In statistical mechanics, the eight-vertex model is a generalization of the ice-type (six-vertex) models. It was discussed by Sutherland[1] and Fan & Wu,[2] and solved by Rodney Baxter in the zero-field case.[3]

Description

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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 the ice-type model (1-6), sinks (7), and sources (8).

 
The eight allowed vertices.

We consider a   lattice, with   vertices and   edges. Imposing periodic boundary conditions requires that the states 7 and 8 occur equally often, as do states 5 and 6, and thus can be taken to have the same energy. For the zero-field case the same is true for the two other pairs of states. Each vertex   has an associated energy   and Boltzmann weight

 

giving the partition function over the lattice as

 

where the outer summation is over all allowed configurations of vertices in the lattice. In this general form the partition function remains unsolved.

Solution in the zero-field case

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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 may be assigned arbitrary weights

 

The solution is based on the observation that rows in transfer matrices commute, for a certain parametrization of these four Boltzmann weights. This came about as a modification of an alternate solution for the six-vertex model which makes use of elliptic theta functions.

Commuting transfer matrices

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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 parametrization of the weights given as

 

for fixed modulus   and   and variable  . Here snh is the hyperbolic analogue of sn, given by

 

and   and   are theta functions of modulus  . The associated transfer matrix   thus is a function of   alone; for all  ,  

 

The matrix function  

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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

  (1)

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

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The commutation of matrices in (1) allow them to be diagonalised, and thus eigenvalues can be found. The partition function is calculated from the maximal eigenvalue, resulting in a free energy per site of

 

for

 

where   and   are the complete elliptic integrals of moduli   and  . The eight vertex model was also solved in quasicrystals.

Equivalence with an Ising model

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There is a natural correspondence between the eight-vertex model, and the Ising model with 2-spin and 4-spin nearest neighbor 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

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Notes

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  1. ^ Sutherland, Bill (1970). "Two‐Dimensional Hydrogen Bonded Crystals without the Ice Rule". Journal of Mathematical Physics. 11 (11). AIP Publishing: 3183–3186. Bibcode:1970JMP....11.3183S. doi:10.1063/1.1665111. ISSN 0022-2488.
  2. ^ Fan, Chungpeng; Wu, F. Y. (1970-08-01). "General Lattice Model of Phase Transitions". Physical Review B. 2 (3). American Physical Society (APS): 723–733. Bibcode:1970PhRvB...2..723F. doi:10.1103/physrevb.2.723. ISSN 0556-2805.
  3. ^ Baxter, R. J. (1971-04-05). "Eight-Vertex Model in Lattice Statistics". Physical Review Letters. 26 (14). American Physical Society (APS): 832–833. Bibcode:1971PhRvL..26..832B. doi:10.1103/physrevlett.26.832. ISSN 0031-9007.

References

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