Vertex model

A vertex model is a type of statistical mechanics model in which the Boltzmann weights are associated with a vertex in the model (representing an atom or particle). This contrasts with a nearest-neighbour model, such as the Ising model, in which the energy, and thus the Boltzmann weight of a statistical microstate is attributed to the bonds connecting two neighbouring particles. The energy associated with a vertex in the lattice of particles is thus dependent on the state of the bonds which connect it to adjacent vertices. It turns out that every solution of the Yang-Baxter equation with spectral parameters in a tensor product of vector spaces ${\displaystyle V\otimes V}$ yields an exactly-solvable vertex model.

Although the model can be applied to various geometries in any number of dimensions, with any number of possible states for a given bond, the most fundamental examples occur for two dimensional lattices, the simplest being a square lattice where each bond has two possible states. In this model, every particle is connected to four other particles, and each of the four bonds adjacent to the particle has two possible states, indicated by the direction of an arrow on the bond. In this model, each vertex can adopt ${\displaystyle 2^{4}}$ possible configurations. The energy for a given vertex can be given by ${\displaystyle \varepsilon _{ij}^{k\ell }}$,

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