low-frequency vibrational modes anomalies and rigidity: a key to understanding the glass and the electronic properties of flexible materials from a topological perspective

Using rigidity (constraint) theory of glasses, the effects of low frequency vibrational modes anomaliesin the glass transition are studied. It is discussed how the possibility of tailoring by chemicaldoping the number of low frequency modes gives clues about how to determine the glass transitiontemperature and glass formation ability. In particular, we present the effects of floppy modes inthe specific heat, entropy , internal energy below glass transition, as well as a discussion of thethermodynamical effects above the glass transition. All the previous results can be extended toinclude the Boson peak, since it can also be understood from a rigidity point of view as a dilutionof bonds in an over-constrained network. Finally, we discuss how a new subject is emerging: floppymodes effects in the electronic properties of flexible systems. Such relationship provides a naturalconnection with topological insulators and two dimensional materials like graphene.