# Role of disorder in determining the vibrational properties of mass-spring networks

## TL;DR

This study investigates how different types of disorder affect vibrational properties in mass-spring networks, revealing the complex origins of the boson peak and its relation to Ioffe-Regel limits.

## Contribution

It systematically analyzes the effects of four fundamental disorders on vibrational spectra, clarifying the conditions under which the boson peak relates to Ioffe-Regel limits.

## Key findings

- Boson peak and Ioffe-Regel limits emerge with disorder.
- Increasing disorder lowers characteristic frequencies.
- Relationship between boson peak and Ioffe-Regel limit is condition-dependent.

## Abstract

By introducing four fundamental types of disorders into a two-dimensional triangular lattice separately, we determine the role of each type of disorder in the vibration of the resulting mass-spring networks. We are concerned mainly with the origin of the boson peak and the connection between the boson peak and the transverse Ioffe-Regel limit. For all types of disorders, we observe the emergence of the boson peak and Ioffe-Regel limits. With increasing disorder, the boson peak frequency $\omega_{BP}$, transverse Ioffe-Regel frequency $\omega_{IR}^T$, and longitudinal Ioffe-Regel frequency $\omega_{IR}^L$ all decrease. We find that there are two ways for the boson peak to form: developing from and coexisting with (but remaining independent of) the transverse van Hove singularity without and with local coordination number fluctuation. In the presence of a single type of disorder, $\omega_{IR}^T\ge \omega_{BP}$, and $\omega_{IR}^T\approx \omega_{BP}$ only when the disorder is sufficiently strong and causes spatial fluctuation of the local coordination number. Moreover, if there is no positional disorder, $\omega_{IR}^T\approx \omega_{IR}^L$. Therefore, the argument that the boson peak is equivalent to the transverse Ioffe-Regel limit is not general. Our results suggest that both local coordination number and positional disorder are necessary for the argument to hold, which is actually the case for most disordered solids such as marginally jammed solids and structural glasses. We further combine two types of disorders to cause disorder in both the local coordination number and lattice site position. The density of vibrational states of the resulting networks resembles that of marginally jammed solids well. However, the relation between the boson peak and the transverse Ioffe-Regel limit is still indefinite and condition-dependent.