Disordered harmonic chain with random masses and springs: a combinatorial approach

Abstract

We study harmonic chains with i.i.d. random spring constants $K_n$ and i.i.d. random masses $m_n$. We introduce a new combinatorial approach which allows to derive a compact approximate expression for the complex Lyapunov exponent, in terms of the solutions of two transcendental equations involving the distributions of the spring constants and the masses. Our result makes easy the asymptotic analysis of the low frequency properties of the eigenmodes (spectral density and localization) for arbitrary disorder distribution, as well as their high frequency properties. We apply the method to the case of power-law distributions $p(K)=\mu K^{-1+\mu }$ with $0<K<1$ and $q(m)=\nu m^{-1-\nu }$ with $m>1$ (with $\mu ,\nu >0$). At low frequency, the spectral density presents the power law $\varrho (\omega \to 0)\sim {\omega }^{2\eta -1}$, where the exponent $\eta$ exhibits first order phase transitions on the line $\mu =1$ and on the line $\nu =1$. The exponent of the non disordered chain ($\eta =1/2$) is recovered when $⟨K_n^{-1}⟩$ and $⟨m_n⟩$ are both finite, i.e. $\mu >1$ and $\nu >1$. The Lyapunov exponent (inverse localization length) shows also a power-law behaviour $\gamma ({\omega }^2\to 0)\sim {\omega }^{2\zeta }$, where the exponent $\zeta$ exhibits several phase transitions~: the exponent is $\zeta =\eta$ for $\mu <1$ or $\nu <1$ ($⟨K_n^{-1}⟩$ or $⟨m_n⟩$ infinite) and $\zeta =1$ when $\mu >2$ and $\nu >2$ ($⟨K_n^{-2}⟩$ and $⟨m_n^2⟩$ both finite). In the intermediate region it is given by $\zeta =\min (\mu ,\nu )/2$. On the transition lines, $\varrho (\omega )$ and $\gamma ({\omega }^2)$ receive logarithmic corrections. Finally, we also consider the Anderson model with random couplings (random spring chain for ``Dyson type I'' disorder).

28 pages, 7 figures, revtex

References in corpus (19)

• Anomalous Localization in Low-Dimensional Systems with Correlated Disorder
• Bosonic Excitations in Random Media
• Delocalization in harmonic chains with long-range correlated random masses
• Functionals of the Brownian motion, localization and metric graphs
• Lyapunov exponents, one-dimensional Anderson localisation and products of random matrices
• Exponential number of equilibria and depinning threshold for a directed polymer in a random potential
• The Lyapunov exponent of products of random $2\times 2$ matrices close to the identity
• Ground state energy of noninteracting fermions with a random energy spectrum
• One-dimensional disordered quantum mechanics and Sinai diffusion with random absorbers
• Supersymmetric quantum mechanics with Levy disorder in one dimension
• Scaling of the localization length in linear electronic and vibrational systems with long-range correlated disorder
• Effect of boundaries on the spectrum of a one-dimensional random mass Dirac Hamiltonian
• Lifshitz tails on the Bethe lattice: a combinatorial approach
• Localization for one-dimensional random potentials with large local fluctuations
• Fluctuations of the product of random matrices and generalized Lyapunov exponent
• Localization properties of harmonic chains with correlated mass and spring disorder: Analytical approach
• Anomalous scaling of heterogeneous elastic lines: a new picture from sample to sample fluctuations
• The generalized Lyapunov exponent for the one-dimensional Schrödinger equation with Cauchy disorder: some exact results
• Aspects of the disordered harmonic chain