
In
mathematical physics
Mathematical physics refers to 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 t ...
, a lattice model is a
mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...
of a physical system that is defined on a
lattice, as opposed to a
continuum, such as the continuum of
space
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually con ...
or
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why diffe ...
. 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 phases which arise from electromagnetic forces between atoms. More generally, the s ...
, where the
atom
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas ...
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, macr ...
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 natural phenomena. This is in contrast to experimental physics, which uses experi ...
, 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 , often taken to be a lattice in -dimensional Euclidean space or the -dimensional torus if the lattice is periodic. Concretely, 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 are sometimes referred to as sites.
- A spin-variable space . The configuration space of possible system states is then the space of functions . For some models, we might instead consider instead the space of functions where is the edge set of the graph defined above.
- An energy functional , which might depend on a set of additional parameters or 'coupling constants' .
Examples
The
Ising model is given by the usual cubic lattice graph
where
is an infinite cubic lattice in
or a period
cubic lattice in
, and
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
. The energy functional is
:
The spin-variable space can often be described as a
coset. For example, for the Potts model we have
. In the limit
, we obtain the XY model which has
. Generalising the XY model to higher dimensions gives the
-vector model which has
.
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
in
dimensions. Then the configuration space
is also finite. We can define the
partition function
:
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
. 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,
but for dimension
, 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
of functions
is replaced by the
convex hull of the spin space
, when
has a realisation in terms of a subset of
. We'll denote this by
. This arises as in going to the mean value of the field, we have
.
As the number of lattice sites
, the possible values of
fill out the convex hull of
. By making a suitable approximation, the energy functional becomes a function of the mean field, that is,
The partition function then becomes
:
As
, that is, in the
thermodynamic limit, the
saddle point approximation tells us the integral is asymptotically dominated by the value at which
is minimised:
:
where
is the argument minimising
.
A simpler, but less mathematically rigorous approach which nevertheless sometimes gives correct results comes from linearising the theory about the mean field
. Writing configurations as
, truncating terms of
then summing over configurations allows computation of the partition function.
Such an approach to the periodic Ising model in
dimensions provides insight into
phase transitions.
Spatially varying mean field
Suppose the
continuum limit of the lattice
is
. Instead of averaging over all of
, we average over neighbourhoods of
. This gives a spatially varying mean field
. We relabel
with
to bring the notation closer to field theory. This allows the partition function to be written as a
path integral
:
where the free energy
Wick rotated
In physics, Wick rotation, named after Italian physicist Gian Carlo Wick, is a method of finding a solution to a mathematical problem in Minkowski space from a solution to a related problem in Euclidean space by means of a transformation that sub ...
version of the action in
quantum field theory.
Examples
Condensed matter physics
- Ising model
- Potts model
Chiral Potts model
The chiral Potts model is a spin model on a planar lattice in statistical mechanics studied by Helen Au-Yang Perk and Jacques Perk, among others. It may be viewed as a generalization of the Potts model, and as with the Potts model, the model is de ...
- XY model
- Classical Heisenberg model
- n-vector model
- Vertex model
Toda lattice The Toda lattice, introduced by , is a simple model for a one-dimensional crystal in solid state physics
Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and ...
Polymer physics
Bond fluctuation model The BFM (bond fluctuation model or bond fluctuation method) is a lattice model for simulating the conformation and dynamics of polymer systems. There are two versions of the BFM used: The earlier version was first introduced by I. Carmesin and Kur ...
- 2nd model
High energy physics
See also
*
Crystal structure
In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric patterns t ...
*
Scaling limit
In mathematical physics and mathematics, the continuum limit or scaling limit of a lattice model refers to its behaviour in the limit as the lattice spacing goes to zero. It is often useful to use lattice models to approximate real-world processe ...
*
QCD matter
Quark matter or QCD matter (quantum chromodynamic) refers to any of a number of hypothetical phases of matter whose degrees of freedom include quarks and gluons, of which the prominent example is quark-gluon plasma. Several series of conferences ...
*
Lattice gas
Lattice gas automata (LGCA), or lattice gas cellular automata, are a type of cellular automaton used to simulate fluid flows, pioneered by HPP model, Hardy–Pomeau–de Pazzis and Uriel Frisch, Frisch–Brosl Hasslacher, Hasslacher–Yves Pomeau, ...
References
*
{{DEFAULTSORT:Lattice Model (Physics)
Theoretical physics