- 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 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 Hamiltoniancan be written in the form
for a 1-dimensional model consisting of "N" dipoles, subject to the periodic boundary condition . 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 and . As such in 1-dimension, the Hamiltonian is given by
where the on the right-hand side indicates the external
magnetic field, with periodic boundary conditions, and at spin , spin matrices given by:
The Hamiltonian then acts upon the
tensor product, of dimension . The objective is to determine the spectrum of the Hamiltonian, from which the partition functioncan be calculated, from which the thermodynamicsof the system can be studied. The most widely known type of Heisenberg model is probably the XXZ Heisenberg model, which occurs in the case . 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 and the dimension of the space. For positive the ground state is always ferromagnetic. At negative 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 orderis present.A system of half-integer spins exhibits quasi-long range order.
Heisenberg model (classical)
Dmrg of Heisenberg model
* R.J. Baxter, "Exactly solved models in statistical mechanics", London, Academic Press, 1982
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