A quantum theory for ferromagnetic and antiferromagnetic Heisenberg spin chains at low temperatures

Abstract

We present a quantum theory for both the ferromagnetic and antiferromagnetic Heisenberg spin chains, valid at low temperatures, in which Mori projection algebra is applied to the calculation of the spin relaxation function. The spin relaxation function is shown to have the functional form of a damped harmonic oscillator, which depends on the spin density, the second moment of the spin current density, and the spin damping function. These quantities are calculated from the spin dynamics of the Heisenberg spin chain, were the spin operators are expanded to leading order in the reciprocal of the spin (i.e., 1/S) and then second-quantized according to the formalism of either Holstein-Primakoff or Dyson-Maleev. In addition, the quantum self-energy corrections for both the ferro- and antiferro-magnons are calculated using finite-temperature Green's functions to first order in the magnon occupation number. By using the spin relaxation function, we also have determined the spin wave resonances, the damping at resonance, and the line-shape of the Van Hove scattering function (generally called the dynamical structure or form factor) for both the ferromagnetic and antiferromagnetic Heisenberg spin chains. Specifically, we have studied the resonance, the damping at resonance, and the line-shape as a function of spin, temperature, exchange energy, and momentum (i.e., S, T, J(,1), and q). The resonances tend to be thermally very stable, and display a marked momentum dependence in the quantum regime (i.e., T/J(,1)S (LESSTHEQ) 1). Also, the damping at resonance demonstrates a linear temperature dependence, and displays a somewhat less marked momentum dependence in the quantum regime. Line-shapes vary greatly between the ferromagnetic and antiferromagnetic Van Hove scattering functions: the ferromagnetic line-shape is symmetric, whereas the antiferromagnetic line-shape is asymmetric. The statics of the Heisenberg spin chain were also studied, employing our earlier calculations, on static spin pair correlations, which could be directly substituted into Mikeska's expression for the quantum static spin pair correlation function. Comparison of the Fourier transforms of this function with simulations for finite S = 1/2 antiferromagnetic Heisenberg spin chains gives excellent agreement with Mikeska's theory.