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Physical Review Letters

Propagating Energy Modes in the Classical Heisenberg Chain: "Magnons" and "Second Magnons"

J M Loveluck, Ewald Balcar

Published Jun 4, 1979• 32 citations
Publisher

Abstract

The ferromagnetically coupled classical Heisenberg chain in an applied magnetic field has been studied by computer simulation. The results indicate the presence of a second collective mode, in addition to the damped spin-wave-like modes which have previously been observed in the absence of a magnetic field. For intermediate wavelengths, the mode manifests itself by well-defined oscillations in the energy-density correlation function, and by a second peak in the spectrum of the longitudinal spin-density correlation functions.

DOI
10.1103/physrevlett.42.1563

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The ferromagnetically coupled classical Heisenberg chain in an applied magnetic field has been studied by computer simulation. The results indicate the presence of a second collective mode, in addition to the damped spin-wave-like modes which have previously been observed in the absence of a magnetic field. For intermediate wavelengths, the mode manifests itself by well-defined oscillations in the energy-density correlation function, and by a second peak in the spectrum of the longitudinal spin-density correlation functions.
We use the spin functional renormalization group recently developed by two of us [Krieg and Kopietz, Phys. Rev. B 99, 060403(R) (2019)] to calculate the magnetization M(H,T)M(H,T)M(H,T) and the damping of magnons due to classical longitudinal fluctuations of quantum Heisenberg ferromagnets. In order to guarantee that for vanishing magnetic field $H\ensuremath{\rightarrow}0$, the magnon spectrum is gapless when the spin rotational invariance is spontaneously broken, we use a Ward identity to express the magnon self-energy in terms of the magnetization. In two dimensions, our approach correctly predicts the absence of long-range magnetic order for H=0H=0H=0 at finite temperature TTT. The magnon spectrum then exhibits a gap from which we obtain the transverse correlation length. We also calculate the wave-function renormalization factor of the magnons. As a mathematical byproduct, we derive a recursive form of the generalized Wick theorem for spin operators in frequency space which facilitates the calculation of arbitrary time-ordered connected correlation functions of an isolated spin in a magnetic field.
A theoretical study of the low-temperature dynamics of the antiferromagnetic Heisenberg chain in an external field is presented. The time- and frequency-dependent transverse and longitudinal spin fluctuations and energy fluctuations are calculated by means of a continued fraction representation. The frequency moments are evaluated analytically by a computer program. The transfer-operator technique is used to calculate all static correlation functions. For modest magnetic fields, a two-peak structure in the dynamic form factor of the transverse spin fluctuations is observed. Apart from the usual spin-wave excitations, the longitudinal-spin-fluctuation spectra show a broad second resonance for some temperatures and fields. The energy fluctuations display a dominant oscillatory time dependence.
We discuss spin-12\frac{1}{2}21​ Heisenberg antiferromagnet on simple square lattice in magnetic field HHH using a recently proposed bond-operator technique. It is well known that magnetically ordered phases of quantum magnets are well described at least qualitatively by the conventional spin-wave theory that only introduces quantum corrections into the classical solution of the problem. We observe that quantum fluctuations change drastically dynamical properties of the considered model at HHH close to its saturation value: the dynamical structure factor shows anomalies corresponding to Green's function poles which have no counterparts in the spin-wave theory. That is, quantum fluctuations produce multiple short-wavelength magnon modes not changing qualitatively the long-wavelength spin dynamics. Our results are in agreement with previous quantum Monte Carlo simulations and exact diagonalization of finite clusters.
We have studied the dynamics of the classical Heisenberg chain at finite and infinite temperatures by computer simulation of an array of 4000 spins. The evaluation of the spin-spin and energy-energy wave-vector- and time-dependent correlation functions out to time $10{J}^{\ensuremath{-}1}$ made it possible to determine the spin and energy diffusion constants. At infinite temperature we find ${D}_{S}=(1.33\ifmmode\pm\else\textpm\fi{}0.10)JS{a}^{2}$ and ${D}_{E}=(3\ifmmode\pm\else\textpm\fi{}1)JS{a}^{2}$. Values of DS{D}_{S}DS​ have also been determined, with less precision, for ferromagnetic and antiferromagnetic interactions at temperatures kBT∣J∣=1.0and0.5\frac{{k}_{B}T}{|J|}=1.0 \mathrm{and} 0.5∣J∣kB​T​=1.0and0.5. Our results are compared with various theoretical estimates. In addition, we calculate the time-dependent spin and energy self-correlation functions.