Concept

# Quantum Heisenberg model

Summary
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 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 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 , 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 and 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.
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