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Register a new userDelocalization in harmonic chains with long-range correlated random masses
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We study the nature of collective excitations in harmonic chains with masses exhibiting long-range correlated disorder with power spectrum proportional to $1/k^{alpha}$, where $k$ is the wave-vector of the modulations on the random masses landscape. Using a transfer matrix method and exact diagonalization, we compute the localization length and participation ratio of eigenmodes within the band of allowed energies. We find extended vibrational modes in the low-energy region for $alpha > 1$. In order to study the time evolution of an initially localized energy input, we calculate the second moment $M_2(t)$ of the energy spatial distribution. We show that $M_2(t)$, besides being dependent of the specific initial excitation and exhibiting an anomalous diffusion for weakly correlated disorder, assumes a ballistic spread in the regime $alpha>1$ due to the presence of extended vibrational modes.
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We study the one-dimensional quantum Heisenberg ferromagnet with exchange couplings exhibiting long-range correlated disorder with power spectrum proportional to $1/k^{alpha}$, where $k$ is the wave-vector of the modulations on the random coupling landscape. By using 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 $alpha > 1$, the wave-packet mean-square displacement changes from a long-time super-diffusive behavior for $alpha <1$ to a long-time ballistic behavior for $alpha > 1$. At the vicinity of $alpha =1$, the mobility edge separating the extended and localized phases is shown to scale with the degree of correlation as $E_cpropto (alpha -1)^{1/3}$.
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 $alpha$, 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 $alpha$. At $alpha_c$ 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.
In the previous paper, we studied the random-mass Dirac fermion in one dimension by using the transfer-matrix methods and by introducing an imaginary vector potential in order to calculate the localization lengths. Especially we considered effects of the nonlocal but short-range correlations of the random mass. In this paper, we shall study effects of the long-range correlations of the random mass especially on the delocalization transition. The results depend on how randomness is introduced in the Dirac mass.
We consider heat transport in one-dimensional harmonic chains with isotopic disorder, focussing our attention mainly on how disorder correlations affect heat conduction. Our approach reveals that long-range correlations can change the number of low-frequency extended states. As a result, with a proper choice of correlations one can control how the conductivity $kappa$ scales with the chain length $N$. We present a detailed analysis of the role of specific long-range correlations for which a size-independent conductivity is exactly recovered in the case of fixed boundary conditions. As for free boundary conditions, we show that disorder correlations can lead to a conductivity scaling as $kappa sim N^{varepsilon}$, with the scaling exponent $varepsilon$ being arbitrarily small (although not strictly zero), so that normal conduction is almost recovered even in this case.
We address the general problem of heat conduction in one dimensional harmonic chain, with correlated isotopic disorder, attached at its ends to white noise or oscillator heat baths. When the low wavelength $mu$ behavior of the power spectrum $W$ (of the fluctuations of the random masses around their common mean value) scales as $W(mu)sim mu^beta$, the asymptotic thermal conductivity $kappa$ scales with the system size $N$ as $kappa sim N^{(1+beta)/(2+beta)}$ for free boundary conditions, whereas for fixed boundary conditions $kappa sim N^{(beta-1)/(2+beta)}$; where $beta>-1$, which is the usual power law scaling for one dimensional systems. Nevertheless, if $W$ does not scale as a power law in the low wavelength limit, the thermal conductivity may not scale in its usual form $kappasim N^{alpha}$, where the value of $alpha$ depends on the particular one dimensional model. As an example of the latter statement, if $W(mu)sim exp(-1/mu)/mu^2$, $kappa sim N/(log N)^3$ for fixed boundary conditions and $kappa sim N/log(N)$ for free boundary conditions, which represent non-standard scalings of the thermal conductivity.
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