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We have carried out neutron spectroscopic measurements on single crystals of La_{1.6−x}Nd_{0.4}Sr_{x}CuO_{4} from 0.12≤x≤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 and 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 (π, π) wave vectors characterizing parallel spin stripe structures and persistant paramagnon scattering away from this wave vector, respectively. Our results for 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 (π, π), 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.

Three-dimensional, commensurate long-range magnetic order in La_{2}CuO_{4} quickly evolves to quasi-two-dimensional, incommensurate correlations upon doping with mobile holes, and superconductivity follows for x as small as 0.05 in the La_{2−x}Sr_{x}/Ba_{x}CuO_{4} family of superconductors. The onset of superconductivity in these systems is known to be coincident with a remarkable rotation of the incommensurate spin order from “diagonal stripes” below x = 0.05 to “parallel stripes” above. However, little is known about the spin correlations at optimal and high doping levels, around and beyond the proposed quantum critical point for the pseudogap phase, p^{*}. Here, we present elastic and inelastic neutron scattering measurements on single crystals of La_{1.6−x}Nd_{0.4}Sr_{x}CuO_{4} with x = 0.125, 0.19, 0.24, and 0.26 and show that two-dimensional, quasistatic, parallel spin stripes have an onset at temperatures such that the parallel spin stripe phase extends beyond p^{*} and envelops the entirety of superconducting ground states in this system. We also show that the elastic order parameter for parallel spin stripes at optimum doping, x = 0.19, displays an inflection point at superconducting T_{c}, while the low-energy dynamic spectral weight of parallel stripe fluctuations grows with decreasing temperature and saturates below T_{c}.