Spin fluctuations associated with the collapse of the pseudogap in a cuprate superconductor

Theories of the origin of superconductivity in cuprates are dependent on an understanding of their normal state which exhibits various competing orders. Transport and thermodynamic measurements on La$_{2-x}$Sr$_x$CuO$_4$ show signatures of a quantum critical point, including a peak in the electronic specific heat $C$ versus doping $p$, near the doping $p^{\star}$ where the pseudogap collapses. The fundamental nature of the fluctuations associated with this peak is unclear. Here we use inelastic neutron scattering to show that close to $T_c$ and near $p^{\star}$, there are very-low-energy collective spin excitations with characteristic energies $\hbar \Gamma \approx$~5 meV. Cooling and applying a 8.8~T magnetic field creates a mixed state with a stronger magnetic response below 10~meV. We conclude that the low-energy spin-fluctuations are due to the collapse of the pseudogap combined with an underlying tendency to magnetic order. We show that the large specific heat near $p^{\star}$ can be understood in terms of collective spin fluctuations. The spin fluctuations we measure exist across the superconducting phase diagram and may be related to the strange metal behaviour observed in overdoped cuprates.