Ergodicity breaking in the Ising model

Question

I initially asked a similar question here on MSE where it didn't get much attraction. As was suggested in a comment, I'm asking here although I'm reformulating the question so that both questions can be considered different (in particular, in the MSE question I'm asking for a proof of ergodicity breaking).

Let's consider the Ising model in dimension $d \geq 2$ with constant coupling $J= 1$ between direct neighbors and without external field.

It is common to read in physics texts that, under the critical temperature, the symmetry of the problem (w.r.t. magnetization inversion) and its ergodicity are broken.

For instance in this article:

Ergodicity is globally broken because the trajectory remains confined within the same component in which it has originated, but continues to hold separately within each component.

However, the Ising model is generally presented with the associated Boltzmann distribution, without any notion of dynamic (there is actually no reference to time) and the term trajectory isn't well defined then.

My question is: how do we define ergodicity (and ergodicity breaking) in such context?

Answer 1

That's a good point.

Usually when you say this kind of thing, you assume the dynamics to be local. For example using a monte carlo dynamics (1 monte carlo move == 1 time unit for example) either the usual one where you just try to flip a spin or the less known Glauber/Kawasaki dynamics where you exchange neighbors. In the last case the number of spin up and down is conserved so it's less relevant to your question. The point being that, these algorithms are local and hence they explore the potential landscape locally, meaning that you can describe the evolution of the magnetization using a Langevin equation for example, with some noise and a potential barrier to cross. The magnetization explores the bottom of one of the minimum of the free energy and only noise can help it escape its minimum to go to the other one. However, the larger the system is, the deeper becomes the well or the smaller becomes the noise (you can see it both way). Therefore, you end up stucked. These algorithms leads to an equilibrium probability distribution because they respect a very important condition called detailed balance.

You can also designed algorithms, that still reproduce the stationary distribution, but flip spins in a highly non local way. They dont respect detailed balance, but do respect a less stringent condition, global balance. Look at the Wolff or the Swendsen–Wang algorithms. Anyway, these things are non local and hence, can really easily make you pass from one minimum of the free energy to the other. For example, if your whole system at 0 temperature is made of spin up, one monte carlo move will change all spin to down.

Finally, in the real world, you would also have some dynamics and it would most likely ressemble a monte carlo dynamics where some spins are flipped by thermal agitations, and returned to their initial state if the temperature is small.

But you are right to say that ergodicity breaking is not something associated only to the static problem (altough, in the real world, the dynamics is always local in some way, hence it is noy so wrong to say that ergodicity is broken if the free energy well is very deep)