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Phasons are collective low-energy modes that appear in disparate condensedmatter systems such as quasicrystals, incommensurate structures, fluctuatingcharge density waves, and Moir\'e superlattices. They share severalsimilarities with acoustic phonon modes, but they are not protected by anyexact translational symmetry. As a consequence, they are subject to awavevector independent damping, and they develop a finite pinning frequency,which destroy their acoustic linearly propagating dispersion. Under a few andsimple well-motivated assumptions, we compute the phason density of states, andwe derive the phason heat capacity as a function of the temperature. Finally,imagining a hypothetical s-wave pairing channel with electrons, we compute thecritical temperature $T_c$ of the corresponding superconducting state as afunction of phason damping using the Eliashberg formalism. We find that forlarge phason damping, the heat capacity is linear in temperature, showing adistinctive glass-like behavior. Additionally, we observe that the phasondamping can strongly enhance the effective Eliashberg coupling, and we reveal asharp non-monotonic dependence of the superconducting temperature $T_c$ on thephason damping, with a maximum located at the underdamped to overdampedcrossover scale. Our simple computations confirm the potential role ofoverdamped modes in explaining the glassy properties of incommensuratestructures, but also in possibly inducing strongly-coupled superconductivitytherein, and enhancing the corresponding $T_c$.

Glassy heat capacity from overdamped phasons and a hypothetical phason-induced superconductivity in incommensurate structures