The Quantum Valley Hall Effect (QVHE) is a topological phenomenon that can emerge in electronic or phononic bands of hexagonal materials. Typically, this effect requires a material to possess both an energy gap in its phonon or electron dispersion and a specific type of line defect in real space cite{Yao2009}.

In this work, we show that the phonon spectrum of graphane (fully hydrogenated graphene) can host topologically protected states along grain boundaries if they reverse the sublattices (textit{i.e.}, the hydrogenation pattern is swapped). These states are a manifestation of the QVHE. However, unlike any other version of the QVHE, there is no need to open a global band gap in the spectrum. This finding can be generalized to other systems, whether electrons or phonons, broadening the range of materials that could exhibit such topological phenomena.

Our calculations are based on tight-binding-like Hamiltonian models and supported by density functional calculations.

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bibitem{Yao2009} Yao, W., Yang, S. A., & Niu, Q. (2009). Edge states in graphene: From gapped flat-band to gapless chiral modes. textit{Physical Review Letters}, 102(9), 096801.
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