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

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 $\varrho(\omega\to0)\sim\omega^{2\eta-1}$, 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 $\langle K_n^{-1}\rangle$ and $\langle m_n\rangle$ are both finite, i.e. $\mu>1$ and $\nu>1$. The Lyapunov exponent (inverse localization length) shows also a power-law behaviour $\gamma(\omega^2\to0)\sim\omega^{2\zeta}$, where the exponent $\zeta$ exhibits several phase transitions~: the exponent is $\zeta=\eta$ for $\mu<1$ or $\nu<1$ ($\langle K_n^{-1}\rangle$ or $\langle m_n\rangle$ infinite) and $\zeta=1$ when $\mu>2$ and $\nu>2$ ($\langle K_n^{-2}\rangle$ and $\langle m_n^2\rangle$ both finite). In the intermediate region it is given by $\zeta=\mathrm{min}(\mu,\nu)/2$. 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).