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Physical review. B, Condensed matter

Energy fluctuations in a classical Heisenberg chain

Hans A. De Raedt, Jan Fivez, Bart De Raedt

Published Aug 1, 1981• 5 citations
Publisher

Abstract

The dynamic structure factor of the energy density fluctuations in the classical Heisenberg chain is studied theoretically. Using exact expressions for the first six frequency moments and the knowledge of the exact $T=0$ result, simple approximations are used to calculate the spectrum. We show rigorously that the time-dependent correlation function is different for ferro-and antiferromagnetic coupling if $T\ensuremath{\ne}0$.

DOI
10.1103/physrevb.24.1562

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The dynamic structure factor of the energy density fluctuations in the classical Heisenberg chain is studied theoretically. Using exact expressions for the first six frequency moments and the knowledge of the exact T=0T=0T=0 result, simple approximations are used to calculate the spectrum. We show rigorously that the time-dependent correlation function is different for ferro-and antiferromagnetic coupling if T\ensuremath≠0T\ensuremath{\ne}0T\ensuremath=0.
The time dependence of the energy fluctuations in a classical Heisenberg chain is studied. The exact low-temperature limit is determined, and the time dependence is given by J0(4sin(12q)t){J}_{0}(4sin(\frac{1}{2}q)t)J0​(4sin(21​q)t). At nonzero temperatures approximate results are obtained by means of the continued-fraction method. Analytic expressions for the second and fourth frequency moments are given.
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.
Rigorous results are presented for the dynamics of the nearest-neighbour Heisenberg chain, for fixed but arbitrary wavevector and frequency, to lowest non-trivial order in the temperature. The authors obtain the two-spin correlation function, in the antiferromagnet for all wavevectors, and in the ferromagnet for wavelengths shorter than a coherence length. The energy correlation function at T=0 is obtained exactly, and is identical in the two systems. They find a violation of dynamical scaling in the form of the spectral function in the antiferromagnet for a range of wavevectors. An extension of the perturbative results for the ferromagnet to treat wavelengths longer than a coherence length shows that the diffusion coefficient vanishes logarithmically, in contrast to the scaling prediction that it is constant. An approximate value for the coefficient of the logarithmic term is obtained which agrees well with the results of simulations.
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.
We compute the exact four-spinon contribution to the zero-temperature dynamical structure factor of the spin-1/2 Heisenberg isotropic antiferromagnet in zero magnetic field, directly in the thermodynamic limit. We make use of the expressions for matrix elements of local spin operators obtained by Jimbo and Miwa using the quantum affine symmetry of the model, and of their adaptation to the isotropic case by Abada, Bougourzi and Si-Lakhal (correcting some overall factors). The four-spinon contribution to the first frequency moment sum rule at fixed momentum is calculated. This shows, as expected, that most of the remaining correlation weight above the known two-spinon part is carried by four-spinon states. Our results therefore provide an extremely accurate description of the exact structure factor.