Spin model

Short description: Mathematical model used to explain magnetism

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 a far reaching and unifying topic that cuts across many fields.

## Contents

* 1 Introduction
* 2 Quantum
* 3 See also
* 4 References
* 5 Bibliography
* 6 External links

## 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. 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. 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[1] 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 transitions.

## 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, and quantum computing.[2] The physical observables in these quantum models are actually operators in a Hilbert space acting on state vectors as opposed to the physical observables in the corresponding classical spin models - like the Ising model - which are commutative variables.

## See also

* ANNNI model
* Bethe ansatz
* Ising model
* Classical Heisenberg model
* Quantum Heisenberg model
* Hubbard model
* J1 J2 model
* Kuramoto model
* Magnetism
* Majumdar–Ghosh model
* Potts model
* t-J model
* Quantum rotor model
* Spin
* Spin stiffness
* Spin waves
* XY model
* Yang–Baxter equation
* Z N model

## References

1. ↑ Nolting, Wolfgang; Ramakanth, Anupuru (2009) (in en). Quantum Theory of Magnetism. Berlin Heidelberg: Springer-Verlag. ISBN 9783540854159. https://www.springer.com/gp/book/9783540854159.
2. ↑ Michael Nielsen and Isaac Chuang (2000). Quantum Computation and Quantum Information. Cambridge: Cambridge University Press. ISBN 0-521-63503-9. OCLC 174527496.

## Bibliography

* Bethe, H. (March 1931). "Zur Theorie der Metalle". Zeitschrift für Physik 71 (3–4): 205–226. doi:10.1007/BF01341708. Bibcode: 1931ZPhy...71..205B.
* R.J. Baxter, Exactly solved models in statistical mechanics, London, Academic Press, 1982 [1]
* Affleck, Ian; Marston, J. Brad (1 March 1988). "Large-n limit of the Heisenberg-Hubbard model: Implications for high-Tc superconductors". Physical Review B 37 (7): 3774–3777. doi:10.1103/PhysRevB.37.3774. PMID 9944997. Bibcode: 1988PhRvB..37.3774A.

## External links

* Introduction to classical and Ising Spin Models
* Institute of Quantum Information Caltech

* v
* t
* e

Statistical mechanics

Theory|

* Principle of maximum entropy
* ergodic theory

Statistical thermodynamics|

* Ensembles
* partition functions
* equations of state
* thermodynamic potential:
* U
* H
* F
* G
* Maxwell relations

Models|

* Ferromagnetism models
* Ising
* Potts
* Heisenberg
* percolation
* Particles with force field
* depletion force
* Lennard-Jones potential

Mathematical approaches|

* Boltzmann equation
* H-theorem
* Vlasov equation
* BBGKY hierarchy
* stochastic process
* mean field theory and conformal field theory

Critical phenomena|

* Phase transition
* Critical exponents
* correlation length
* size scaling

Entropy|

* Boltzmann
* Shannon
* Tsallis
* Rényi
* von Neumann

Applications|

* Statistical field theory
* elementary particle
* superfluidity
* condensed matter physics
* complex system
* chaos
* information theory
* Boltzmann machine

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