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

TL;DR

This work develops a novel combinatorial framework to analyze a disordered harmonic chain with i.i.d. random masses $m_n$ and spring constants $K_n$, enabling a compact approximate expression for the complex Lyapunov exponent $\Omega(\lambda)$ via two transcendental fugacity equations. The method yields explicit low- and high-frequency asymptotics and reveals a rich phase diagram for power-law disorder, including phase transitions at $\mu=1$ and $\nu=1$ and universal scaling exponents $\eta$ and $\zeta$ for spectral density and localization length. By connecting the spectral determinant to a fermionic random energy model and exploiting a duality between $1/K_n$ and $m_n$, the approach also extends to Dyson-type random hopping problems (Anderson model with random couplings). The results reproduce and generalize classical findings (e.g., Matsuda–Ishii, Dyson) in new regimes, providing a versatile tool for 1D disordered systems where conventional weak-disorder methods fail due to heavy-tailed disorder. Overall, the work offers a transparent, transport-relevant picture of how disorder statistics control vibrational spectra and localization in 1D chains, with potential applications to nanoscale materials and wave-propagation in random media.

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 $p(K)=μ\,K^{-1+μ}$ with $0<K<1$ and $q(m)=ν\,m^{-1-ν}$ with $m>1$ (with $μ,\:ν>0$). At low frequency, the spectral density presents the power law $\varrho(ω\to0)\simω^{2η-1}$, where the exponent $η$ exhibits first order phase transitions on the line $μ=1$ and on the line $ν=1$. The exponent of the non disordered chain ($η=1/2$) is recovered when $\langle K_n^{-1}\rangle$ and $\langle m_n\rangle$ are both finite, i.e. $μ>1$ and $ν>1$. The Lyapunov exponent (inverse localization length) shows also a power-law behaviour $γ(ω^2\to0)\simω^{2ζ}$, where the exponent $ζ$ exhibits several phase transitions~: the exponent is $ζ=η$ for $μ<1$ or $ν<1$ ($\langle K_n^{-1}\rangle$ or $\langle m_n\rangle$ infinite) and $ζ=1$ when $μ>2$ and $ν>2$ ($\langle K_n^{-2}\rangle$ and $\langle m_n^2\rangle$ both finite). In the intermediate region it is given by $ζ=\mathrm{min}(μ,ν)/2$. On the transition lines, $\varrho(ω)$ and $γ(ω^2)$ receive logarithmic corrections. Finally, we also consider the Anderson model with random couplings (random spring chain for ``Dyson type I'' disorder).

Figures (7)

• Figure 1: IDoS obtained numerically. In orange : the case of exponential distributions (with $\left\langle K \right\rangle=\left\langle 1/m \right\rangle=\infty$) shows a slow convergence towards unity at high frequency. In blue : the IDoS for power-law distribution (with $\left\langle K \right\rangle<\infty$ and $\left\langle 1/m \right\rangle<\infty$) for $\mu=3/4$ and $\nu=5/2$ exhibits a rapid convergence (Lifshitz tail). Black dashed lines are our analytical asymptotic predictions (no adjustable parameter).
• Figure 2: Comparison between the exact result \ref{['eq:ComplexLyapForExpDistributions']} (dashed line) and the approximate result of the combinatorial approach \ref{['eq:ExpCase-Omega']} (continuous line), from a numerical resolution of the transcendental equation for $\theta_*$, i.e. Eqs. (\ref{['eq:ExpCase-EqForPsi']},\ref{['eq:ExpCase-SaddleEq']}). Inset : same with the dominant $\mathcal{O}(\ln(-\lambda))$ term substracted, with comparison to the asymptotics \ref{['eq:OmegaForExpWeights']} (orange dashed line).
• Figure 3: Coefficient $c_\mu$ (blue line) controlling the exact asymptotic behaviour and coefficient $\tilde{c}_\mu$ obtained from our combinatorial method (dashed red line). Inset : relative difference $(c_\mu-\tilde{c}_\mu)/c_\mu$ does not exceed 1.6%.
• Figure 4: Lyapunov exponent obtained numerically for a chain of $10^5$ random masses and spring constants, as a function of $\lambda>0$ for different $\mu$ and $\nu$. Dashed lines correpond to the expected power law $\gamma(\lambda)\sim\lambda^\zeta$ (no adjustable parameter : the prefactor is the one given in the text).
• Figure 5: Exponent $\zeta$ controlling the $\lambda\to0^+$ behaviour of the Lyapunov exponent as a function of $\mu$ for different values of $\nu$.