![LatticemodelAB](https://upload.wikimedia.org/wikipedia/commons/8/86/LatticemodelAB.png)
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
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
, as opposed to a
continuum
Continuum may refer to:
* Continuum (measurement), theories or models that explain gradual transitions from one condition to another without abrupt changes
Mathematics
* Continuum (set theory), the real line or the corresponding cardinal number ...
, 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 consider ...
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 differen ...
. 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 sub ...
, 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, and ...
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 natural phenomena. This is in contrast to experimental physics, which uses experim ...
, for many reasons. Some models are
exactly solvable
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantity, conserved qua ...
, 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 for which a quantitative theory already exists. Historically, computational physics was the first application of modern computers in 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 or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium ...
s. Techniques for solving these include the
inverse scattering transform In mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. The method is a non-linear analogue, and in some sense generalization, of the Fourier transform, which itself is applied to solv ...
and the method of
Lax pair In mathematics, in the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or operators that satisfy a corresponding differential equation, called the ''Lax equation''. Lax pairs were introduced by Peter Lax to discuss sol ...
s, the
Yang–Baxter equation
In physics, the Yang–Baxter equation (or star–triangle relation) is a consistency equation which was first introduced in the field of statistical mechanics. It depends on the idea that in some scattering situations, particles may preserve thei ...
and
quantum group
In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebras) ...
s. The solution of these models has given insights into the nature of
phase transitions
In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of ...
,
magnetization
In classical electromagnetism, magnetization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. Movement within this field is described by direction and is either Axial or Di ...
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 classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
. Physical lattice models frequently occur as an approximation to a continuum theory, either to give an
ultraviolet cutoff
In theoretical physics, cutoff (AE: cutoff, BE: cut-off) is an arbitrary maximal or minimal value of energy, momentum, or length, used in order that objects with larger or smaller values than these physical quantities are ignored in some calculat ...
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
Lattice QCD is a well-established non-perturbative approach to solving the quantum chromodynamics (QCD) theory of quarks and gluons. It is a lattice gauge theory formulated on a grid or lattice of points in space and time. When the size of the lat ...
, a discretization of
quantum chromodynamics
In theoretical physics, quantum chromodynamics (QCD) is the theory 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 ...
. However,
digital physics
Digital physics is a speculative idea that the universe can be conceived of as a vast, digital computation device, or as the output of a deterministic or probabilistic computer program. The hypothesis that the universe is a digital computer was ...
considers nature fundamentally discrete at the Planck scale, which imposes
upper limit to the density of information, aka
Holographic principle
The holographic principle is an axiom in string theories and a supposed property of quantum gravity that states that the description of a volume of space can be thought of as encoded on a lower-dimensional boundary to the region — such as a ...
. More generally,
lattice gauge theory
In physics, lattice gauge theory is the study of gauge theories on a spacetime that has been discretized into a lattice.
Gauge theories are important in particle physics, and include the prevailing theories of elementary particles: quantum elec ...
and
lattice field theory
In physics, lattice field theory is the study of lattice models of quantum field theory, that is, of field theory on a space or spacetime that has been discretised onto a lattice.
Details
Although most lattice field theories are not exactly sol ...
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
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
, 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
In graph theory, a lattice graph, mesh graph, or grid graph is a graph whose drawing, embedded in some Euclidean space , forms a regular tiling. This implies that the group of bijective transformations that send the graph to itself is a latti ...
. 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
The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent ...
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
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
. 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
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantity, conserved qua ...
.
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
In physics and probability theory, Mean-field theory (MFT) or Self-consistent field theory studies the behavior of high-dimensional random (stochastic) models by studying a simpler model that approximates the original by averaging over degrees of ...
. This mean field may be spatially varying, or global.
Global mean field
The configuration space
of functions
is replaced by the
convex hull
In geometry, the convex hull or 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
, 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
In statistical mechanics, the thermodynamic limit or macroscopic limit, of a system is the 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 particles.S.J. Blundel ...
, 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
In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of ...
.
Spatially varying mean field
Suppose the
continuum limit
In mathematical physics and mathematics, the continuum limit or scaling limit of a lattice model (physics), 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 approxi ...
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