Description
As with the ice-type models, the eight-vertex model is a square lattice model, where each state is a configuration of arrows at a vertex. The allowed vertices have an even number of arrows pointing towards the vertex; these include the six inherited from theSolution in the zero-field case
The zero-field case of the model corresponds physically to the absence of external electric fields. Hence, the model remains unchanged under the reversal of all arrows; the states 1 and 2, and 3 and 4, consequently must occur as pairs. The vertices can be assigned arbitrary weights : The solution is based on the observation that rows in transfer matrices commute, for a certain parametrisation of these four Boltzmann weights. This came about as a modification of an alternate solution for the six-vertex model; it makes use of elliptic theta functions.Commuting transfer matrices
The proof relies on the fact that when and , for quantities : the transfer matrices and (associated with the weights , , , and , , , ) commute. Using the star-triangle relation, Baxter reformulated this condition as equivalent to a parametrisation of the weights given as : for fixed modulus and and variable . Here snh is the hyperbolic analogue of sn, given by : and and areThe matrix function
The other crucial part of the solution is the existence of a nonsingular matrix-valued function , such that for all complex the matrices commute with each other and the transfer matrices, and satisfy where : The existence and commutation relations of such a function are demonstrated by considering pair propagations through a vertex, and periodicity relations of the theta functions, in a similar way to the six-vertex model.Explicit solution
The commutation of matrices in () allow them to be diagonalised, and thusEquivalence with an Ising model
There is a natural correspondence between the eight-vertex model, and the Ising model with 2-spin and 4-spin nearest neighbour interactions. The states of this model are spins on faces of a square lattice. The analogue of 'edges' in the eight-vertex model are products of spins on adjacent faces: : The most general form of the energy for this model is : where , , , describe the horizontal, vertical and two diagonal 2-spin interactions, and describes the 4-spin interaction between four faces at a vertex; the sum is over the whole lattice. We denote horizontal and vertical spins (arrows on edges) in the eight-vertex model , respectively, and define up and right as positive directions. The restriction on vertex states is that the product of four edges at a vertex is 1; this automatically holds for Ising 'edges'. Each configuration then corresponds to a unique , configuration, whereas each , configuration gives two choices of configurations. Equating general forms of Boltzmann weights for each vertex , the following relations between the and , , , , define the correspondence between the lattice models: : It follows that in the zero-field case of the eight-vertex model, the horizontal and vertical interactions in the corresponding Ising model vanish. These relations gives the equivalence between the partition functions of the eight-vertex model, and the 2,4-spin Ising model. Consequently a solution in either model would lead immediately to a solution in the other.See also
* Six-vertex model * Transfer-matrix method * Ising modelNotes
References
*{{Citation , last1=Baxter , first1=Rodney J. , title=Exactly solved models in statistical mechanics , url=http://physics.anu.edu.au/theophys/_files/Exactly.pdf , publisher=Academic Press Inc. arcourt Brace Jovanovich Publishers, location=London , isbn=978-0-12-083180-7 , mr=690578 , year=1982 , access-date=2012-08-12 , archive-date=2021-04-14 , archive-url=https://web.archive.org/web/20210414063635/https://physics.anu.edu.au/theophys/_files/Exactly.pdf , url-status=dead Exactly solvable models Statistical mechanics Lattice models