The One-dimensional Heisenberg Magnet

K. Nagata, I. Yamamoto

Andrzej Pękalski

Hans-Peter Eckle

Abstract This chapter introduces the Heisenberg model, a fully quantum mechanical model that describes the magnetism of localized magnetic moments. The one-dimensional version of the Heisenberg model, the Heisenberg quantum spin chain, provides a good picture of magnetic materials that belong to a class of insulating magnetic materials where the interaction of the magnetic moments in one particular direction is much larger than in the perpendicular directions, and which can be described with high accuracy as quasi- one-dimensional magnets. A detailed description of the Heisenberg quantum spin chain is followed by a discussion of its various special cases, in particular the special case of the anisotropic Heisenberg quantum spin chain, the so-called XXZ quantum spin chain. It considers the solution of eigenvalue problem of this quantum spin and leads to Bethe’s conjecture for the wave function.

Lin-Jie Jiang, Huan-Qiang Zhou

The one-dimensional (1D) Heisenberg XXZ magnetic chain with isolated impurities is presented as a new completely integrable system. This system includes the 1D Heisenberg XXZ model and the 1D Heisenberg XXX magnetic chain with isolated impurities as special limiting cases. The model Hamiltonian is diagonalised, and the algebraic Bethe ansatz equations are derived. This makes it possible to investigate the equilibrium thermodynamics of the system.

Minoru Takahashi

A set of integral equations which describes the thermodynamic properties of one-dimensional Heisenberg model is obtained. From these equations one can calculate the energy, entropy and magnetization of the system at given temperature and magnetic field. We have succeeded in reproducing the known exact results in the limiting cases of zero temperature, infinite temperature and infinite magnetic field.