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Vertex modelA 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 nearestneighbour 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 YangBaxter equation in a tensor product of vector spaces yields an exactlysolvable 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 squarelattice 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 2^{4} possible configurations. The energy for a given vertex can be given by , with a state of the lattice is an assignment of a state of each bond, with 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 boundary conditions are imposed on the model. Additional recommended knowledge
DiscussionFor a given state, the Boltzmann weight can be written in terms of the product of the Boltzmann weights of the corresponding vertices where the Boltzmann weights for the vertices are written
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 solution is desired. where β = 1 / kT, T is temperature and k is Boltzmann's constant. The probability that the system is in any given 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 ndimensional vector space V, with a basis , and as and as thereby implying that T can be written as where the indices indicate the factors of the tensor product 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 is the rowtransfer matrix. 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 tau as where λ_{max} 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 , 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 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. This leads to the relationship between statistical mechanics YangBaxter equation, and thus the study of quantum groups. IntegrabilityDefinition: A vertex model is integrable if, such that
This is a parameterized version of the YangBaxter 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 , 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 , the transfer matrix τ(μ) commutes with . This illustrates the role of the YangBaxter 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 YangBaxter equation in the tensor product of two ndimensional vector spaces, there is a corresponding 2dimensional 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 finitedimensional irreducible representations of a given quantum algebra in order to find solvable models coreesponding to it. Notable vertex models
References
Categories: Statistical mechanics  Lattice models 

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Vertex_model". A list of authors is available in Wikipedia. 