The quantum Heisenberg model, developed by Werner Heisenberg, 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.
It is related to the prototypical Ising model, where at each site of a lattice, a spin
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 Magnetism § Quantum-mechanical origin of 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) and on a 1-dimensional periodic lattice, the Hamiltonian can be written in the form where
is the coupling constant and dipoles are represented by classical vectors (or "spins") σj, 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 acting upon the tensor product
on the right-hand side indicates the external magnetic field, with periodic boundary conditions.
The objective is to determine the spectrum of the Hamiltonian, from which the partition function can be calculated and the thermodynamics of the system can be studied.
The spin 1/2 Heisenberg model in one dimension may be solved exactly using the Bethe ansatz.
[1] In the algebraic formulation, these are related to particular quantum affine algebras and elliptic quantum groups in the XXZ and XYZ cases respectively.
[3] The physics of the Heisenberg XXX model strongly depends on the sign of the coupling constant
[4] 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.
A simplified version of Heisenberg model is the one-dimensional Ising model, where the transverse magnetic field is in the x-direction, and the interaction is only in the z-direction: At small g and large g, the ground state degeneracy is different, which implies that there must be a quantum phase transition in between.
It can be solved exactly for the critical point using the duality analysis.
Under periodic boundary conditions, the transformed Hamiltonian can be shown is of a very similar form: but for the
Assuming that there's only one critical point, we can conclude that the phase transition happens at
Following the approach of Ludwig Faddeev (1996), the spectrum of the Hamiltonian for the XXX model
In this context, for an appropriately defined family of operators
(in turn defined using a Lax matrix), which acts on
The FCRs also show there is a large commuting subalgebra given by the generating function
The Bethe vectors are in fact simultaneous eigenvectors for the whole subalgebra.
for the deformation from the XXX model, the BAE (Bethe ansatz equation) is
these are precisely the BAEs for the six-vertex model, after identifying
[6][7] This was originally thought to be coincidental until Baxter showed the XXZ Hamiltonian was contained in the algebra generated by the transfer matrix
The integrability is underpinned by the existence of large symmetry algebras for the different models.
These appear through the transfer matrix, and the condition that the Bethe vectors are generated from a state
corresponds to the solutions being part of a highest-weight representation of the extended symmetry algebras.