In statistical mechanics, the ice-type models or six-vertex models are a family of vertex models for crystal lattices with hydrogen bonds. The first such model was introduced by Linus Pauling in 1935 to account for the residual entropy of water ice. Variants have been proposed as models of certain ferroelectric and antiferroelectric crystals. In 1967, Elliott H. Lieb found the exact solution to a two-dimensional ice model known as "square ice". The exact solution in three dimensions is only known for a special "frozen" state.
Description An ice-type model is a lattice model defined on a lattice of coordination number 4. That is, each vertex of the lattice is connected by an edge to four "nearest neighbours". A state of the model consists of an arrow on each edge of the lattice, such that the number of arrows pointing inwards at each vertex is 2. This restriction on the arrow configurations is known as the ice rule. In graph theoretic terms, the states are Eulerian orientations of an underlying 4-regular undirected graph. The partition function also counts the number of nowhere-zero 3-flows. For two-dimensional models, the lattice is taken to be the square lattice. For more realistic models, one can use a three-dimensional lattice appropriate to the material being considered; for example, the hexagonal ice lattice is used to analyse ice. At any vertex, there are six configurations of the arrows which satisfy the ice rule (justifying the name "six-vertex model"). The valid configurations for the (two-dimensional) square lattice are the following:
The energy of a state is understood to be a function of the configurations at each vertex. For square lattices, one assumes that the total energy E {\displaystyle E} is given by
E = n 1 ϵ 1 + n 2 ϵ 2 + … + n 6 ϵ 6 , {\displaystyle E=n_{1}\epsilon _{1}+n_{2}\epsilon _{2}+\ldots +n_{6}\epsilon _{6},}
for some constants ϵ 1 , … , ϵ 6 {\displaystyle \epsilon _{1},\ldots ,\epsilon _{6}} , where n i {\displaystyle n_{i}} here denotes the number of vertices with the i {\displaystyle i} th configuration from the above figure. The value ϵ i {\displaystyle \epsilon _{i}} is the energy associated with vertex configuration number i {\displaystyle i} . One aims to calculate the partition function Z {\displaystyle Z} of an ice-type model, which is given by the formula
Z = ∑ exp ( − E / k B T ) , {\displaystyle Z=\sum \exp(-E/k_{\rm {B}}T),}
where the sum is taken over all states of the model, E {\displaystyle E} is the energy of the state, k B {\displaystyle k_{\rm {B}}} is the Boltzmann constant, and T {\displaystyle T} is the system's temperature. Typically, one is interested in the thermodynamic limit in which the number N {\displaystyle N} of vertices approaches infinity. In that case, one instead evaluates the free energy per vertex f {\displaystyle f} in the limit as N → ∞ {\displaystyle N\to \infty } , where f {\displaystyle f} is given by
f = − k B T N − 1 log Z . {\displaystyle f=-k_{\rm {B}}TN^{-1}\log Z.}
Equivalently, one evaluates the partition function per vertex W {\displaystyle W} in the thermodynamic limit, where
W = Z 1 / N . {\displaystyle W=Z^{1/N}.}
The values f {\displaystyle f} and W {\displaystyle W} are related by
f = − k B T log W . {\displaystyle f=-k_{\rm {B}}T\log W.}
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