Vertex 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

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 Boltzmann's 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 { v 1 , … , v n } , and R ∈ E n d ( V ⊗ V ) as

and T ∈ E n d ( V ⊗ V ⊗ N ) as

thereby implying that T can be written as

where the indices indicate the factors of the tensor product V ⊗ V ⊗ N on which R operates. Summing over the states of the bonds in the first row with the periodic boundary conditions i 1 = i 1 ′ , gives

where τ = trace V ⁡ ( T ) is the row-transfer matrix.

By summing the contributions over two rows, the result is

which upon summation over the vertical bonds connecting the first two rows gives: ( ( trace V ⁡ ( T ) ) 2 ) j 1 … j N ℓ 1 … ℓ N

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 τ as

where λ m a x is the largest eigenvalue of τ . The approximation follows from the fact that the eigenvalues of τ M are the eigenvalues of τ to the power of M, and as M → ∞ , 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 Z reduces to the problem of finding the maximum eigenvalue of τ . This in it itself is another field of study. However, a standard approach to the problem of finding the largest eigenvalue of τ is to find a large family of operators which commute with τ . 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 if, ∀ μ , ν , ∃ λ 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 λ , μ and ν defined as above, then

as endomorphisms of V ⊗ V ⊗ V ⊗ N , where R ( λ ) acts on the first two vectors of the tensor product.

It follows by multiplying both sides of the above equation on the right by R ( λ ) − 1 and using the cyclic property of the trace operator that the following corollary holds.

Corollary: For an integrable vertex model for which R ( λ ) is invertible ∀ λ , the transfer matrix τ ( μ ) commutes with τ ( ν ) ,   ∀ μ , ν .

This illustrates the role of the Yang-Baxter equation in the solution of solvable lattice models. Since the transfer matrices τ commute for all λ , ν , the eigenvectors of τ 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 { 1 , … , n } , where R is an endomorphism in the space spanned by { | a ⟩ ⊗ | b ⟩ } , 1 ≤ a , b ≤ n . 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.

Notable vertex models