In physics, a lattice model is a physical model that is defined on a lattice, as opposed to 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. Physics (from the Greek, (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the discovery and understanding of the fundamental laws which govern matter, energy, space and time. ... See lattice for other meanings of this term, both within and without mathematics. ... Look up continuum in Wiktionary, the free dictionary. ... Space has been an interest for philosophers and scientists for much of human history. ... In physics, spacetime is a mathematical model that combines three-dimensional space and one-dimensional time into a single construct called the space-time continuum, in which time plays the role of the 4th dimension. ... Condensed matter physics is the field of physics that deals with the macroscopic physical properties of matter. ... Properties In chemistry and physics, an atom (Greek ἄτομος or átomos meaning indivisible) is the smallest particle of a chemical element that retains its chemical properties. ... Quartz crystal In chemistry and mineralogy, a crystal is a solid in which the constituent atoms, molecules, or ions are packed in a regularly ordered, repeating pattern extending in all three spatial dimensions. ... Theoretical physics employs mathematical models and abstractions of physics, as opposed to experimental processes, in an attempt to understand Nature. ... Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem. ... Computational physics is the study and implementation of numerical algorithms in order to solve problems in physics for which a quantitative theory already exists. ...

Examples of lattice models in condensed matter physics include the Ising model, the Potts model, the XY model, the Toda lattice. 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 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. Condensed matter physics is the field of physics that deals with the macroscopic physical properties of matter. ... The Ising model, named after the physicist Ernst Ising, is a mathematical model in statistical mechanics. ... In statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice. ... Like the Ising model, the XY model is one of the many highly simplified models in the branch of physics known as statistical mechanics. ... In mathematics and physics, a soliton is a self-reinforcing solitary wave caused by nonlinear effects in the medium. ... In physics, a phase transition is the transformation of a thermodynamic system from one phase to another. ... This article needs to be cleaned up to conform to a higher standard of quality. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ...

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. More generally, lattice gauge theory and lattice field theory are areas of study. In theoretical physics, cutoff usually represents a particular energy scale or length scale. ... Numerical analysis is the study of approximate methods for the problems of continuous mathematics (as distinguished from discrete mathematics). ... Lattice gauge theory is a method to deal with gauge theory that is useful for computer-assisted calculations. ... Quantum chromodynamics (QCD) is the theory of the strong interaction, a fundamental force describing the interactions of the quarks and gluons found in nucleons (such as the proton and neutron). ... It has been suggested that this article or section be merged into Lattice field theory. ... It has been suggested that Lattice gauge theory be merged into this article or section. ...

See also

• Scaling limit

If we wish to have a lattice model which approximates a continuum quantum field theory in the limit as the lattice spacing goes to zero, then this corresponds to finding a second order phase transition of the model. ...

References

• Peter Mayer, "Lattice Geometries" (2000). (A review of lattice geometries in computational physics).
• Peter Mayer, "Computational Studies of Pure and Dilute Spin Models" (2000) (PhD Thesis)