Solvable 3D classical statistical models and a spin-fermion mapping

In this talk, I will describe an exactly solvable classical statistical model on a 3D lattice. The model has two classical topological phases (one of which has been introduced previously [1]), and a finite-temperature phase transition between them. In contrast to prior 3D solvable statistical models [2], our system is much simpler, yet does not trivially reduce to a 2D model, and thus displays genuinely 3D correlations. Excitingly, this provides the first exactly solvable model in which to explore genuinely 3D critical phenomena; a caveat, however, is that the model Hamiltonian has imaginary terms. The construction and solution of this model are the first application of a new method that we introduce, developing ideas of [3], to map between a locally interacting fermionic Hamiltonian and a locally interacting spin Hamiltonian, in arbitrary dimension, in a simple way based on the use of algebraic isomorphisms. We expect that the spin-fermion mapping can be used not only for exact solutions, but for other applications such as numerical algorithms or perturbative calculations.
References:
[1] C. Castelnovo and C. Chamon, Phys. Rev. B 76, 174416 (2007).
[2] A.B. Zamolodchikov, Commun. Math. Phys. 79, 489-505 (1981).
[3] F. Verstraete and J.I. Cirac, J. Stat. Mech. P09012 (2005).