For quantum mechanical reasons, the dominant coupling between two dipoles may cause nearest-neighbors to have lowest energy when they are aligned. Under this assumption 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 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, of dimension. To define it, recall the Pauli spin-1/2 matrices and for and denote, where is the identity matrix. Given a choice of real-valued coupling constants and, the Hamiltonian is given by where the 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, from which the thermodynamics of the system can be studied. It is common to name the model depending on the values of, and : if, the model is called the Heisenberg XYZ model; in the case of, it is the Heisenberg XXZ model; if, it is the Heisenberg XXX model. The spin 1/2 Heisenberg model in one dimension may be solved exactly using the Bethe ansatz. In the algebraic formulation, these are related to particular Quantum affine algebras and Elliptic Quantum Group in the XXZ and XYZ cases respectively. Other approaches do so without Bethe ansatz. The physics of the Heisenberg XXX 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 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. The duality transition of the Pauli matrices is and, where and are also Pauli matrices which obey the Pauli matrix algebra. Under periodic boundary conditions, the transformed Hamiltonian can be shown is of a very similar form: but for the attached to the spin interaction term. Assuming that there's only one critical point, we can conclude that the phase transition happens at.
Applications
Another important object is entanglement entropy. One way to describe it is to subdivide the unique ground state into a block and the environment. The entropy of the block can be considered as entanglement entropy. At zero temperature in the critical region it scales logarithmically with the size of the block. As the temperature increases the . For large temperatures linear dependencefollows from the second law of thermodynamics.
The Heisenberg model provides an important and tractable theoretical example for applying Density Matrix Renormalisation.
The half-filled Hubbard model in the limit of strong repulsive interactions can be mapped onto a Heisenberg model with representing the strength of the superexchange interaction.
