Journal of Physics Condensed Matter

The Lyapunov exponent for magnons in binary spin-glass chains

S. N. Evangelou, Anzhong Wang

Published Nov 23, 1992• 4 citations

Abstract

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.

DOI: 10.1088/0953-8984/4/47/002

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Journal of Physics Condensed Matter

The Lyapunov exponent for magnons in binary spin-glass chains

S. N. Evangelou, Anzhong Wang

Published Nov 23, 1992• 4 citations

Publisher

Abstract

DOI: 10.1088/0953-8984/4/47/002

Related papers

Magnons in a one-dimensional spin glass: The high-field and zero-field limits

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Anomalous dynamics in two- and three-dimensional Heisenberg-Mattis spin glasses

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

We investigate the spectral and localization properties of unmagnetized Heisenberg-Mattis spin glasses, in space dimensionalities $d=2$ and $3$ , at $T=0$ . 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 $d=2$ 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 $d=3$ 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.We investigate the spectral and localization properties of unmagnetized Heisenberg-Mattis spin glasses, in space dimensionalities $d=2$ and $3$, at $T=0$. 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 $d=2$ 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 $d=3$ 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.

arXiv:cond-mat/0603043Physical Review BJun 13, 20063 citations

{)}^{1/3}

and the ILL 1/L(E)\ensuremath{\sim}(E-H

{)}^{2/3}

for the symmetric distribution of the exchange interactions (concentration c=0.5). In zero field, the DOS and the ILL are calculated using negative-eigenvalue counting for 0E4J and 0.0001E0.01J. For c=0.5, a connection is established between the zero-field and the high-field limits, and for other concentrations, a phenomenological approach is developed in the low-energy regime where it is found that \ensuremath{\rho}(E)=f(c)${\mathit{E}}^{\mathrm{\ensuremath{-}}1/3}$ and 1/L(E)=g(c)

{\mathit{E}}^{2/3}

[{(J/h)}^{1/z}{]}_{\mathrm{av}}=1

. Particularly, whereas the average spin autocorrelation function in imaginary time decays as $[G{]}_{\mathrm{av}}(\ensuremath{\tau})\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}1/z}$, the average energy-density autocorrelations decay with another exponent as $[{G}^{e}{]}_{\mathrm{av}}(\ensuremath{\tau})\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}2\ensuremath{-}1/z}$.

c<~0.1

and

c>~0.9)

. It is found that low-energy magnons have an anomalous power-law (2/3) behavior in the spectrum for random sign-fixed magnitude exchange interactions. Combining these and previous results, a phenomenological fit is established describing the concentration dependence of the spectrum for a wide range of values of

c

. A discussion based on scaling arguments is presented to account for the anomalous power law for arbitrary concentrations. The general case where the magnitudes and the signs of the exchange interactions and the magnitudes of the spins are all random quantities is further investigated to check whether the anomalous power-law behavior holds. For uniform (flat), non-singular distributions of the random quantities, the 2/3 power in the spectrum is obtained but for singular distributions and mixed distributions (e.g., the magnitude of the spin has a singular distribution while the magnitude of the exchange interaction has a regular distribution) significant deviations from the 2/3 power law occur.

We investigate the spectral and localization properties of unmagnetized Heisenberg-Mattis spin glasses, in space dimensionalities

d=2

and

3

, at

T=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=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=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.