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Physical review. B, Condensed matter

Magnon-fracton crossover of the spin-wave density of states in the diluted two-dimensional ferromagnet

S. N. Evangelou

Published Mar 1, 1986• 12 citations
Publisher

Abstract

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 ${p}_{c}$ 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.

DOI
10.1103/physrevb.33.3602

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Journal of Physics A Mathematical and General•Jan 21, 1990•1 citation
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\ensuremath−\ensuremathφ{E}^{\mathrm{\ensuremath{-}}\ensuremath{\varphi}}E\ensuremath−\ensuremathφ. 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.
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.
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.
The two-dimensional antiferromagnetic S=1/2S=1/2S=1/2 Heisenberg model with random site dilution is studied using quantum Monte Carlo simulations. Ground-state properties of the largest connected cluster on $L\ifmmode\times\else\texttimes\fi{}L$ lattices, with L up to 64, are calculated at the classical percolation threshold. In addition, clusters with a fixed number Nc{N}_{c}Nc​ of spins on an infinite lattice at the percolation density are studied for Nc{N}_{c}Nc​ up to 1024. The disorder averaged sublattice magnetization per spin extrapolates to the same nonzero infinite-size value for both types of clusters. Hence, the percolating clusters, which are fractal with dimensionality d=91/48,d=91/48,d=91/48, have antiferromagnetic long-range order. This implies that the order-disorder transition driven by site dilution occurs exactly at the percolation threshold and that the exponents are classical. The same conclusion is reached for the bond-diluted system. The full sublattice magnetization versus site dilution curve is obtained in terms of a decomposition into a classical geometrical factor and a factor containing all the effects of quantum fluctuations. The spin stiffness is shown to obey the same scaling as the conductivity of a random resistor network.
The decay of zero temperature spin wave excitations near the percolation threshold in bond-diluted d-dimensional (d>1) Heisenberg ferromagnets is investigated using a Green function equation-of-motion method. The diagrammatic perturbation theory is convergent in the hydrodynamic limit (k xi <<1) and the following expression is obtained there for the damping, or decay rate: Gamma c (k, xi ) varies as (p-p c ) -vd+(t- beta ) k d+2 ,(k xi <<1). This form satisfies the requirements of the dynamic scaling principle, by which Gamma c (k, xi )=k z g(k xi ) for general k xi , and z=2+(t- beta )/ nu . An explicit expression, applicable in the hydrodynamic regime and consistent with dynamic scaling, is also obtained for the infinite cluster transverse dynamic structure function. The approach is extended to incorporate the ensemble of finite clusters and to treat the case of non-zero temperature and also the experimentally more relevant situation of site dilution.
Quantum phase transition of site-diluted and bond-diluted Heisenberg antiferromagnets on square lattices is studied. By using the novel continuous-time loop algorithm, we perform quantum Monte Carlo simulations on quite larger lattices at extremely lower temperatures than the previous numerical studies. It is found that the antiferromagnetic long-range order at T = 0 persists so long as a cluster of magnetic sites percolates, that is, the critical concentration is equal to the classical percolation threshold, in both of the site-diluted and bond-diluted cases. Furthermore, we find that some critical exponents, such as the magnetization exponent β, are non-classical and strongly depend on the spin size S. On the other hand, we show that the correlation-length exponent ν is universal and is equal to the classical value (ν=4/3).
The authors use a truncated replicated nonlinear sigma model to study crossover in the vicinity of the percolation threshold of a randomly diluted classical m-component Heisenberg model with m>2. They find that the order parameter correlation length xi diverges with temperature T for concentration p of occupied bonds greater than the percolation threshold p c as exp(2 pi rho s (p)/T) where rho s (p) approximately mod p-p c mod phi is the spin stiffness and for p=p c as T - phi vp/ where v p is the percolation correlation length exponent. The numerical values of the exponents phi and v p predicted in their truncation scheme, however, differ significantly from their accepted values.