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 Λ {\displaystyle \Lambda } , often taken to be a lattice in d {\displaystyle d} -dimensional Euclidean space R d {\displaystyle \mathbb {R} ^{d}} or the d {\displaystyle d} -dimensional torus if the lattice is periodic. Concretely, Λ {\displaystyle \Lambda } 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. The vertices of Λ {\displaystyle \Lambda } are sometimes referred to as sites. A spin-variable space S {\displaystyle S} . The configuration space C {\displaystyle {\mathcal {C}}} of possible system states is then the space of functions σ : Λ → S {\displaystyle \sigma :\Lambda \rightarrow S} . For some models, we might instead consider instead the space of functions σ : E → S {\displaystyle \sigma :E\rightarrow S} where E {\displaystyle E} is the edge set of the graph defined above. An energy functional E : C → R {\displaystyle E:{\mathcal {C}}\rightarrow \mathbb {R} } , which might depend on a set of additional parameters or 'coupling constants' { g i } {\displaystyle \{g_{i}\}} .
Examples The Ising model is given by the usual cubic lattice graph G = ( Λ , E ) {\displaystyle G=(\Lambda ,E)} where Λ {\displaystyle \Lambda } is an infinite cubic lattice in R d {\displaystyle \mathbb {R} ^{d}} or a period n {\displaystyle n} cubic lattice in T d {\displaystyle T^{d}} , and E {\displaystyle E} is the edge set of nearest neighbours (the same letter is used for the energy functional but the different usages are distinguishable based on context). The spin-variable space is S = { + 1 , − 1 } = Z 2 {\displaystyle S=\{+1,-1\}=\mathbb {Z} _{2}} . The energy functional is
E ( σ ) = − H ∑ v ∈ Λ σ ( v ) − J ∑ { v 1 , v 2 } ∈ E σ ( v 1 ) σ ( v 2 ) . {\displaystyle E(\sigma )=-H\sum _{v\in \Lambda }\sigma (v)-J\sum _{\{v_{1},v_{2}\}\in E}\sigma (v_{1})\sigma (v_{2}).}
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