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In
mathematical physics Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the de ...
, a lattice model is a
mathematical model A mathematical model is an abstract and concrete, abstract description of a concrete system using mathematics, mathematical concepts and language of mathematics, language. The process of developing a mathematical model is termed ''mathematical m ...
of a physical system that is defined on a lattice, as opposed to a continuum, such as the continuum of
space Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless ...
or
spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
. Lattice models originally occurred in the context of
condensed matter physics Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid State of matter, phases, that arise from electromagnetic forces between atoms and elec ...
, where the
atom Atoms are the basic particles of the chemical elements. An atom consists of a atomic nucleus, nucleus of protons and generally neutrons, surrounded by an electromagnetically bound swarm of electrons. The chemical elements are distinguished fr ...
s of a
crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macros ...
automatically form a lattice. Currently, lattice models are quite popular in
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
, for many reasons. Some models are exactly solvable, and thus offer insight into physics beyond what can be learned from
perturbation theory In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
. Lattice models are also ideal for study by the methods of
computational physics Computational physics is the study and implementation of numerical analysis to solve problems in physics. Historically, computational physics was the first application of modern computers in science, and is now a subset of computational science ...
, 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
soliton In mathematics and physics, a soliton is a nonlinear, self-reinforcing, localized wave packet that is , in that it preserves its shape while propagating freely, at constant velocity, and recovers it even after collisions with other such local ...
s. 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 In classical electromagnetism, magnetization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. Accordingly, physicists and engineers usually define magnetization as the quanti ...
and
scaling behaviour Scaling may refer to: Science and technology Mathematics and physics * Scaling (geometry), a linear transformation that enlarges or diminishes objects * Scale invariance, a feature of objects or laws that do not change if scales of length, energ ...
, as well as insights into the nature of
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
. 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 In theoretical physics, quantum chromodynamics (QCD) is the study of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type of ...
. 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 \Lambda, often taken to be a lattice in d-dimensional Euclidean space \mathbb^d or the d-dimensional torus if the lattice is periodic. Concretely, \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 \Lambda are sometimes referred to as sites. * A spin-variable space S. The configuration space \mathcal of possible system states is then the space of functions \sigma: \Lambda \rightarrow S. For some models, we might instead consider instead the space of functions \sigma: E \rightarrow S where E is the edge set of the graph defined above. * An energy functional E:\mathcal\rightarrow\mathbb, which might depend on a set of additional parameters or 'coupling constants' \.


Examples

The
Ising model The Ising model (or Lenz–Ising model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical models in physics, mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that r ...
is given by the usual cubic lattice graph G = (\Lambda, E) where \Lambda is an infinite cubic lattice in \mathbb^d or a period n cubic lattice in T^d, and 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 = \ = \mathbb_2. The energy functional is :E(\sigma) = -H\sum_\sigma(v) - J\sum_\sigma(v_1)\sigma(v_2). The spin-variable space can often be described as a
coset In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
. For example, for the Potts model we have S = \mathbb_n. In the limit n\rightarrow \infty, we obtain the XY model which has S = SO(2). Generalising the XY model to higher dimensions gives the n-vector model which has S = S^n = SO(n+1)/SO(n).


Solvable models

We specialise to a lattice with a finite number of points, and a finite spin-variable space. This can be achieved by making the lattice periodic, with period n in d dimensions. Then the configuration space \mathcal is also finite. We can define the partition function :Z = \sum_\exp(-\beta E(\sigma)) and there are no issues of convergence (like those which emerge in field theory) since the sum is finite. In theory, this sum can be computed to obtain an expression which is dependent only on the parameters \ and \beta. 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. Examples of exactly solvable models are the periodic 1D Ising model, and the periodic 2D Ising model with vanishing external magnetic field, H = 0, but for dimension d>2, 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

The configuration space \mathcal of functions \sigma is replaced by the
convex hull In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ...
of the spin space S, when S has a realisation in terms of a subset of \mathbb^m. We'll denote this by \langle\mathcal\rangle. This arises as in going to the mean value of the field, we have \sigma \mapsto \langle \sigma \rangle := \frac\sum_\sigma(v). As the number of lattice sites N = , \Lambda, \rightarrow \infty, the possible values of \langle \sigma \rangle fill out the convex hull of S. By making a suitable approximation, the energy functional becomes a function of the mean field, that is, E(\sigma)\mapsto E(\langle \sigma \rangle). The partition function then becomes :Z = \int_d\langle\sigma\rangle e^\Omega(\langle\sigma\rangle) =: \int_d\langle\sigma\rangle e^. As N\rightarrow \infty, that is, in the
thermodynamic limit In statistical mechanics, the thermodynamic limit or macroscopic limit, of a system is the Limit (mathematics), limit for a large number of particles (e.g., atoms or molecules) where the volume is taken to grow in proportion with the number of ...
, the saddle point approximation tells us the integral is asymptotically dominated by the value at which f(\langle\sigma\rangle) is minimised: :Z \sim e^ where \langle\sigma\rangle_0 is the argument minimising f. A simpler, but less mathematically rigorous approach which nevertheless sometimes gives correct results comes from linearising the theory about the mean field \langle\sigma\rangle. Writing configurations as \sigma(v)=\langle\sigma\rangle + \Delta\sigma(v), truncating terms of \mathcal(\Delta\sigma^2) then summing over configurations allows computation of the partition function. Such an approach to the periodic Ising model in d dimensions provides insight into phase transitions.


Spatially varying mean field

Suppose the continuum limit of the lattice \Lambda is \mathbb^d. Instead of averaging over all of \Lambda, we average over neighbourhoods of \mathbf\in\mathbb^d. This gives a spatially varying mean field \langle\sigma\rangle:\mathbb^d\rightarrow \langle\mathcal\rangle. We relabel \langle\sigma\rangle with \phi to bring the notation closer to field theory. This allows the partition function to be written as a path integral :Z = \int \mathcal\phi e^ where the free energy F phi/math> is a Wick rotated version of the action in
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
.


Examples


Condensed matter physics

*
Ising model The Ising model (or Lenz–Ising model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical models in physics, mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that r ...
* 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 In crystallography, crystal structure is a description of ordered arrangement of atoms, ions, or molecules in a crystalline material. Ordered structures occur from intrinsic nature of constituent particles to form symmetric patterns that repeat ...
* Continuum limit * QCD matter * Lattice gas


References

* {{DEFAULTSORT:Lattice Model (Physics)