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).[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 yields an exactly-solvable vertex model.
A 2-dimensional 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 possible configurations. The energy for a given vertex can be given by ,
A vertex in the square lattice vertex model
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 or domain wall[3] boundary conditions may be imposed on the model.
A vertexmodel is a type of statistical mechanics model in which the Boltzmann weights are associated with a vertex in the model (representing an atom...
Look up vertex in Wiktionary, the free dictionary. Vertex, vertices or vertexes may refer to: Vertex (geometry), a point where two or more curves, lines...
six-vertexmodel, after identifying γ = 2 η {\displaystyle \gamma =2\eta } , where η {\displaystyle \eta } is the anisotropy parameter of the six-vertex model...
In graph theory, a vertex cover (sometimes node cover) of a graph is a set of vertices that includes at least one endpoint of every edge of the graph....
statistical lattice models 8-vertexmodel Gaudin model Ising model in 1- and 2-dimensions Ice-type model of Lieb Quantum Heisenberg model Hitchin system Mathematical...
evolved toward a unified shader model. Shaders are simple programs that describe the traits of either a vertex or a pixel. Vertex shaders describe the attributes...
abstractions (such as Shader Model 1.x) used very different instruction sets for vertex and pixel shaders, with vertex shaders having much more flexible...
Standard Model of particle physics, there are six types of quarks and six types of leptons. In statistical mechanics, the six-vertexmodel has six possible...
particles. In statistical mechanics, the eight-vertexmodel has 8 possible configurations of arrows at each vertex. Messier object M8, a magnitude 5.0 nebula...
generalization of the Potts model, and as with the Potts model, the model is defined by configurations which are assignments of spins to each vertex of a graph, where...
solved models, in particular vertexmodels such as the six-vertexmodel and eight-vertexmodel, and the chiral Potts model and hard hexagon model. A recurring...
of times each vertex topples during each iteration is also independent of the choice of toppling order. To generalize the sandpile model from the rectangular...
In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string...
Face-vertex meshes improve on VV-mesh for modeling in that they allow explicit lookup of the vertices of a face, and the faces surrounding a vertex. The...
determinant.[citation needed] They are also closely related to the six-vertexmodel with domain wall boundary conditions from statistical mechanics. They...
onto a rotated 3D model, such as zbrush or mudbox, also sometimes able to modify vertex attributes. 3D scene A collection of 3D models and lightsources...
explicitly for artistic purposes. Vertex normals are used in Gouraud shading, Phong shading and other lighting models. Using vertex normals, much smoother shading...
constraint that the expected degree of each vertex matches the degree of the vertex in the original graph. The null model is the basic concept behind the definition...
is just a vertex coloring of its line graph, and a face coloring of a plane graph is just a vertex coloring of its dual. However, non-vertex coloring problems...
XXZ spin chain XYZ spin chain 6-vertexmodel 8-vertexmodel Kondo Model Anderson impurity model Chiral Gross–Neveu model Ishimori equation Kadomtsev–Petviashvili...
orientation, from one vertex to another vertex. A path in a directed graph is a sequence of edges having the property that the ending vertex of each edge in...
six hexagons meet at each vertex, and four edges meet at each vertex. Viewed in perspective outside of a Poincaré disk model, the image above shows one...
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