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 spins 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 ${\displaystyle \sigma _{i}\in \{\pm 1\}}$ represents a microscopic magnetic dipole to which the magnetic moment is either up or down. Except the coupling between magnetic dipole moments, there is also a multipolar version of Heisenberg model called the multipolar exchange interaction.

For quantum mechanical reasons (see exchange interaction or the subchapter "quantum-mechanical origin of magnetism" in the article on 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 can be written in the form

where ${\displaystyle J}$ is the coupling constant for a 1-dimensional model consisting of N dipoles, represented by classical vectors (or "spins") σ_j, subject to the periodic boundary condition ${\displaystyle \sigma _{N+1}=\sigma _{1}}$. 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 ${\displaystyle J_{x},J_{y},}$ and ${\displaystyle J_{z}}$. As such in 3-dimensions, the Hamiltonian is given by

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