Spin-wave theory in a randomly disordered lattice: A Heisenberg ferromagnet

TL;DR

This paper develops a theoretical framework using spin-wave theory and quantum statistical methods to analyze the effects of random nonmagnetic impurities on the magnon spectrum and thermal properties of a Heisenberg ferromagnet on a cubic lattice.

Contribution

It introduces a novel approach combining Dyson-Maleev transformation and Green function techniques to study impurity effects in disordered ferromagnetic lattices, providing explicit formulas for physical quantities as functions of impurity concentration.

Findings

Magnon energy increases with impurity concentration for low-lying states.

Spectral density and magnon lifetime depend on impurity concentration.

Derived closed-form expressions for physical quantities below percolation threshold.

Abstract

Starting from the hamiltonian for the Heisenberg ferromagnet which comprise randomly distributed nonmagnetic ions as impurities in a Bravais lattice, we express the spin operators by means of the Dyson-Maleev transformation in terms of the Bose operators of the second quantization. Then by using methods of quantum statistical field theory, we derive the partition function and the free energy for the system. We adopt the Matsubara thermal perturbation method to a portion of the hamiltonian which describes the interaction between magnons and the stationary field of nonmagnetic ions. Upon averaging over all possible distributions of impurities, we express the free energy of the system as a function of the mean impurity concentration. Subsequently, we set up the double-time single particle Green function at temperature T in the momentum space in terms of magnon operators and derive the equation of motion for the Green function through the Heisenberg equation of motion and then solve the resulting equation. From this, we calculate the self-energy and then the spectral density for the system. We apply the formalism to the case of the simple cubic lattice and compute the density of states, the spectral density function and the lifetime of the magnons as a function of energy for several values of the mean concentration of nonmagnetic ions in the lattice. We calculate magnon energy spectrum as a function of average impurity concentration fraction c, which shows that for low lying states, the excitation energy increases continuously with c in the studied range 0.1 < c < 0.7. We use the spectral density function to compute some thermal quantities. We have obtained closed form expressions for the configurationally averaged physical quantities of interest in a unified fashion as functions of c to any order of c applicable below a critical percolation concentration.