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 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 formula_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
Heisenberg group a group, as is shown by the multiplication: and The Heisenberg group is a simply-connected Lie group whose Lie algebra consists of matrices where By letting e, ..., e be the canonical basis of R, and setting the associated Lie algebra can be characterized by the canonical commutation relations, where "p", ..., "p", "q", ..., "q", "z" are the algebra generators. In particular, "z" is a "central" element of the Heisenberg Lie algebra. Note that the Lie algebra of the Heisenberg group is nilpotent. The exponential map is given by the following expression The exponential map of a nilpotent Lie