I am currently revising the lecture notes in which different spin systems are analyzed, focussing on the occurrence (or absence) of phase transitions. Different techniques are applied to analyze the different models and - seemingly - also different notions of phase transitions. These notions are not mathematically precisely defined and the question is how to connect them.

I will briefly outline the analysis of three different models and their notions of phase transitions.

  1. Absence of phase transitions via smoothness of the free energy: 1-D Ising Model. Here it is shown that the free energy per spin is a smooth (in general not analytic) function of the temperature and the magnetic field.

  2. Phase transitions via magnetization: 1-dimensional chain of non-interacting spins in a magnetic field. Here it is shown that the average magnetization (averaged over all spins, not as in a expected value) converges a.s. to tanh(hb) in the thermodynamical limit where h is the magnetic field and b is the inverse temperature.

  3. Phase transition via magnetization: Curie-Weiß-model without magnetic field. Here it is shown that for high temperatures the average magnetization converges to zero. For low temperatures however the magnetization is either positive or negative, each with proability 1/2

  4. Phase transition via existence of Gibbs measure: 2D Ising model without magnetic field. Here it is shown that in the thermodynamical limit there exists a Gibbs measure such that above some critical temperature the expected direction of every spin is equal and non-zero. This means that the expected average magnetization must also be non-zero.

I am wondering how to connect these different concept and how these relate to the real-world intuition we have for phase transitions.


2 Answers 2


One can provide two characterizations of first-order phase transitions in Ising models at inverse temperature $\beta$ and external magnetic field $h$:

Thermodynamic characterization: there is a first-order phase transition at $(\beta^*,h^*)$ if the free energy $h \mapsto f(\beta^*,h)$ is not differentiable at $h^*$.

Probabilistic characterization: there is a first-order phase transition at $(\beta^*,h^*)$ if there are multiple infinite-volume Gibbs measures for these values of the parameters.

These two characterizations turn out to be equivalent. In the Ising model, this is easily seen to be a consequence of the FKG inequality. Namely, one has the following equivalences: $$ \text{unique Gibbs measure at }(\beta,h) \quad\Leftrightarrow\quad \langle\sigma_0\rangle^+_{\beta,h} = \langle\sigma_0\rangle^-_{\beta,h} $$ where $\langle\cdot\rangle^+_{\beta,h}$, resp. $\langle\cdot\rangle^-_{\beta,h}$, denote expectation with respect to the infinite-volume Gibbs measures obtained as limits of measures with $+$, resp. $-$, boundary condition.

Now, another consequence of the FKG inequality is that $\langle\sigma_0\rangle^+_{\beta,h}$ and $\langle\sigma_0\rangle^-_{\beta,h}$ coincide with the right- and left-derivatives of the free energy: $$ \langle\sigma_0\rangle^+_{\beta,h} = \frac{\partial}{\partial h^+} f(\beta,h) \qquad \langle\sigma_0\rangle^-_{\beta,h} = \frac{\partial}{\partial h^-} f(\beta,h) $$ In particular, these quantities coincide with the average magnetization density in the $+$ and $-$ phases. Note also that whenever the left- and right-derivatives of the free energy do not coincide, the magnetization density (which is its derivative) has a discontinuity.

The above should answer your questions:

  1. In the 1d Ising model, analyticity of the free energy as a function of $h$ implies that its right and left derivative coincide. In particular, there is a unique infinite-volume Gibbs measure and the magnetization vanishes (by symmetry) when $h=0$.
  2. In the case of noninteracting spins, the magnetization density is again a smooth function of $h$, which implies uniqueness of the Gibbs measure at all $h$.
  3. At low temperature, for the Ising model in dimension $2$ or more (as well as for the Curie-Weiss that you mention), the limits of the magnetization as $h\downarrow 0$ and $h\uparrow 0$ do not coincide. This implies that the infinite-volume $+$ and $-$ states are distinct. In particular, if you sample an Ising configuration in a large system with, say, free or periodic boundary condition, then the magnetization density is either close to $\langle\sigma_0\rangle^+_{\beta,0}$ or to $\langle\sigma_0\rangle^-_{\beta,0}$, each possibility occurring with probability $1/2$ (by symmetry).
  4. Again, in dimensions $2$ and larger, there are multiple infinite-volume Gibbs measures (in fact, infinitely many of them). Of those, there are two extremal ones that are the physically relevant ones: those obtained as limits with $+$ and $-$ boundary condition respectively. Under those measures, the average spin magnetization at $h=0$ and $\beta>\beta_c$ is positive, resp. negative.

(All this is discussed in detail and with complete proofs in Chapter 3 of our book.)


As for your 1.,2.,3., magnetization is a derivative of free energy with respect to magnetic field, so smoothness of free energy and smoothness of magnetization are closely related.


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