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Physical Review Letters

Scaling Treatment of Critical and "Chaotic" Dynamics of the Dilute Heisenberg Chain

Robin B. Stinchcombe

Published Jan 17, 1983• 32 citations
Publisher

Abstract

A lattice rescaling method is applied to the equations of motion of the dilute Heisenberg chain and leads via a probabilistic integral equation to an iterative map for the characteristic frequency $\ensuremath{\beta}$ and concentration $p$. Dilution induces a crossover in the $\ensuremath{\beta}$ scaling from "chaotic" (ergodic and mixing) behavior, corresponding to the sampling of the pure band, to periodic orbits corresponding to isolated cluster response. A dynamic scaling form is obtained for the critical dynamics by fixed-point analysis.

DOI
10.1103/physrevlett.50.200

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A lattice rescaling method is applied to the equations of motion of the dilute Heisenberg chain and leads via a probabilistic integral equation to an iterative map for the characteristic frequency \ensuremathβ\ensuremath{\beta}\ensuremathβ and concentration ppp. Dilution induces a crossover in the \ensuremathβ\ensuremath{\beta}\ensuremathβ scaling from "chaotic" (ergodic and mixing) behavior, corresponding to the sampling of the pure band, to periodic orbits corresponding to isolated cluster response. A dynamic scaling form is obtained for the critical dynamics by fixed-point analysis.
Critical spin wave dynamics in the dilute Heisenberg chain near the percolation threshold is treated by two complementary approaches. The first exploits the renormalisation group transformation of parameters under a length scaling achieved by decimation on the equations of motion of the random system, the second obtains from the contributions from chain segments of all possible sizes the average dynamic response using a continuum approach valid for small wavevector k and long percolation correlation length xi p. Both approaches yield identical dynamic scaling forms for the dynamic response, with dynamic exponent z=2, and details of the crossover of characteristic frequency between hydrodynamic and critical form as k xi p varies from 0 to infinity . A detailed expression for the scaling function for the dynamic response is also obtained.
The critical dynamics of Heisenberg spin systems at the percolation threshold is obtained from the transformation of the equations of motion under a length scaling acheived by climination of sites. The method is applied exactly to ddd-dimensional nonrandom fractals representing the infinite-cluster backbone and approximatcly to a truly random twodimensional system to yield dynamic scaling forms for response function and characteristic frequency, and associated static and dynamic critical exponents.
The transverse dynamic response of the dilute classical Heisenberg antiferromagnetic chain is calculated near the percolation threshold as k to 0 and k to pi . In the critical limit the universal dynamic structure function, which is calculated in closed form, exhibits dynamic scaling with exponents z=1, eta T =1. This function displays the crossover from spin wave (k- pi ) xi p >>1 to hydrodynamic (k- pi ) xi p <<1 response. The relevance of the results to the experimental situation is discussed.
We study a model equation of motion describing the dynamics of isotropic antiferromagnets near the critical point. We perform a renormalization-group analysis of this equation of motion correct to $O(\ensuremath{\epsilon})$, where $\ensuremath{\epsilon}=4\ensuremath{-}d$, finding recursion relations and analyzing their fixed points and their stability. For the one stable fixed point we obtain the dynamical index $z=\frac{d}{2}=2\ensuremath{-}\frac{\ensuremath{\epsilon}}{2}$. We also calculate the response and correlation functions in the scaling region using perturbation theory correct to $O(\ensuremath{\epsilon})$. Our calculation gives analytical forms for the correlation functions for the staggered magnetization and the magnetization. These correlation functions can be written in the dynamical scaling form. The shape function that characterizes the frequency spectrum for the staggered magnetization is Lorentzian in the hydrodynamical regime, but shows fluctuation induced peaks at and near the N\ifmmode\dagger\else\textdagger\fi{}eel temperature. The shape function for the magnetization is essentially Lorentzian for all $x=q\ensuremath{\xi}$ (where qqq is the wave number and $\ensuremath{\xi}$ is the correlation length) and shows a narrow lydrodynamical width as $x\ensuremath{\rightarrow}0$.