Vertex Model - Integrability

Integrability

Definition: A vertex model is integrable if, such that

This is a parameterized version of the Yang-Baxter equation, corresponding to the possible dependence of the vertex energies,and hence the Boltzmann weights R on external parameters, such as temperature, external fields, etc.

The integrability condition implies the following relation.

Proposition: For an integrable vertex model, with and defined as above, then

as endomorphisms of, where acts on the first two vectors of the tensor product.

It follows by multiplying both sides of the above equation on the right by and using the cyclic property of the trace operator that the following corollary holds.

Corollary: For an integrable vertex model for which is invertible, the transfer matrix commutes with .

This illustrates the role of the Yang-Baxter equation in the solution of solvable lattice models. Since the transfer matrices commute for all, the eigenvectors of are common, and hence independent of the parameterization. It is a recurring theme which appears in many other types of statistical mechanical models to look for these commuting transfer matrices.

From the definition of R above, it follows that for every solution of the Yang-Baxter equation in the tensor product of two n-dimensional vector spaces, there is a corresponding 2-dimensional solvable vertex model where each of the bonds can be in the possible states, where R is an endomorphism in the space spanned by . This motivates the classification of all the finite-dimensional irreducible representations of a given Quantum algebra in order to find solvable models coreesponding to it.