Dynamic Parallel Spin Stripes from the 1/8 anomaly to the End of Superconductivity in La$_{1.6-x}$Nd$_{0.4}$Sr$_x$CuO$_4$

We have carried out new neutron spectroscopic measurements on single crystals of La$_{1.6-x}$Nd$_{0.4}$Sr$_x$CuO$_4$ from 0.12 to 0.26 using time-of-flight techniques. These measurements allow us to follow the evolution of parallel spin stripe fluctuations with energies less than 33 meV, from x=0.12 to 0.26. Samples at these hole-doping levels are known to display static (on the neutron scattering time scale) parallel spin stripes at low temperature, with onset temperatures and intensities which decrease rapidly with increasing x. Nonetheless, we report remarkably similar dynamic spectral weight for the corresponding dynamic parallel spin stripes, between 5 meV to 33 meV, from the 1/8 anomaly near x=0.12, to optimal doping near x=0.19 to the quantum critical point for the pseudogap phase near x=0.24, and finally to the approximate end of superconductivity near x=0.26. This observed dynamic magnetic spectral weight is structured in energy with a peak near 17 meV at all dopings studied. Earlier neutron and resonant x-ray scattering measurements on related cuprate superconductors have reported both a disappearance with increasing doping of magnetic fluctuations at ($\pi$, $\pi$) wavevectors characterizing parallel spin stripe structures, and persistant paramagnon scattering away from this wavevector, respectively. Our new results on La$_{1.6-x}$Nd$_{0.4}$Sr$_x$CuO$_4$ from 0.12 < x <0.26 clearly show persistent parallel spin stripe fluctuations at and around at ($\pi$, $\pi$), and across the full range of doping studied. These results are also compared to recent theory. Together with a rapidly declining x-dependence to the static parallel spin stripe order, the persistent parallel spin stripe fluctuations show a remarkable similarity to the expectations of a quantum spin glass, random t-J model, recently introduced to describe strong local correlations in cuprates.