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arXiv:cond-mat/0703444 · Physical Review B

Distribution of local Lyapunov exponents in spin-glass dynamics

S. L. A. de Queiroz, Robin B. Stinchcombe

Published Nov 16, 2007• 1 citation
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Abstract

We investigate the statistical properties of local Lyapunov exponents which characterize magnon localization in the one-dimensional Heisenberg-Mattis spin glass (HMSG) at zero temperature by means of a connection to a suitable version of the Fokker-Planck (FP) equation. We consider the local Lyapunov exponents (LLEs), in particular, the case of instantaneous LLE. We establish a connection between the transfer-matrix recursion relation for the problem and an FP equation governing the evolution of the probability distribution of the instantaneous LLE. The closed-form (stationary) solutions to the FP equation are in excellent accord with numerical simulations for both the unmagnetized and magnetized versions of the HMSG. Scaling properties for nonstationary conditions are derived from the FP equation in a special limit (in which diffusive effects tend to vanish), and also shown to provide a close description to the corresponding numerical-simulation data.

DOI
10.1103/physrevb.76.184421

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We investigate the statistical properties of local Lyapunov exponents which characterize magnon localization in the one-dimensional Heisenberg-Mattis spin glass (HMSG) at zero temperature by means of a connection to a suitable version of the Fokker-Planck (FP) equation. We consider the local Lyapunov exponents (LLEs), in particular, the case of instantaneous LLE. We establish a connection between the transfer-matrix recursion relation for the problem and an FP equation governing the evolution of the probability distribution of the instantaneous LLE. The closed-form (stationary) solutions to the FP equation are in excellent accord with numerical simulations for both the unmagnetized and magnetized versions of the HMSG. Scaling properties for nonstationary conditions are derived from the FP equation in a special limit (in which diffusive effects tend to vanish), and also shown to provide a close description to the corresponding numerical-simulation data.
We investigate the spectral and localization properties of unmagnetized Heisenberg-Mattis spin glasses, in space dimensionalities d=2d=2d=2 and 333, at T=0T=0T=0. We use numerical transfer-matrix methods combined with finite-size scaling to calculate Lyapunov exponents, and eigenvalue-counting theorems, coupled with Gaussian elimination algorithms, to evaluate densities of states. In d=2d=2d=2 we find that all states are localized, with the localization length diverging as ${\ensuremath{\omega}}^{\ensuremath{-}1}$, as energy $\ensuremath{\omega}\ensuremath{\rightarrow}0$. Logarithmic corrections to density of states behave in accordance with theoretical predictions. In d=3d=3d=3 the density-of-states dependence on energy is the same as for spin waves in pure antiferromagnets, again in agreement with theoretical predictions, though the corresponding amplitudes differ.
The authors present a calculation of the Lyapunov exponent for transverse magnon modes in a Heisenberg spin-glass chain with positive and negative exchange interactions, both analytically as a microcanonical average and also numerically in the usual canonical statistical ensemble. In the positive- omega regime they find that the localization length obeys xi ( omega ) varies as omega -1 at omega to 0 if p>q, where p, q=1-p are the +1 and -1 spin concentrations, respectively. It is shown that the previously reported xi ( omega ) varies as omega -2/3 singularity, related to the anomalous dispersion law omega varies as k 3/2 , holds only in the limit of equal concentrations p=q=0.5. The corresponding densities of states diverge as rho ( omega ) varies as omega -1/2 and varies as omega -1/3 for p>q and p=q, respectively.
We study integrability breaking and transport in a discrete space-time lattice with a local integrability-breaking perturbation. We find a singular distribution of the Lyapunov spectrum where the majority of Lyapunov exponents vanish in the thermodynamic limit. The subextensive sequence of nonzero exponents, converging in the thermodynamic limit, corresponds to Lyapunov vectors that are exponentially localized with localization lengths proportional to inverse Lyapunov exponents. Moreover, we investigate the transport behavior of the system by considering the spin-spin and current-current spatiotemporal correlation functions. Our results indicate that the overall transport behavior, similar to the purely integrable case, conforms to Kardar-Parisi-Zhang scaling in the thermodynamic limit and at vanishing magnetization. The same dynamical exponent z=3/2 governs the effect of local perturbation spreading in the bulk.
The maximum Lyapunov exponent (MLE) is evaluated as a function of temperature in the isotropic Heisenberg chain. At low temperatures the MLE varies almost quadratically with temperature; it corresponds closely with ${\mathrm{\ensuremath{\tau}}}_{1\mathrm{/}2}^{\mathrm{\ensuremath{-}}1}$, the rate at which the local self-correlation decays to half its initial value. At higher temperatures, the MLE saturates at a value close to 1/2 in the limit of large chain lengths. The strong stochasticity threshold (defined by the change of slope of the MLE) parallels closely the transition from predominantly ballistic to predominantly diffusive behavior of the self-correlation and the concomitant steep increase in ${\mathrm{\ensuremath{\tau}}}_{1\mathrm{/}2}^{\mathrm{\ensuremath{-}}1}$. The complete Lyapunov spectrum has been derived for a chain of 18 spins; deviations from linearity occurring at infinite temperature suggest that the chaoticity of the system is incomplete. Finally, it is suggested that a systematic study of finite-size effects might be useful in deciding the issue of anomalous versus conventional spin diffusion.
We discuss the problem of static chaos in spin glasses. In the case of magnetic-field perturbations, we propose a scaling theory for the spin-glass phase. Using the mean-field approach we argue that some pure states are suppressed by the magnetic field and their free-energy cost is determined by the finite-temperature fixed-point exponents. In this framework, numerical results suggest that mean-field chaos exponents are probably exact in finite dimensions. If we use the droplet approach, numerical results suggest that the zero-temperature fixed-point exponent \ensuremath{\theta} is very close to (d-3)/2. In both approaches d=3 is the lower critical dimension in agreement with recent numerical simulations.
Closed non-linear equations are derived through mode-coupling approximations for the correlation functions and susceptibilities for the Edwards-Anderson model of a classical Heisenberg system with nearest-neighbour random exchange couplings. These equations, which exhibit full dynamical stability, lead to a self-consistent treatment of spin-density propagation and spin-current relaxation. They are found to describe a transition from a paramagnetic to a spin glass phase, which is viewed as one where the dynamics changes from ergodic to a non-ergodic behaviour. Near the critical temperature T c the low-frequency correlation functions are obtained as dynamical scaling laws governed by a critical frequency omega c , which slows down proportionally to epsilon 2 , with epsilon =(T-T c )/T c . Approaching the critical point from the paramagnetic side the diffusivity vanishes linearly with epsilon . The susceptibility exhibits a symmetric cusp. The drastic changes of the spin dynamics at the transition point show up most clearly for the spin-current spectra. The numerical solution of the equations is discussed.