A vertex model is a type of statistical mechanics model in which the Boltzmann weightss 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 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 possible configurations. The energy, with the state of the lattice being an assignment of a state of each bond, and the total energy of the state being the sum of the vertex energies. As the energy is often divergent for an infinite lattice, the model is studied for a finite lattice as the lattice approaches infinite size. Periodic or domain wall boundary conditions may be imposed on the model.

Discussion

For a given state of the lattice, the Boltzmann weight can be written as the product over the vertices of the Boltzmann weights of the corresponding vertex states

where the Boltzmann weights for the vertices are written

1. ,

and the i, j, k, l range over the possible statuses of each of the four edges attached to the vertex. The vertex states of adjacent vertices must satisfy compatibility conditions along the connecting edges (bonds) in order for the state to be admissible.

The probability of the system being in any given state at a particular time, and hence the properties of the system are determined by the partition function, for which an analytic form is desired.

where β = 1/kT, T is temperature and k is the Boltzmann constant. The probability that the system is in any given state (microstate) is given by

so that the average value of the energy of the system is given by

In order to evaluate the partition function, firstly examine the states of a row of vertices.

The external edges are free variables, with summation over the internal bonds. Hence, form the row partition function

This can be reformulated in terms of an auxiliary n-dimensional vector space V, with a basis as

as

thereby implying that T can be written as

where the indices indicate the factors of the tensor product on which R, gives

is the row-transfer matrix.

By summing the contributions over two rows, the result is

for M rows, this gives

as

is the largest eigenvalue to the power of M, the power of the largest eigenvalue becomes much larger than the others. As the trace. This implies that the eigenspaces are common, and restricts the possible space of solutions. Such a family of commuting operators is usually found by means of the Yang–Baxter equation, which thus relates statistical mechanics to the study of quantum groups.

Integrability

Definition: A vertex model is integrable 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 defined as above, then

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

and using the cyclic property of the trace operator that the following corollary holds.

Corollary.

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, where R. This motivates the classification of all the finite-dimensional irreducible representations of a given Quantum algebra in order to find solvable models corresponding to it.

Notable vertex models

• Six-vertex model

• Eight-vertex model

• Nineteen-vertex model (Izergin-Korepin model)

References