Vertex model
[1][2] 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
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.
Periodic or domain wall[3] boundary conditions may be imposed on the model.
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 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
Summing over the states of the bonds in the first row with the periodic boundary conditions
By summing the contributions over two rows, the result is which upon summation over the vertical bonds connecting the first two rows gives:
for M rows, this gives and then applying the periodic boundary conditions to the vertical columns, the partition function can be expressed in terms of the transfer matrix
, the power of the largest eigenvalue becomes much larger than the others.
As the trace is the sum of the eigenvalues, the problem of calculating
reduces to the problem of finding the maximum eigenvalue of
However, a standard approach to the problem of finding the largest eigenvalue of
is to find a large family of operators which commute with
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.
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.
acts on the first two vectors of the tensor product.
and using the cyclic property of the trace operator that the following corollary holds.
This illustrates the role of the Yang–Baxter equation in the solution of solvable lattice models.
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
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.
