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.
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 Boltzmann's 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
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