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 (several sequential spins) and the environment (the rest of the ground state). The entropy of the block can be considered as entanglement entropy. At zero temperature in the critical region (thermodynamic limit) it scales logarithmically with the size of the block. As the temperature increases the logarithmic dependence changes into a linear function. For large temperatures linear dependence follows from the second law of thermodynamics. The Heisenberg model provides an important and tractable theoretical example for applying density matrix renormalisation. The six-vertex model can be solved using the algebraic Bethe ansatz for the Heisenberg spin chain (see Baxter, "Exactly Solved Models in Statistical Mechanics"). 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. See also Classical Heisenberg model DMRG of the Heisenberg model Quantum rotor model t-J model J1 J2 model Majumdar–Ghosh model AKLT model Multipolar exchange interaction References R.J. Baxter, Exactly solved models in statistical mechanics, London, Academic Press, 1982 Notes Spin models Quantum magnetism Quantum lattice models Magnetic ordering Werner Heisenberg