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| The '''Heisenberg model''' is a [[statistical mechanics|statistical mechanical]] [[mathematical model|model]] used in the study of [[critical point (thermodynamics)|critical point]]s and [[phase transition]]s of magnetic systems, in which the [[spin (physics)|spin]]s of the magnetic systems are treated [[quantum mechanics|quantum mechanically]]. In the prototypical [[Ising model]], defined on a d-dimensional lattice, at each lattice site, a spin <math>\sigma_i \in \{ \pm 1\}</math>represents a microscopic magnetic dipole to which the magnetic moment is either up or down.
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| ==Overview==
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| For quantum mechanical reasons (see [[exchange interaction]] or the subchapter "quantum-mechanical origin of magnetism" in the article on [[magnetism#Quantum-mechanical_origin_of_magnetism|magnetism]]), the dominant coupling between two dipoles may cause nearest-neighbors to have lowest energy when they are ''aligned''. Under this assumption (so that magnetic interactions only occur between adjacent dipoles) the [[Hamiltonian (quantum mechanics)|Hamiltonian]] can be written in the form | |
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| :<math>\hat H = -J \sum_{j =1}^{N} \sigma_j \sigma_{j+1} - h \sum_{j =1}^{N} \sigma_j </math>
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| where <math>J</math> is the coupling constant for a 1-dimensional model consisting of ''N'' dipoles, represented by classical vectors (or "spins") σ<sub>j</sub>, subject to the periodic boundary condition <math>\sigma_{N+1} = \sigma_1 </math>.
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| The Heisenberg model is a more realistic model in that it treats the spins quantum-mechanically, by replacing the spin by a quantum operator ([[Pauli matrices|Pauli spin-1/2 matrices]] at spin 1/2), and the coupling constants <math>J_x, J_y,</math> and <math>J_z</math>. As such in 3-dimensions, the Hamiltonian is given by
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| :<math>\hat H = -\frac{1}{2} \sum_{j=1}^{N} (J_x \sigma_j^x \sigma_{j+1}^x + J_y \sigma_j^y \sigma_{j+1}^y + J_z \sigma_j^z \sigma_{j+1}^z - h\sigma_j^{z}) </math>
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| where the <math>h</math> on the right-hand side indicates the external [[magnetic field]], with periodic [[boundary conditions]], and at spin <math>s=1/2</math>, spin matrices given by
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| :<math>
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| \sigma^x =
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| \begin{pmatrix}
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| 0&1\\
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| 1&0
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| \end{pmatrix}
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| </math>
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| :<math>
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| \sigma^y =
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| \begin{pmatrix}
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| 0&-i\\
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| i&0
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| \end{pmatrix}
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| </math>
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| :<math>
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| \sigma^z =
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| \begin{pmatrix}
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| 1&0\\
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| 0&-1
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| \end{pmatrix}
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| </math>
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| The Hamiltonian then acts upon the [[tensor product]] <math>(\mathbb{C}^2)^{\otimes N}</math>, of dimension <math>2^N</math>. The objective is to determine the spectrum of the Hamiltonian, from which the [[partition function (statistical mechanics)|partition function]] can be calculated, from which the [[thermodynamics]] of the system can be studied. The most widely known type of Heisenberg model is the [[Heisenberg XXZ model]], which occurs in the case <math>J = J_x = J_y \neq J_z = \Delta</math>. The spin 1/2 Heisenberg model in one dimension may be solved exactly using the [[Bethe ansatz]],<ref>http://www.iop.org/EJ/abstract/0305-4470/25/15/007</ref> while other approaches do so without Bethe ansatz.<ref>http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0103-97332001000400008</ref>
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| The physics of the Heisenberg model strongly depends on the sign of the coupling constant
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| <math>J</math> and the dimension of the space. For positive <math>J</math> the ground state is always [[ferromagnetism|ferromagnetic]]. At negative <math>J</math> the ground state is [[antiferromagnetism|antiferromagnetic]] in two and three dimensions, it is from this ground state that the [[Hubbard model]] is given.<ref>http://math.arizona.edu/~tgk/qs.html</ref> In one dimension the nature of correlations in the antiferromagnetic Heisenberg model depends on the spin of the magnetic dipoles. If the spin is integer then only [[short-range order]] is present.
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| A system of half-integer spins exhibits [[quasi-long range order]].
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| ==Applications==
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| * Another important object is entanglement entropy. One way to describe it is to subdivide the unique ground state into a block (several sequential spins) and the environment (the rest of the ground state). The entropy of the block can be considered as entanglement entropy. At zero temperature in the critical region (in thermodynamic limit) it scales logarithmically with the size of the block. As the temperature increases the [http://arxiv.org/pdf/cond-mat/0311056 logarithmic dependence changes into a linear function]. For large temperature linear dependence follows from the second law of thermodynamics.
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| ==See also==
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| *[[Classical Heisenberg model]]
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| *[[Dmrg of Heisenberg model]]
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| *[[Quantum rotor model]]
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| *[[t-J model]]
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| *[[J1 J2 model]]
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| *[[Majumdar-Ghosh Model]]
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| *[[AKLT Model]]
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| == References ==
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| * R.J. Baxter, ''Exactly solved models in statistical mechanics'', London, Academic Press, 1982
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| * H. Bethe, Zur Theorie der Metalle, ''Zeitschrift für Physik A'', 1931 {{doi|10.1007/BF01341708}}
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| ==Notes==
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| {{reflist}}
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| [[Category:Spin models]]
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| [[Category:Quantum magnetism]]
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| [[Category:Quantum Lattice models]]
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| [[Category:Magnetic ordering]]
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Irwin Butts is what my spouse enjoys to call me although I don't really like becoming known as like that. For years I've been operating as a payroll clerk. To collect cash is what his family and him enjoy. His family lives in South Dakota but his spouse wants them to transfer.
Feel free to visit my web page Xrambo.com