Spin model
Encyclopedia
A spin model is a mathematical model
used in physics primarily to explain magnetism
. Spin models may either be classical or quantum
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 and computability theory
in theoretical computer science
. The theory of spin models is far reaching and unifying topic that cuts across many fields.
s, the magnetic dipole moments of individual atoms produce magnetic fields that cancel one another, because each dipole points in a random direction. Ferromagnetic materials below their Curie temperature, however, exhibit magnetic domains 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
. For instance, the Ising model
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. These models have many interesting properties, which have led to interesting results in the theory of phase transition
s.
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 not only in the natural sciences and engineering disciplines A mathematical model is a...
used in physics primarily to explain magnetism
Magnetism
Magnetism is a property of materials that respond at an atomic or subatomic level to an applied magnetic field. Ferromagnetism is the strongest and most familiar type of magnetism. It is responsible for the behavior of permanent magnets, which produce their own persistent magnetic fields, as well...
. Spin models may either be classical or quantum
Quantum spin model
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, and quantum computing...
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 and computability theory
Computability theory
Computability theory, also called recursion theory, is a branch of mathematical logic that originated in the 1930s with the study of computable functions and Turing degrees. The field has grown to include the study of generalized computability and definability...
in theoretical computer science
Theoretical computer science
Theoretical computer science is a division or subset of general computer science and mathematics which focuses on more abstract or mathematical aspects of computing....
. The theory of spin models is far reaching and unifying topic that cuts across many fields.
Introduction
In ordinary materialMaterial
Material is anything made of matter, constituted of one or more substances. Wood, cement, hydrogen, air and water are all examples of materials. Sometimes the term "material" is used more narrowly to refer to substances or components with certain physical properties that are used as inputs to...
s, the magnetic dipole moments of individual atoms produce magnetic fields that cancel one another, because each dipole points in a random direction. Ferromagnetic materials below their Curie temperature, however, exhibit magnetic domains 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
Condensed matter physics deals with the physical properties of condensed phases of matter. These properties appear when a number of atoms at the supramolecular and macromolecular scale interact strongly and adhere to each other or are otherwise highly concentrated in a system. The most familiar...
. For instance, the Ising model
Ising model
The Ising model is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables called spins that can be in one of two states . The spins are arranged in a graph , and each spin interacts with its nearest neighbors...
describes spins (dipoles) that have only two possible states, up and down, whereas in the Heisenberg model
Heisenberg model
The Heisenberg model can refer to two models in statistical mechanics:*Heisenberg model , a classical nearest neighbour spin model*Heisenberg model , a model where the spins are treated quantum mechanically using Pauli matrices....
the spin vector is allowed to point in any direction. These models have many interesting properties, which have led to interesting results in the theory of phase transition
Phase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
s.
See also
- Ising modelIsing modelThe Ising model is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables called spins that can be in one of two states . The spins are arranged in a graph , and each spin interacts with its nearest neighbors...
- Classical Heisenberg modelHeisenberg model (classical)The Classical Heisenberg model is the n = 3 case of the n-vector model, one of the models used in statistical physics to model ferromagnetism, and other phenomena.-Definition:...
- Quantum Heisenberg modelHeisenberg model (quantum)The Heisenberg model is a statistical mechanical model used in the study of critical points and phase transitions of magnetic systems, in which the spin of the magnetic systems are treated quantum mechanically...
- Quantum rotor modelQuantum rotor modelThe 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 magnetic dipole moments...
- Kuramoto modelKuramoto modelThe Kuramoto model, first proposed by Yoshiki Kuramoto , is a mathematical model used to describe synchronization. More specifically, it is a model for the behavior of a large set of coupled oscillators...
- ANNNI modelANNNI modelIn statistical physics, the axial next-nearest-neighbor Ising model, usually known as the ANNNI model, is a variant of the Ising model in which competing ferromagnetic and...
- Potts modelPotts modelIn 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 phenomena of solid state physics...
- XY modelXY modelThe classical XY model is a model of statistical mechanics. It is the special case of the n-vector model for n=2.-Definition:...
- Hubbard modelHubbard modelThe Hubbard model is an approximate model used, especially in solid state physics, to describe the transition between conducting and insulating systems...
- t-J modelT-J modelThe t-J model was first derived in 1977 from the Hubbard model by Józef Spałek. The model describes strongly correlated electron systems. It is used to calculate high temperature superconductivity states in doped antiferromagnets.The t-J Hamiltonian is:...
- J1 J2 modelJ1 J2 modelThe J1-J2 model is a quantum spin model like the Heisenberg model but also includes a term for the interaction between next-nearest neighbor spins.-Hamiltonian:-See also:*Spin model*Heisenberg model *Hubbard model*t-J model*Majumdar-Ghosh Model...
- Bethe ansatzBethe ansatzIn physics, the Bethe ansatz is a method for finding the exact solutions of certain one-dimensional quantum many-body models. It was invented by Hans Bethe in 1931 to find the exact eigenvalues and eigenvectors of the one-dimensional antiferromagnetic Heisenberg model Hamiltonian...
- Yang–Baxter equation
- MagnetismMagnetismMagnetism is a property of materials that respond at an atomic or subatomic level to an applied magnetic field. Ferromagnetism is the strongest and most familiar type of magnetism. It is responsible for the behavior of permanent magnets, which produce their own persistent magnetic fields, as well...
- SpinSpin (physics)In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
- Spin waves
- Spin stiffnessSpin stiffnessThe spin-stiffness or spin rigidity or helicity modulus or the "superfluid density" is a constant which represents the change in the ground state energy of a spin system as a result of introducing a slow in plane twist of the spins...
- Majumdar-Ghosh ModelMajumdar-Ghosh ModelThe Majumdar-Ghosh model is an extension of the one-dimensional quantum Heisenberg spin model in which an extra interaction is added coupling spins two lattice spacings apart such that second-neighbor coupling half as strong the first. It is therefore a special case of the J1 J2 model...