Eight-vertex model

Last updated March 24, 2021

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

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

Eightvertex2 Eightvertex2.png — Eightvertex2

We consider a $N\times N$ lattice, with $N^{2}$ vertices and $2N^{2}$ 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 $j$ has an associated energy $\epsilon _{j}$ and Boltzmann weight $w_{j}=e^{-{\frac {\epsilon _{j}}{kT}}}$ , giving the partition function over the lattice as

Z=\sum \exp \left(-{\frac {\sum _{j}n_{j}\epsilon _{j}}{kT}}\right)

where the 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

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

{\begin{aligned}w_{1}=w_{2}&=a\\w_{3}=w_{4}&=b\\w_{5}=w_{6}&=c\\w_{7}=w_{8}&=d.\end{aligned}}

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 $\Delta '=\Delta$ and $\Gamma '=\Gamma$ , for quantities

{\begin{aligned}\Delta &={\frac {a^{2}+b^{2}-c^{2}-d^{2}}{2(ab+cd)}}\\\Gamma &={\frac {ab-cd}{ab+cd}}\end{aligned}}

the transfer matrices $T$ and $T'$ (associated with the weights $a$ , $b$ , $c$ , $d$ and $a'$ , $b'$ , $c'$ , $d'$ ) commute. Using the star-triangle relation, Baxter reformulated this condition as equivalent to a parametrisation of the weights given as

a:b:c:d=\operatorname {snh} (\eta -u):\operatorname {snh} (\eta +u):\operatorname {snh} (2\eta ):k\operatorname {snh} (2\eta )\operatorname {snh} (\eta -u)\operatorname {snh} (\eta +u)

for fixed modulus $k$ and $\eta$ and variable $u$ . Here snh is the hyperbolic analogue of sn, given by

{\begin{aligned}\operatorname {snh} (u)&=-i\operatorname {snh} (iu)\\{\text{where }}\operatorname {snh} (u)&={\frac {H(u)}{k^{1/2}\Theta (u)}}\end{aligned}}

and $H(u)$ and $\Theta (u)$ are Jacobi elliptic functions of modulus $k$ . The associated transfer matrix $T$ thus is a function of $u$ alone; for all $u$ , $v$

T(u)T(v)=T(v)T(u).

The matrix function $Q(u)$

The other crucial part of the solution is the existence of a nonsingular matrix-valued function $Q$ , such that for all complex $u$ the matrices $Q(u),Q(u')$ commute with each other and the transfer matrices, and satisfy

\zeta (u)T(u)Q(u)=\phi (u-\eta )Q(u+2\eta )+\phi (u+\eta )Q(u-2\eta )

(1)

where

{\begin{aligned}\zeta (u)&=[c^{-1}H(2\eta )\Theta (u-\eta )\Theta (u+\eta )]^{N}\\\phi (u)&=[\Theta (0)H(u)\Theta (u)]^{N}.\end{aligned}}

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 ( 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

{\begin{aligned}f=\epsilon _{5}-2kT\sum _{n=1}^{\infty }{\frac {\sinh ^{2}((\tau -\lambda )n)(\cosh(n\lambda )-\cosh(n\alpha ))}{n\sinh(2n\tau )\cosh(n\lambda )}}\end{aligned}}

for

{\begin{aligned}\tau &={\frac {\pi K'}{2K}}\\\lambda &={\frac {\pi \eta }{iK}}\\\alpha &={\frac {\pi u}{iK}}\end{aligned}}

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

Equivalence 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 $\sigma =\pm 1$ on faces of a square lattice. The analogue of 'edges' in the eight-vertex model are products of spins on adjacent faces:

{\begin{aligned}\alpha _{ij}&=\sigma _{ij}\sigma _{i,j+1}\\\mu _{ij}&=\sigma _{ij}\sigma _{i+1,j}.\end{aligned}}

The most general form of the energy for this model is

{\begin{aligned}\epsilon &=-\sum _{ij}(J_{h}\mu _{ij}+J_{v}\alpha _{ij}+J\alpha _{ij}\mu _{ij}+J'\alpha _{i+1,j}\mu _{ij}+J''\alpha _{ij}\alpha _{i+1,j})\end{aligned}}

where $J_{h}$ , $J_{v}$ , $J$ , $J'$ describe the horizontal, vertical and two diagonal 2-spin interactions, and $J''$ 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 $\mu$ , $\alpha$ 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 $\sigma$ configuration then corresponds to a unique $\mu$ , $\alpha$ configuration, whereas each $\mu$ , $\alpha$ configuration gives two choices of $\sigma$ configurations.

Equating general forms of Boltzmann weights for each vertex $j$ , the following relations between the $\epsilon _{j}$ and $J_{h}$ , $J_{v}$ , $J$ , $J'$ , $J''$ define the correspondence between the lattice models:

{\begin{aligned}\epsilon _{1}&=-J_{h}-J_{v}-J-J'-J'',\quad \epsilon _{2}=J_{h}+J_{v}-J-J'-J''\\\epsilon _{3}&=-J_{h}+J_{v}+J+J'-J'',\quad \epsilon _{4}=J_{h}-J_{v}+J+J'-J''\\\epsilon _{5}&=\epsilon _{6}=J-J'+J''\\\epsilon _{7}&=\epsilon _{8}=-J+J'+J''.\end{aligned}}

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 $Z_{I}=2Z_{8V}$ 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.

Notes

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

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References

Baxter, Rodney J. (1982), Exactly solved models in statistical mechanics (PDF), London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], ISBN 978-0-12-083180-7, MR 0690578

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[1] Sutherland, Bill (1970). "Two‐Dimensional Hydrogen Bonded Crystals without the Ice Rule". Journal of Mathematical Physics. AIP Publishing. 11 (11): 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. American Physical Society (APS). 2 (3): 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. American Physical Society (APS). 26 (14): 832–833. Bibcode:1971PhRvL..26..832B. doi:10.1103/physrevlett.26.832. ISSN 0031-9007.

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Eight-vertex model

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