A 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 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 V ⊗ V {\displaystyle V\otimes V} 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 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 {\displaystyle 2^{4}} possible configurations. The energy for a given vertex can be given by ε i j k ℓ {\displaystyle \varepsilon _{ij}^{k\ell }} , 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
exp ( − β ε ( state ) ) = ∏ vertices exp ( − β ε i j k ℓ ) {\displaystyle \exp(-\beta \varepsilon ({\mbox{state}}))=\prod _{\mbox{vertices}}\exp(-\beta \varepsilon _{ij}^{k\ell })}
where the Boltzmann weights for the vertices are written
R i j k ℓ = exp ( − β ε i j k ℓ ) {\displaystyle R_{ij}^{k\ell }=\exp(-\beta \varepsilon _{ij}^{k\ell })} , 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.
Z = ∑ states exp ( − β ε ( state ) ) {\displaystyle \mathbb {Z} =\sum _{\mbox{states}}\exp(-\beta \varepsilon ({\mbox{state}}))}
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
exp ( − β ε ( state ) ) Z {\displaystyle {\frac {\exp(-\beta \varepsilon ({\mbox{state}}))}{\mathbb {Z} }}}
so that the average value of the energy of the system is given by
⟨ ε ⟩ = ∑ states ε exp ( − β ε ) ∑ states exp ( − β ε ) = k T 2 ∂ ∂ T ln Z {\displaystyle \langle \varepsilon \rangle ={\frac {\sum _{\mbox{states}}\varepsilon \exp(-\beta \varepsilon )}{\sum _{\mbox{states}}\exp(-\beta \varepsilon )}}=kT^{2}{\frac {\partial }{\partial T}}\ln \mathbb {Z} }
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
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