how your examples are relevant. $\endgroup$ – Yvan Velenik Jul 15 '18 at 7:46
  • $\begingroup$ @YvanVelenik not sure how proteins aren't continuous system an neither why the Mermin Wagner theorem has a version for Heisenberg model? And also how this related to Noeather's theorem? $\endgroup$ – 0x90 Jul 15 '18 at 7:51
  • $\begingroup$ You are misunderstanding the meaning of continuous symmetry. This has nothing to do with the fact that the model is defined on a lattice or in the continuum. What is relevant is whether the Hamiltonian is invariant under the action of a continuous symmetry group. For example, the Hamiltonian of the Heisenberg model is invariant under a simultaneous rotation of all the spins by the same amount. As far as I can tell, there are no direct relations between Noether theorem and Mermin-Wagner. $\endgroup$ – Yvan Velenik Jul 15 '18 at 8:15
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    $\begingroup$ Note also that even when a model's Hamiltonian is invariant under the action of a continuous symmetry group, the model might still undergo a phase transition in dimension 2: the only claim is that the continuous symmetry cannot be broken, but there may be additional symmetries that are broken, for example. $\endgroup$ – Yvan Velenik Jul 15 '18 at 8:18
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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).

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  • $\begingroup$ What do you mean by "an unbounded (local) state space,"? $\endgroup$ – 0x90 Jul 24 '18 at 12:56
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    $\begingroup$ Think of an Ising or Potts type model where the degrees of freedom ("spins") sit on a lattice. By bounded local state space I mean that each such "spin" can take one of a finite number of states (e.g $\pm 1$ for the Ising model). An example of a model with an ubounded local state space is a model of an interface or solid-on-solid where each lattice site has a "height" variable $h_i \in \mathbb{N}$. In this case the transfer matrix of the model is of infinite size and phase transitions may occor (see the example of the Chui-Weeks's model in the paper of Cuesta and Sanchez). $\endgroup$ – Ori Jul 24 '18 at 13:36
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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.

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  • $\begingroup$ What do you mean in here: Materials can undergo phase transitions, single molecules can't. A trough in an potential is not a phase. $\endgroup$ – 0x90 Aug 25 '18 at 5:29
  • $\begingroup$ @0x90 What do you find confusing about it? $\endgroup$ – knzhou Aug 25 '18 at 5:39
  • $\begingroup$ why single molecules can't undergo a phase transition? And why a basin in the potential isn't a phase transition? $\endgroup$ – 0x90 Aug 25 '18 at 5:44
  • $\begingroup$ @0x90 What definition of "phase transition" are you using? $\endgroup$ – knzhou Aug 25 '18 at 5:53
  • $\begingroup$ transition between state of matter. How do you define phase transition? $\endgroup$ – 0x90 Aug 25 '18 at 13:45

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