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Physical Review Letters

Classical Ferromagnetic Heisenberg Chain in a Magnetic Field

U. Balucani, A. Rettori, V. Tognetti

Published Mar 8, 1982• 13 citations
Publisher

Abstract

The dynamics of classical ferromagnetic Heisenberg chains in a magnetic field is investigated. Contrary to previous theories, in the longitudinal spectral shapes temperature is found to play a crucial role through a dynamical spin-energy coupling whose effect is competitive with the field. Thermal disorder ultimately causes a nearly isotropic situation, leading to a new interpretation of the features of recent computer experiments.

DOI
10.1103/physrevlett.48.703

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The dynamics of classical ferromagnetic Heisenberg chains in a magnetic field is investigated. Contrary to previous theories, in the longitudinal spectral shapes temperature is found to play a crucial role through a dynamical spin-energy coupling whose effect is competitive with the field. Thermal disorder ultimately causes a nearly isotropic situation, leading to a new interpretation of the features of recent computer experiments.
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.
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Magnetic-field tuning of equal-time longitudinal and transverse correlations is investigated for the antiferromagnetic isotropic spin-1/2 Heisenberg chain. Full exact diagonalization of a ten-site periodic ring is used at kBT/J=0.15 over 0≤h/J≤2.6, where h=gμBB. The field breaks spin-rotational symmetry, increases uniform magnetization, redistributes longitudinal and transverse correlation weight, and drives the system toward the fully polarized state above the saturation field hs=2J. Below saturation, longitudinal correlations acquire a uniform m² background and field-dependent oscillatory structure, whereas transverse antiferromagnetic correlations weaken as the density of reversed spins decreases. The finite-temperature magnetization rounds the zero-temperature level crossings and saturation threshold. The numerical results are interpreted using the Tomonaga-Luttinger-liquid description and checked against exact limiting values. The analysis provides a reproducible finite-chain account of field-controlled spin correlations and identifies the distinctions required when connecting static calculations to scattering experiments.
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.
The thermodynamic Bethe-Ansatz equations for the isotropic ferromagnetic S=12S=\frac{1}{2}S=21​ Heisenberg chain have been solved numerically. At low temperatures, I find a power-law dependence on TTT for the specific heat and the susceptibility with critical exponents $\ensuremath{\alpha}=\ensuremath{-}0.49\ifmmode\pm\else\textpm\fi{}0.02$, $\ensuremath{\gamma}=2.00\ifmmode\pm\else\textpm\fi{}0.02$, and $\ensuremath{\Delta}\ensuremath{\simeq}\ensuremath{\gamma}$. The exponent $\ensuremath{\alpha}$ is compatible with effective spin waves and $\ensuremath{\gamma}$ and $\ensuremath{\Delta}$ are the exponents of the classical chain. Amplitudes and corrections to scaling are obtained and differences with previous results are discussed.
We study the low-energy sector of the Heisenberg necklace model. Using the field-theory methods, we estimate how the coupling of the electronic spins with the paramagnetic Kondo spins affects the overall spin dynamics and evaluate its dependence on a magnetic field. We are motivated by the experimental realizations of the spin-1/2 Heisenberg chains in SrCuO2{\mathrm{SrCuO}}_{2}SrCuO2​ and Sr2CuO3{\mathrm{Sr}}_{2}{\mathrm{CuO}}_{3}Sr2​CuO3​ cuprates, which remain one-dimensional Luttinger liquids down to temperatures much lower than the in-chain exchange coupling JJJ. We consider the perturbation of the energy spectrum caused by the interaction $\ensuremath{\gamma}$ with nuclear spins (I=3/2I=3/2I=3/2) present on the same sites. We find that the resulting necklace model has a characteristic energy scale, $\mathrm{\ensuremath{\Lambda}}\ensuremath{\sim}{J}^{1/3}{(\ensuremath{\gamma}I)}^{2/3}$, at which the coupling between (nuclear) spins of the necklace and the spins of the Heisenberg chain becomes strong. This energy scale is insensitive to a magnetic field BBB. For ${\ensuremath{\mu}}_{B}B>\mathrm{\ensuremath{\Lambda}}$ we find two gapless bosonic modes that have different velocities, whose ratio at strong fields approaches a universal number, 2+1\sqrt{2}+12​+1.