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arXiv:cond-mat/0206038 · Physical review. B, Condensed matter

Delocalization and spin-wave dynamics in ferromagnetic chains with long-range correlated random exchange

F. A. B. F. de Moura, Mauricio D. Coutinho-Filho, Ernesto P. Raposo, M. L. Lyra

Published Jul 8, 2002• 60 citations
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Abstract

We study the one-dimensional quantum Heisenberg ferromagnet with exchange couplings exhibiting long-range correlated disorder with power spectrum proportional to ${1/k}^{\ensuremath{\alpha}},$ where k is the wave vector of the modulations on the random coupling landscape. By using the renormalization group, integration of the equations of motion, and exact diagonalization, we compute the spin-wave localization length and the mean-square displacement of the wave packet. We find that, associated with the emergence of extended spin waves in the low-energy region for \ensuremath{\alpha}>1, the wave-packet mean-square displacement changes from a long-time super diffusive behavior for \ensuremath{\alpha}1 to a long-time ballistic behavior for \ensuremath{\alpha}>1. At the vicinity of \ensuremath{\alpha}=1, the mobility edge separating the extended and localized phases is shown to scale with the degree of correlation as ${E}_{c}\ensuremath{\propto}(\ensuremath{\alpha}\ensuremath{-}{1)}^{1/3}.$

DOI
10.1103/physrevb.66.014418

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We study one-magnon excitations in a random ferromagnetic Heisenberg chain with long-range correlations in the coupling constant distribution. By employing an exact diagonalization procedure, we compute the localization length of all one-magnon states within the band of allowed energies E. The random distribution of coupling constants was assumed to have a power spectrum decaying as $S(k)\ensuremath{\propto}{1/k}^{\ensuremath{\alpha}}.$ We found that for $\ensuremath{\alpha}<1,$ one-magnon excitations remain exponentially localized with the localization length $\ensuremath{\xi}$ diverging as 1/E.1/E.1/E. For $\ensuremath{\alpha}=1$ a faster divergence of $\ensuremath{\xi}$ is obtained. For any $\ensuremath{\alpha}>1,$ a phase of delocalized magnons emerges at the bottom of the band. We characterize the scaling behavior of the localization length on all regimes and relate it with the scaling properties of the long-range correlated exchange coupling distribution.
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While there are well-established methods to study delocalization transitions of single particles in random systems, it remains a challenging problem how to characterize many-body delocalization transitions. Here, we use a generalized real-space renormalization group technique to study the anisotropic Heisenberg model with long-range interactions, decaying with a power α , which are generated by placing spins at random positions along the chain. This method permits a large-scale finite-size scaling analysis. We examine the full distribution function of the excitation energy gap from the ground state and observe a crossover with decreasing α . At α c the full distribution coincides with a critical function. Thereby, we find strong evidence for the existence of a many-body localization transition in disordered antiferromagnetic spin chains with long-range interactions.
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We calculate the ground-state two-spin correlation functions of spin-12\frac{1}{2}21​ quantum Heisenberg chains with random exchange couplings using the real-space renormalization-group scheme. We extend the conventional scheme to take account of the contribution of local higher multiplet excitations in each decimation step. This extended scheme can provide highly accurate numerical data for large systems. The random average of staggered spin correlations of the chains with random antiferromagnetic (AF) couplings shows algebraic decay like 1/r2,{1/r}^{2},1/r2, which verifies the Fisher's analytic results. For chains with random ferromagnetic (FM) and AF couplings, the random average of generalized staggered correlations is found to decay more slowly than a power law, in the form close to 1/ln(r).1/\mathrm{ln}(r).1/ln(r). The difference between the distribution functions of the spin correlations of the random AF chains and of the random FM-AF chains is also discussed.
We study the ``flux-noise'' spectrum of random-bond quantum Heisenberg spin systems using a real-space renormalization group (RSRG) procedure that accounts for both the renormalization of the system Hamiltonian and of a generic probe that measures the noise. For spin chains, we find that the dynamical structure factor Sq(f){S}_{q}(f)Sq​(f), at finite wave vector qqq, exhibits a power-law behavior both at high and low frequencies fff, with exponents that are connected to one another and to an anomalous dynamical exponent through relations that differ at T=0T=0T=0 and $T=\ensuremath{\infty}$. The low-frequency power-law behavior of the structure factor is inherited by any generic probe with a finite bandwidth and is of the form $1/{f}^{\ensuremath{\alpha}}$ with $0.5<\ensuremath{\alpha}<1$. An analytical calculation of the structure factor, assuming a limiting distribution of the RG flow parameters (spin size, length, bond strength) confirms numerical findings. More generally, we demonstrate that this form of the structure factor, at high temperatures, is a manifestation of anomalous diffusion which directly follows from a generalized spin-diffusion propagator. We also argue that 1/f1/f1/f-noise is intimately connected to many-body-localization at finite temperatures. In two dimensions, the RG procedure is less reliable; however, it becomes convergent for quasi-one-dimensional geometries where we find that one-dimensional $1/{f}^{\ensuremath{\alpha}}$ behavior is recovered at low frequencies; the latter configurations are likely representative of paramagnetic spin networks that produce $1/{f}^{\ensuremath{\alpha}}$ noise in SQUIDs.