Solvable statistical models on two- and three-dimensional lattices
Exactly solvable models play an important role in statistical
mechanics. The major progress in that field
has been achieved for one- and two-dimensional systems
[ R.J. Baxter, Exactly solvable models in Statistical Mechanics
(Academic Press, London, 1982)].
In three dimensions, exact results had been
scarce. In particular, the attempts to find a physical solution of
tetrahedron equation (integrability condition for
isotropic statistical models on cubic lattice, analogous to
Yang-Baxter relation in 2D) have failed.
We looked from the other viewpoint, namely at multiplane 3D statistical
models, consisting of interacting planes. The interaction is a free
parameter. For inter-plane interaction turned off, the system splits into
solvable 2D planes. One finds at the expense
of losing isotropy, a new class of
exactly solvable physical 3D models [7-10 ].
This approach turns out to be also useful at the level of associated
quantum and classical spin chains
[5,6 ],
and for the vertex
model of traffic [17 ].