Statistical Mechanics models whose Hamiltonian are Euclidean Field Theory actions

Question

In QFT, one is interested in studying functional integrals of the form: \begin{eqnarray} \langle \mathcal{O}_{1},...,\mathcal{O}_{n}\rangle = \int e^{\frac{i}{\hbar}S(\phi)}\mathcal{O}_{1}(\phi)\cdots\mathcal{O}_{n}(\phi)D\phi \tag{1}\label{1} \end{eqnarray} In Euclidean Field Theory, the action $S=S(\phi)$ can be assumed to be of the form: \begin{eqnarray} S(\phi) = \int_{\mathbb{R}^{d}}d^{d}x\bigg{[}\frac{1}{2}\nabla\phi(x)^{2}+\frac{m_{\phi}}{2}\phi(x)^{2}+\lambda\phi(x)^{4}\bigg{]}=\int_{\mathbb{R}^{d}}d^{d}x\bigg{[}\phi(x)(-\Delta+m^{2}_{\phi})\phi(x)+\lambda \phi(x)^{4}\bigg{]} \tag{2}\label{2} \end{eqnarray} where the second equality holds if $\phi$ decays sufficiently fast when $|x|\to \infty$.

There is a natural connection between QFT and Statistical Mechanics. For instance, by Wick rotation we can replace $i/\hbar$ by $-\beta$ and (\ref{1}) becomes expectations in a Statistical Mechanics model with Hamiltonian $H= S$. Moreover, many models in statistical mechanics have Hamiltonians which can be put in the general form (\ref{2}). However, everytime I think about a Statistical Mechanics system which has a Hamiltonian of the form (\ref{2}), the only example that comes to my mind is the Ising model. So, I'd like to collect more models for which the Hamiltonian is of the form (\ref{2}). What are some good examples of such models?

Answer 1

Since we can deform your theory by any number of RG-irrelevant interactions while still remaining in the Ising universality class in the scaling limit, there are an infinite number of such models. So for example, you can add longer range interactions to the Ising model (provided they are not too-long range). Or you can add more complicated interactions which preserve the $\mathbb{Z}_2$ symmetry.

If you are curious about experimentally relevant models, Section 3.1 of this review article goes over several examples in detail, including uniaxial magnets, binary mixtures, and the critical point in liquid-vapor transitions in fluids.

Of course, one expects many more systems in this universality class. If you have a microscopic model with a $\mathbb{Z}_2$ symmetry, and you see a continuous transition to a phase where it is spontaneously broken, you'd expect it to be in this universality class. For example, consider taking an XY model and placing it in a field breaking the symmetry down from O(2) to $\mathbb{Z}_2$: $$ H = - J \sum_{\langle i j \rangle} \cos\left( \theta_i - \theta_j \right) - h \sum_{i} \cos\left( 2 \theta_i \right). $$ Here, we have an angular variable $\theta_i \in [0,2 \pi)$ at every lattice site of a hypercubic lattice. For $h=0$ this has phase transitions in the same universality class as the XY model, but upon adding $h>0$ they cross over into the Ising universality class, where the low-temperature phase settles into the phase where $\theta_i = 0$ or $\pi$ on each site. Of course, a Hamiltonian like this (or a Heisenberg Hamiltonian with an extra field) probably better approximates the microscopic Hamiltonians for a real uniaxial magnet.