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Disordered harmonic chains with random masses and springs: A combinatorial approach.

Maximilien Bernard^1,2, Christophe Texier¹

¹LPTMS, Université Paris-Saclay, CNRS, 91405 Orsay, France.

Physical Review. E

|February 20, 2026

Summary

This study introduces a new combinatorial method to analyze disordered harmonic chains. The approach yields a general formula for the Lyapunov exponent, simplifying the study of spectral density and localization properties.

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Area of Science:

Condensed Matter Physics
Statistical Mechanics
Disordered Systems

Background:

Harmonic chains with random spring constants and masses exhibit complex behaviors.
Understanding spectral density and localization is crucial for disordered systems.

Purpose of the Study:

To develop a general approximate formula for the complex Lyapunov exponent in disordered harmonic chains.
To enable asymptotic analysis of low- and high-frequency properties of eigenmodes.
To investigate phase transitions in spectral density and localization exponents.

Main Methods:

A novel combinatorial approach is introduced.
Derivation of a compact formula for the complex Lyapunov exponent.
Application to power-law distributions for spring constants and masses.

Main Results:

The spectral density exhibits a low-frequency power law ϱ(ω→0)∼ω^{2η-1} with phase transitions at μ=1 and ν=1.
The Lyapunov exponent shows power-law behavior γ(ω^{2}→0)∼ω^{2ζ}, with transitions at ζ=η, ζ=1, and ζ=min(μ,ν)/2.
Logarithmic corrections appear on transition lines; the Anderson model is also considered.

Conclusions:

The combinatorial method provides a versatile tool for analyzing disordered harmonic chains.
The study reveals detailed insights into spectral density and localization phenomena.
Phase transitions and critical exponents are characterized for various disorder distributions.