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Physical review. B, Condensed matter

Fracton density of states by the maximum-entropy method

S. N. Evangelou, N.I. Papanicolaou, Eleftherios N. Economou

Published May 1, 1991• 4 citations
Publisher

Abstract

We present a study of the one-magnon density of states in the three-dimensional dilute Heisenberg ferromagnet. The ensemble-averaged density of states is evaluted by calculating a small number of spectral moments and employing a maximum-entropy criterion. Finite-size-lattice data, via common numerical simulation techniques, are also presented. The results demonstrate the fracton excitations at the percolation concentration ${\mathit{p}}_{\mathit{c}}$ as expected from the Alexander-Orbach conjecture and show the presence of the magnon-to-fracton crossover. At the percolation threshold ${\mathit{p}}_{\mathit{c}}$, we estimate a value for the spectral dimension of ${\mathit{d}}_{\mathit{s}}$\ensuremath{\approxeq}1.52. The corresponding problem of phonons in random materials is also discussed in connection with the available experimental data.

DOI
10.1103/physrevb.43.11171

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Single-Length-Scaling Analysis for Antiferromagnetic Fractons in Dilute Heisenberg System RbMn0.4Mg0.6F3

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The dynamic structure factors S ( q ,ω) of an ideal percolating network, the three-dimensional (3 d ) dilute Heisenberg antiferromagnet RbMn 0.4 Mg 0.6 F 3 , obtained from high resoluton (Δ E = 17.5 µeV) inelastic neutron scattering (INS) experiments are analyzed for the first time within the framework of the single-length-scaling postulate (SLSP). The analysis confirms the validity of the SLSP and is also used to extract the values of the key exponents governing the spin dynamics, the dynamic exponent (\(z_{\text{AF}} = D_{\text{f}}/\tilde{d}_{\text{AF}}\)) being 2.5 ±0.1 and the spectral dimension \(\tilde{d}_{\text{AF}}\) for antiferromagnetic (AFM) fractons taking a value of unity.

Journal of the Physical Society of Japan•Dec 25, 2008•3 citations
We present a study of the one-magnon density of states in the three-dimensional dilute Heisenberg ferromagnet. The ensemble-averaged density of states is evaluted by calculating a small number of spectral moments and employing a maximum-entropy criterion. Finite-size-lattice data, via common numerical simulation techniques, are also presented. The results demonstrate the fracton excitations at the percolation concentration pc{\mathit{p}}_{\mathit{c}}pc​ as expected from the Alexander-Orbach conjecture and show the presence of the magnon-to-fracton crossover. At the percolation threshold pc{\mathit{p}}_{\mathit{c}}pc​, we estimate a value for the spectral dimension of ds{\mathit{d}}_{\mathit{s}}ds​\ensuremath{\approxeq}1.52. The corresponding problem of phonons in random materials is also discussed in connection with the available experimental data.
A numerical study of the spin-wave density of states on the infinite percolating cluster of the site diluted Heisenberg ferromagnet is presented. The Gaussian elimination algorithm is employed for the accurate determination of the integrated density of states as a function of concentration in large two-dimensional strips. At the percolation threshold the energy dependence of the density of states is \ensuremath{\rho}(E)\ensuremath{\propto}${E}^{\mathrm{\ensuremath{-}}\ensuremath{\varphi}}$. We obtained the value of \ensuremath{\varphi}=0.32\ifmmode\pm\else\textpm\fi{}0.02 consistent with the presence of fracton excitations at all length scales, in agreement with previous estimates. For concentrations above pc{p}_{c}pc​ the density of states is constant in the magnon or long-wavelength regime while a crossover to the fracton regime occurs at shorter wavelengths. The magnon-fracton crossover in energy is smooth and no indication of a fracton-edge peak is found for this system.
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.
Scaling curves for the density of states and localization length of fractons near a percolation threshold are obtained for dilute and superconducting lattices in both two and three dimensions. These are used to verify the predictions of single-parameter scaling and to show that the energy dependence of the inverse localization length is consistent with known values of the spectral dimensionalities of these structures. The results demonstrate that eigenstates of an infinite, percolating, dilute structure exhibit superlocalization of the form \ensuremath{\psi}(r)\ensuremath{\sim}${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\beta}}\mathit{r}\mathrm{\ensuremath{\delta}}}$ with values of \ensuremath{\delta}=1.13\ifmmode\pm\else\textpm\fi{}0.06 and 1.39\ifmmode\pm\else\textpm\fi{}0.07 in two and three dimensions, respectively.
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.
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 dynamic structure factors S ( q ,ω) of an ideal percolating network, the three-dimensional (3 d ) dilute Heisenberg antiferromagnet RbMn 0.4 Mg 0.6 F 3 , obtained from high resoluton (Δ E = 17.5 µeV) inelastic neutron scattering (INS) experiments are analyzed for the first time within the framework of the single-length-scaling postulate (SLSP). The analysis confirms the validity of the SLSP and is also used to extract the values of the key exponents governing the spin dynamics, the dynamic exponent (zAF=Df/d~AFz_{\text{AF}} = D_{\text{f}}/\tilde{d}_{\text{AF}}zAF​=Df​/d~AF​) being 2.5 ±0.1 and the spectral dimension d~AF\tilde{d}_{\text{AF}}d~AF​ for antiferromagnetic (AFM) fractons taking a value of unity.