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 is given by
for some constants , where here denotes the number of vertices with the th configuration from the above figure. The value is the energy associated with vertex configuration number .
One aims to calculate the partition function of an ice-type model, which is given by the formula
where the sum is taken over all states of the model, is the energy of the state, is the Boltzmann constant, and is the system's temperature.
Typically, one is interested in the thermodynamic limit in which the number of vertices approaches infinity. In that case, one instead evaluates the free energy per vertex in the limit as , where is given by
Equivalently, one evaluates the partition function per vertex in the thermodynamic limit, where
The values and are related by
Physical justification
Several real crystals with hydrogen bonds satisfy the ice model, including ice and potassium dihydrogen phosphate (KDP). Indeed, such crystals motivated the study of ice-type models.
In ice, each oxygen atom is connected by a bond to four other oxygens, and each bond contains one hydrogen atom between the terminal oxygens. The hydrogen occupies one of two symmetrically located positions, neither of which is in the middle of the bond. Pauling argued that the allowed configuration of hydrogen atoms is such that there are always exactly two hydrogens close to each oxygen, thus making the local environment imitate that of a water molecule, . Thus, if we take the oxygen atoms as the lattice vertices and the hydrogen bonds as the lattice edges, and if we draw an arrow on a bond which points to the side of the bond on which the hydrogen atom sits, then ice satisfies the ice model. Similar reasoning applies to show that KDP also satisfies the ice model.
In recent years, ice-type models have been explored as descriptions of pyrochlore spin ice and artificial spin ice systems, in which geometrical frustration in the interactions between bistable magnetic moments ("spins") leads to "ice-rule" spin configurations being favoured. Recently such analogies have been extended to explore the circumstances under which spin-ice systems may be accurately described by the Rys F-model.
Specific choices of vertex energies
On the square lattice, the energies associated with vertex configurations 1-6 determine the relative probabilities of states, and thus can influence the macroscopic behaviour of the system. The following are common choices for these vertex energies.
The ice model
When modeling ice, one takes , as all permissible vertex configurations are understood to be equally likely. In this case, the partition function equals the total number of valid states. This model is known as the ice model (as opposed to an ice-type model).