Real space solution of inhomogeneous elastic wave equation with localized vibration and flat dispersion relation

TL;DR

This paper presents a real space solution to the inhomogeneous elastic wave equation, revealing localized vibrations and flat dispersion relations associated with the Boson peak, thereby clarifying its origin in disordered systems.

Contribution

It introduces a non-perturbative real space method to analyze vibrational properties, unifying previous conflicting theories of the Boson peak in a simple fluctuating shear modulus model.

Findings

Observation of localized exponential decay vibrations in soft spots

Confirmation of shear modulus fluctuation length dependence on BP frequency

Identification of flat dispersion relation linked to Boson peak

Abstract

The low frequency vibrational anomaly known as Boson peak (BP) have been studied extensively in various disordered systems, however its origin and theoretical description are still under debate. In this work, as one of the simplest model for describing vibrational properties in disordered systems, inhomogeneous elastic wave equation, is solved in real space without using perturbative approach as previous works. In real space solution, the BP associated flat dispersion relation can be obtained, localized vibration in exponential decay in soft spot can be observed, and the fluctuation length of shear modulus dependent BP frequency is also confirmed. These features have been reported in recent progresses but missed within perturbative approach. This work unify divergent and controversial conclusions of BP within a simple model of fluctuating shear modulus under clear visualization.