Antiferromagnetic fluctuations in the very underdoped high-temperature superconductor HgBa$_2$CuO$_{4+\delta}$

We report inelastic neutron scattering measurements of magnetic fluctuations over a large energy and momentum range in the high-temperature cuprate superconductor HgBa$_2$CuO$_{4+\delta}$ (Hg1201) at two low doping levels (UD45: T$\mathrm{_c}$$\approx$45K, p$\approx$0.058; UD55: T$\mathrm{_c}$$\approx$55K, p$\approx$0.063). In both samples, the ``hourglass'' dispersion, thought to be universal among the cuprates, is not observed. Instead, the antiferromagnetic spectrum is commensurate just above the magnetic gap ($\sim$10 meV in both samples) and disperses outwards into a ring of scattering above $\sim$50 meV. The magnetic resonance is prominently observed in UD45 (at $\sim$20meV), but is small or non-existent in higher-doped UD55. This result runs counter to the heretofore accepted notion that the resonance is most prominent in the compounds with the highest optimal T$\mathrm{_c}$. Additionally, we find that the previously reported Ising-like dispersionless excitations in optimal and moderately underdoped Hg1201 is no longer observed in UD45. We conclude that there exists a crossover near p$\sim$0.06 between two distinct regimes.