In mathematical physics, a lattice model is a mathematical model of a physical system that is defined on a lattice, as opposed to a continuum, such as the continuum of space or spacetime. Lattice models originally occurred in the context of condensed matter physics, where the atoms of a crystal automatically form a lattice. Currently, lattice models are quite popular in theoretical physics, for many reasons. Some models are exactly solvable, and thus offer insight into physics beyond what can be learned from perturbation theory. Lattice models are also ideal for study by the methods of computational physics, as the discretization of any continuum model automatically turns it into a lattice model. The exact solution to many of these models (when they are solvable) includes the presence of solitons. Techniques for solving these include the inverse scattering transform and the method of Lax pairs, the Yang–Baxter equation and quantum groups. The solution of these models has given insights into the nature of phase transitions, magnetization and scaling behaviour, as well as insights into the nature of quantum field theory. Physical lattice models frequently occur as an approximation to a continuum theory, either to give an ultraviolet cutoff to the theory to prevent divergences or to perform numerical computations. An example of a continuum theory that is widely studied by lattice models is the QCD lattice model, a discretization of quantum chromodynamics. However, digital physics considers nature fundamentally discrete at the Planck scale, which imposes upper limit to the density of information, aka Holographic principle. More generally, lattice gauge theory and lattice field theory are areas of study. Lattice models are also used to simulate the structure and dynamics of polymers.

Mathematical description

A number of lattice models can be described by the following data:

• A lattice is often the cubic lattice. If two points on the lattice are considered 'nearest neighbours', then they can be connected by an edge, turning the lattice into a lattice graph are sometimes referred to as sites.

• is the edge set of the graph defined above.

• .

Examples

The Ising model is the edge set of nearest neighbours (the same letter is used for the energy functional. The energy functional is

The spin-variable space can often be described as a coset.

Solvable models

is also finite. We can define the partition function

. In practice, this is often difficult due to non-linear interactions between sites. Models with a closed-form expression for the partition function are known as exactly solvable.

, the Ising model remains unsolved.

Mean field theory

Due to the difficulty of deriving exact solutions, in order to obtain analytic results we often must resort to mean field theory. This mean field may be spatially varying, or global.

Global mean field

is replaced by the convex hull.

The partition function then becomes

, that is, in the thermodynamic limit, the saddle point approximation is minimised:

.

then summing over configurations allows computation of the partition function.

dimensions provides insight into phase transitions.

Spatially varying mean field

Suppose the continuum limit to bring the notation closer to field theory. This allows the partition function to be written as a path integral

is a Wick rotated version of the action in quantum field theory.

Examples

Condensed matter physics

• Ising model

• ANNNI model

• Potts model

• Chiral Potts model

• XY model

• Classical Heisenberg model

• n-vector model

• Vertex model

• Toda lattice

• cellular automata

Polymer physics

• Bond fluctuation model

• 2nd model

High energy physics

• QCD lattice model

See also

• Crystal structure

• Continuum limit

• QCD matter

• Lattice gas

References