A spin 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 ...
used in physics primarily to explain
magnetism
Magnetism is the class of physical attributes that occur through a magnetic field, which allows objects to attract or repel each other. Because both electric currents and magnetic moments of elementary particles give rise to a magnetic field, ...
. Spin models may either be
classical or
quantum
In physics, a quantum (: quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a property can be "quantized" is referred to as "the hypothesis of quantization". This me ...
mechanical in nature. Spin models have been studied in quantum field theory as examples of
integrable models. Spin models are also used in
quantum information theory
Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both t ...
and
computability theory
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since ex ...
in
theoretical computer science
Theoretical computer science is a subfield of computer science and mathematics that focuses on the Abstraction, abstract and mathematical foundations of computation.
It is difficult to circumscribe the theoretical areas precisely. The Associati ...
. The theory of spin models is a far reaching and unifying topic that cuts across many fields.
Introduction
In ordinary materials, the magnetic dipole moments of individual atoms produce magnetic fields that cancel one another, because each dipole points in a random direction.
Ferromagnet
Ferromagnetism is a property of certain materials (such as iron) that results in a significant, observable magnetic permeability, and in many cases, a significant magnetic coercivity, allowing the material to form a permanent magnet. Ferromag ...
ic materials below their
Curie temperature
In physics and materials science, the Curie temperature (''T''C), or Curie point, is the temperature above which certain materials lose their permanent magnetic properties, which can (in most cases) be replaced by induced magnetism. The Curie ...
, however, exhibit
magnetic domain
A magnetic domain is a region within a magnetic material in which the magnetization is in a uniform direction. This means that the individual magnetic moments of the atoms are aligned with one another and they point in the same direction. When c ...
s in which the atomic dipole moments are locally aligned, producing a macroscopic, non-zero magnetic field from the domain. These are the ordinary "magnets" with which we are all familiar.
The study of the behavior of such "spin models" is a thriving area of research in
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 ...
. For instance, 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 ...
describes spins (dipoles) that have only two possible states, up and down, whereas in the
Heisenberg model the spin vector is allowed to point in any direction. In certain magnets, the magnetic dipoles are only free to rotate in a 2D plane, a system which can be adequately described by the so-called
xy-model.
The lack of a unified theory of magnetism forces scientist to model magnetic systems theoretically with one, or a combination of these spin models in order to understand the intricate behavior of atomic magnetic interactions.
Numerical implementation of these models has led to several interesting results, such as quantitative research in the theory of
phase transition
In physics, chemistry, and other related fields like biology, 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 Sta ...
s.
Quantum
A quantum spin model is a
quantum Hamiltonian model that describes a system which consists of spins either interacting or not and are an active area of research in the fields of
strongly correlated electron systems,
quantum information theory
Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both t ...
, and
quantum computing
A quantum computer is a computer that exploits quantum mechanical phenomena. On small scales, physical matter exhibits properties of wave-particle duality, both particles and waves, and quantum computing takes advantage of this behavior using s ...
.
The
physical observables in these quantum models are actually operators in a
Hilbert space
In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
acting on state vectors as opposed to the physical observables in the corresponding classical spin models - like the Ising model - which are
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...
variables.
See also
*
ANNNI model
In statistical physics, the axial (or anisotropic) next-nearest neighbor Ising model, usually known as the ANNNI model, is a variant of the Ising model. In the ANNNI model, competing ferromagnetic and antiferromagnetic exchange interactions couple ...
*
Bethe ansatz
In physics, the Bethe ansatz is an ansatz for finding the exact wavefunctions of certain quantum many-body models, most commonly for one-dimensional lattice models. It was first used by Hans Bethe in 1931 to find the exact eigenvalues and eigenv ...
*
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 ...
*
Classical Heisenberg model
*
Quantum Heisenberg model
*
Hubbard model
The Hubbard model is an Approximation, approximate model used to describe the transition between Conductor (material), conducting and Electrical insulation, insulating systems.
It is particularly useful in solid-state physics. The model is named ...
*
J1 J2 model
*
Kuramoto model
The Kuramoto model (or Kuramoto–Daido model), first proposed by , is a mathematical model used in describing synchronization. More specifically, it is a model for the behavior of a large set of coupled oscillators. Its formulation was motivated b ...
*
Magnetism
Magnetism is the class of physical attributes that occur through a magnetic field, which allows objects to attract or repel each other. Because both electric currents and magnetic moments of elementary particles give rise to a magnetic field, ...
*
Majumdar–Ghosh model
*
Potts model
In statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice. By studying the Potts model, one may gain insight into the behaviour of ferromagnets and certain other phenom ...
*
t-J model
*
Quantum rotor model The quantum rotor model is a mathematical model for a quantum system. It can be visualized as an array of rotating electrons which behave as rigid rotors that interact through short-range dipole-dipole magnetic forces originating from their magnet ...
*
Spin
Spin or spinning most often refers to:
* Spin (physics) or particle spin, a fundamental property of elementary particles
* Spin quantum number, a number which defines the value of a particle's spin
* Spinning (textiles), the creation of yarn or thr ...
*
Spin stiffness
*
Spin waves
*
XY model
The classical XY model (sometimes also called classical rotor (rotator) model or O(2) model) is a lattice model of statistical mechanics. In general, the XY model can be seen as a specialization of Stanley's ''n''-vector model for .
Definition
...
*
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 their ...
*
Z N model
References
Bibliography
*
* R.J. Baxter, ''Exactly solved models in statistical mechanics'', London, Academic Press, 198
*
External links
Introduction to classical and Ising Spin ModelsQuantum Field Theory of Many-Body Systems Institute of Quantum Information
Caltech
{{Statistical mechanics topics
Magnetism
Statistical mechanics