# Heisenberg model (quantum)

Heisenberg 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. In the prototypical Ising model, defined on a d-dimensional lattice, at each lattice site, a spin $sigma_i in \left\{ pm 1\right\}$represents a microscopic magnetic dipole to which the magnetic moment is either up or down.

For quantum mechanical reasons (see exchange interaction), 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 can be written in the form

:$hat H = -J sum_\left\{j =1\right\}^\left\{N\right\} sigma_j sigma_\left\{j+1\right\} - h sum_\left\{j =1\right\}^\left\{N\right\} sigma_j$

for a 1-dimensional model consisting of "N" dipoles, subject to the periodic boundary condition $sigma_\left\{N+1\right\} = sigma_1$. 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 spin-1/2 matrices at spin 1/2), and the coupling constants $J_x, J_y,$ and $J_z$. As such in 1-dimension, the Hamiltonian is given by

:$hat H = -frac\left\{1\right\}\left\{2\right\} sum_\left\{j=1\right\}^\left\{N\right\} \left(J_x sigma_j^x sigma_\left\{j+1\right\}^x + J_y sigma_j^y sigma_\left\{j+1\right\}^y + J_z sigma_j^z sigma_\left\{j+1\right\}^z - hsigma_j^\left\{z\right\}\right)$

where the $h$ on the right-hand side indicates the external magnetic field, with periodic boundary conditions, and at spin $s=1/2$, spin matrices given by:

:

:

The Hamiltonian then acts upon the tensor product $\left(mathbb\left\{C\right\}^2\right)^\left\{otimes N\right\}$, of dimension $2^N$. The objective is to determine the spectrum of the Hamiltonian, from which the partition function can be calculated, from which the thermodynamics of the system can be studied. The most widely known type of Heisenberg model is probably the XXZ Heisenberg model, which occurs in the case $J = J_x = J_y eq J_z = Delta$. The spin 1/2 Heisenberg model in one dimension may be solved exactly using the Bethe ansatz.

The physics of the Heisenberg model strongly depends on the sign of the coupling constant$J$ and the dimension of the space. For positive $J$ the ground state is always ferromagnetic. At negative $J$ the ground state is antiferromagnetic in two and three dimensions. 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.A system of half-integer spins exhibits quasi-long range order.

ee also

*Heisenberg model (classical)
*Dmrg of Heisenberg model
*Ising model
*t-J model

References

* R.J. Baxter, "Exactly solved models in statistical mechanics", London, Academic Press, 1982

Wikimedia Foundation. 2010.

### См. также в других словарях:

• Heisenberg model (classical) — The 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.It can be formulated as follows: take a d dimensional lattice, and a set of spins of the unit …   Wikipedia

• Heisenberg model — The Heisenberg model can refer to two models in statistical mechanics:*Heisenberg model (classical), a classical nearest neighbour spin model *Heisenberg model (quantum), a model where the spins are treated quantum mechanically using Pauli… …   Wikipedia

• Dmrg of Heisenberg model — This example presents the infinite DMRG algorithm. It is about S = 1 antiferromagnetic Heisenberg chain, but the recipe can be applied for every translationally invariant one dimensional lattice. DMRG is a renormalization group technique because… …   Wikipedia

• Classical Heisenberg model — 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. Contents 1 Definition 2 Properties 3 See also …   Wikipedia

• Quantum gravity — is the field of theoretical physics attempting to unify quantum mechanics, which describes three of the fundamental forces of nature (electromagnetism, weak interaction, and strong interaction), with general relativity, the theory of the fourth… …   Wikipedia

• Quantum mechanics — For a generally accessible and less technical introduction to the topic, see Introduction to quantum mechanics. Quantum mechanics …   Wikipedia

• quantum mechanics — quantum mechanical, adj. Physics. a theory of the mechanics of atoms, molecules, and other physical systems that are subject to the uncertainty principle. Abbr.: QM Cf. nonrelativistic quantum mechanics, relativistic quantum mechanics. [1920 25]… …   Universalium

• Quantum decoherence — Quantum mechanics Uncertainty principle …   Wikipedia

• Heisenberg's microscope — exists only as a thought experiment, one that was proposed by Werner Heisenberg, criticized by his mentor Niels Bohr, and subsequently served as the nucleus of some commonly held ideas, and misunderstandings, about Quantum Mechanics. Basic ideas… …   Wikipedia

• Quantum mysticism — Claims Quantum mechanics can be interpreted according to paranormal, spiritual, or mystical ideas Related scientific disciplines Physics, Psychology Year proposed ca. 1920 Original proponents Niels Bohr, Arthur Eddington …   Wikipedia

### Поделиться ссылкой на выделенное

##### Прямая ссылка:
Нажмите правой клавишей мыши и выберите «Копировать ссылку»