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Fractals

DYNAMICAL STRUCTURE FACTOR OF HEISENBERG ANTIFERROMAGNETS WITH FRACTAL STRUCTURES

Takamichi Terao, Kousuke Yakubo, Tsuneyoshi Nakayama

Published Dec 1, 1993• 0 citations
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Abstract

The dynamical structure factor S(q, ω) of Heisenberg antiferromagnets with random fractal structures is investigated in terms of a dynamical scaling argument. It is found that the asymptotic behavior of S(q, ω) can be characterized by the dynamic exponent za and a new exponent y. We have performed numerical calculations of S(q, ω) in order to confirm its predicted scaling behaviors for d=2 percolating antiferromagnets. The calculated values for the exponents y and za are consistent with our theoretical prediction.

DOI
10.1142/s0218348x93000964

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The dynamical structure factor S(q, ω) of Heisenberg antiferromagnets with random fractal structures is investigated in terms of a dynamical scaling argument. It is found that the asymptotic behavior of S(q, ω) can be characterized by the dynamic exponent za and a new exponent y. We have performed numerical calculations of S(q, ω) in order to confirm its predicted scaling behaviors for d=2 percolating antiferromagnets. The calculated values for the exponents y and za are consistent with our theoretical prediction.
The scaling behavior of the dynamical structure factor S(q,\ensuremath{\omega}) of percolating Heisenberg antiferromagnets is investigated in terms of a dynamic scaling argument and numerical calculations. It is found, based on the single-length-scaling postulate, that the asymptotic behavior of S(q,\ensuremath{\omega}) can be characterized by only two exponents: the dynamic exponent za{\mathit{z}}_{\mathit{a}}za​ and a new exponent y. This theoretical prediction is confirmed by direct numerical calculations. The values of the exponents za{\mathit{z}}_{\mathit{a}}za​ and y are determined for the case of d=2 percolating Heisenberg antiferromagnets.
The fracton dynamics of percolating Heisenberg antiferromagnets are studied in terms of a dynamic scaling argument and large-scale numerical analysis. It is shown that the spectral (or fracton) dimension for antiferromagnetic fractons, d~AF\tilde{d}_{\rm AF}d~AF​, is bounded by the relation d~AF≤1\tilde{d}_{\rm AF}\le 1d~AF​≤1. The densities of states (DOS) are calculated for very large percolating antiferromagnets at the critical concentration p c formed on d =2, 3, and 4 simple cubic lattices. It is concluded, from the calculated DOS's, that the spectral dimensions d~AF\tilde{d}_{\rm AF}d~AF​ are very close to unity for any Euclidean dimension d (≥2). We have also calculated the DOS for percolating antiferromagnets above p c in order to clarify the crossover behavior from extended magnons to fractons. In addition, the dynamical structure factor S ( q ,ω) is investigated both analytically and numerically. It is found, postulating a single-length scaling, that the asymptotic behaviors can be characterized by two exponents, the dynamical exponent z AF and a new exponent y . The numerical results for S ( q ,ω) support the validity of the single-length-scale postulate.
We report the observations of the dynamical structure factors S(q,ω) for antifferomagnetic fractons in diluted three-dimensional (3d) and two-dimensional (2d) Heisenberg systems, RbMn0.4Mg0.6F3 and Rb2Mn0.598Mg0.402F4, with a magnetic concentration close to the percolation concentration, which were obtained by means of high-resolution (ΔE = 17.5 μeV) inelastic neutron scattering experiments. The peak intensity A(q) and the dispersion relation E(q) show the clear scaling laws following to A(q) ∝ q−y with y = 2.9 ± 0.1 and E(q) ∝ qz with z = 2.5 ± 0.1 for the 3d system, and y = 2.9 ± 0.2 and z = 1.8 ± 0.2 for the 2d system. The validity of the single-length-scaling postulate (SLSP) for S(q,ω) are demonstrated, for the first time. In addition, we show that the spectral dimension of antiferromagnetic fractons is unity independent of the embedding Euclidean dimension of the systems. These values are consistent with the theoretical predictions.
The dynamical structure factors, S ( q ,ω), for antiferromagnetic fractons in diluted three-dimensional (3D) and two-dimensional (2D) Heisenberg systems, RbMn 0.4 Mg 0.6 F 3 and Rb 2 Mn 0.598 Mg 0.402 F 4 , respectively, with a magnetic concentration close to the percolation concentration, were observed through high-resolution (Δ E = 17.5 µeV) inelastic neutron scattering experiments well below T N . The exponents describing the peak intensity ( A ( q ) ∝ q - y ) and the dispersion relation ( E ( q ) ∝ q z ) were determined to be y = 2.9 ±0.1 and z = 2.5 ±0.1 for the 3D system and y = 2.9 ±0.2 and z = 1.8 ±0.2 for the 2D system, respectively. In addition, the validity of the single-length-scaling postulate (SLSP) was confirmed for S ( q ,ω) by using the above values of y and z . We experimentally demonstrated, for the first time, that the spectral dimension, d~AF=1\tilde{d}_{\text{AF}} = 1d~AF​=1, is universal and independent of the embedding Euclidean dimensions of the systems.
The dynamic structure factor of very large percolating clusters (bond and site), in two dimensions and three dimensions are calculated using the spectral moments method. Interactions are represented by the scalar model. Numerical results are presented and interpreted in terms of scaling arguments given by Alexander, Courtens and Vacher (ACV) in 1993. Concerning the q lambda <<1 limit, our results confirm and supplement previous numerical work and are in agreement with the scaling behaviour theoretically deduced by ACV. Concerning the q lambda >>1 limit, we have shown that the dynamic structure factor complies with the very nice asymptotic behaviour g(q, omega )=q y H(q lambda ( omega )) where the scaling function H(x) is of power-law form x - tau ' in agreement with the theory. However, our results indicate a scaling behaviour with exponents that differ from those deduced by theory. The values obtained for tau ' are 1.20 for the two-dimensional bond and site percolating lattices, and 1.00 for the three-dimensional percolating lattices. Comparisons with previous simulations are reported.
The critical spin-wave dynamics of dilute ferromagnetic and antiferromagnetic Heisenberg systems near the percolation threshold is studied on a two-dimensional regular (fractal) model for bond percolation, which captures the relevant geometric features of the cluster structure. The dynamic critical exponent and spectral dimension for both the ferromagnetic and antiferromagnetic systems are calculated. The treatment of the dynamics in these systems requires the use of generalised scaling techniques for sublattice systems, which involve an extension of the parameter space.