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The cuprates exhibit a prominent charge-density-wave (CDW) instability withwavevector along [100], i.e., the Cu-O bond direction. Whereas CDW order ismost prominent at moderate doping and low temperature, there exists increasingevidence for dynamic charge correlations throughout a large portion of thetemperature-doping phase diagram. In particular, signatures of incipient chargeorder have been observed as phonon softening and/or broadening near the CDWwavevector approximately half-way through the Brillouin zone. Most of this workhas focused on moderately-doped cuprates, for which the CDW order is robust, oron optimally-doped samples, for which the superconducting transitiontemperature ($T_c$) attains its maximum. Here we present a time-of-flightneutron scattering study of phonons in simple-tetragonal$\text{HgBa}_2\text{CuO}_{4+\delta}$ ($T_c = 55$ K) at a low doping level whereprior work showed the CDW order to be weak. We employ and showcase a newsoftware-based technique that mines the large number of measured Brillouinzones for useful data in order to improve accuracy and counting statistics.Density-functional theory has not provided an accurate description of phononsin $\text{HgBa}_2\text{CuO}_{4+\delta}$, yet we find the right set ofparameters to qualitatively reproduce the data. The notable exception is adispersion minimum in the longitudinal Cu-O bond-stretching branch along [100].This discrepancy suggests that, while CDW order is weak, there existsignificant dynamic charge correlations in the optic phonon range at lowdoping, near the edge of the superconducting dome.

Phonon spectrum of underdoped $\text{HgBa}_2\text{CuO}_{4+\delta}$ investigated by neutron scattering