# Simplest exactly solved model displaying a phase transition?

The classical example of an exactly solved model which displays a phase transition is the 2D Ising model. However, all the proofs I've seen of this have been very long and complicated.

So, I wanted to know whether there were any other exactly solved models with phase transition, which were easier to solve, or that the 2D Ising model is the simplest such model that we know of.

• You may look into the Kitaev honeycomb model which is exactly solvable, though the phase transition is a quantum one. Sep 7 '21 at 4:38
• The integrability of the planar Ising model is a nice feature (and provides detailed information on the phase transition), but is by no means required to prove the existence of the phase transition (actually, in any dimension larger or equal $2$). Note that the first proof of a phase transition in this model is due to Peierls in 1936, several years before Onsager's computation of the free energy of the model. Sep 7 '21 at 6:02

The simplest model demonstrating a phase transition is probably the Ising model with an interaction constant that is the same for all spin pairs:

$$H=-J\sum_{i,j}S_i S_j$$.

I will try to find a reference later.

• In 1D I assume? Sep 6 '21 at 23:06
• @ConnorBehan the dimensionality doesn't matter since the interactions are not dependent on the separation. Sep 6 '21 at 23:13
• @ConnorBehan it is infinite-dimensional, since all spins are coupled to all spins. It is typiclaly referred to as the infinite range Ising model. Sep 7 '21 at 11:33

The quantum Ising model in a transverse field $$H=\sum_n \left(\hat \sigma_{z,n} \hat \sigma_{z, n+1} +\lambda \hat \sigma_{x,n}\right)$$ is easily solvable and has a phase transition. It's a one-dimensional quantum model but, through the usual quantum-classical map that takes the hamiltonian to the transfer matrix, it is equivalent to the classical Ising model in 2d.

• Could you post more details or a link to an accessible tutorial (both for solving and for mapping to the 2D Ising)? Sep 7 '21 at 11:53
• It's treated in many books and papers. The best way to find it is google "transverse ising model" The wiki article is en.wikipedia.org/wiki/Transverse-field_Ising_model Sep 7 '21 at 13:36

Exactly solvable models in Statistical mechanics by Baxter is the place to look.

The other answers have already pointed the infinite range Ising model and 1D Ising model as exactly solvable (although the latter has phase transition at zero temperature). Besides the infinite and the 1D case, the Ising (and more generally Potts) model is also solvable on a Bethe lattice, as discussed extensively in the book cited above.

There are also offshoots of these models with additional features: e.g., Sherrington-Kirkpatrick model is the infinite range Ising model with random couplings, which is useful for understanding spin glasses.

However, the specific features of phase transitions in these models differ, and thus the model of choice depends on what one is looking for.

Remark: As more exotic case of exactly solvable models one could mention soluton of Kondo and Anderson impurity models by Bethe ansatz.