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Register a new userNon-standard anomalous heat conduction in harmonic chains with correlated isotopic disorder
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Added by
J. A. Mendez-Bermudez
Publication date
2019
fields
Physics
and research's language is
English
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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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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 consider heat transport in one-dimensional harmonic chains attached at its ends to Langevin heat baths. The harmonic chain has mass impurities where the separation $d$ between any two successive impurities is randomly distributed according to a power-law distribution $P(d)sim 1/d^{alpha+1}$, being $alpha>0$. In the regime where the first moment of the distribution is well defined ($1<alpha<2$) the thermal conductivity $kappa$ scales with the system size $N$ as $kappasim N^{(alpha-3)/alpha}$ for fixed boundary conditions, whereas for free boundary conditions $kappasim N^{(alpha-1)/alpha}$ if $Ngg1$. When $alpha=2$, the inverse localization length $lambda$ scales with the frequency $omega$ as $lambdasim omega^2 ln omega$ in the low frequency regime, due to the logarithmic correction, the size scaling law of the thermal conductivity acquires a non-closed form. When $alpha>2$, the thermal conductivity scales as in the uncorrelated disorder case. The situation $alpha<1$ is only analyzed numerically, where $lambda(omega)sim omega^{2-alpha}$ which leads to the following asymptotic thermal conductivity: $kappa sim N^{-(alpha+1)/(2-alpha)}$ for fixed boundary conditions and $kappa sim N^{(1-alpha)/(2-alpha)}$ for free boundary conditions.
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.
We study critical behavior of the diluted 2D Ising model in the presence of disorder correlations which decay algebraically with distance as $sim r^{-a}$. Mapping the problem onto 2D Dirac fermions with correlated disorder we calculate the critical properties using renormalization group up to two-loop order. We show that beside the Gaussian fixed point the flow equations have a non trivial fixed point which is stable for $0.995<a<2$ and is characterized by the correlation length exponent $ u= 2/a + O((2-a)^3)$. Using bosonization, we also calculate the averaged square of the spin-spin correlation function and find the corresponding critical exponent $eta_2=1/2-(2-a)/4+O((2-a)^2)$.
We use large-scale Monte Carlo simulations to test the Weinrib-Halperin criterion that predicts new universality classes in the presence of sufficiently slowly decaying power-law-correlated quenched disorder. While new universality classes are reasonably well established, the predicted exponents are controversial. We propose a method of growing such correlated disorder using the three-dimensional Ising model as benchmark systems both for generating disorder and studying the resulting phase transition. Critical equilibrium configurations of a disorder-free system are used to define the two-value distributed random bonds with a small power-law exponent given by the pure Ising exponent. Finite-size scaling analysis shows a new universality class with a single phase transition, but the critical exponents $ u_d=1.13(5), eta_d=0.48(3)$ differ significantly from theoretical predictions. We find that depending on details of the disorder generation, disorder-averaged quantities can develop peaks at two temperatures for finite sizes. Finally, a layer model with the two values of bonds spatially separated to halves of the system genuinely has multiple phase transitions and thermodynamic properties can be flexibly tuned by adjusting the model parameters.
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