Magnetic scaling in cuprate superconductors

We determine the magnetic phase diagram for the YBa$_2$Cu$_3$O$_{6+x}$ and La$_{2-x}$Sr$_x$CuO$_4$ systems from various NMR experiments. We discuss the possible interpretation of NMR and neutron scattering experiments in these systems in terms of both the non-linear $\sigma$-model of nearly localized spins and a nearly antiferromagnetic Fermi liquid description of magnetically coupled quasiparticles. We show for both the 2:1:4 and 1:2:3 systems that bulk properties, such as the spin susceptibiltiy, and probes at the antiferromagnetic wavevector $(\pi, \pi)$, such as $^{63}T_1$, the $ ^{63}Cu$ spin relaxation time, both display a crossover at a temperature $T_{cr}$, which increases linearly with decreasing hole concentration, from a non-universal regime to a $z=1$ scaling regime characterized by spin pseudogap behavior. We pursue the consequences of the ansatz that $T_{cr}$ corresponds to a fixed value of the antiferromagnetic correlation length, $\xi$, and show how this enables one to extract the magnitude and temperature dependence of $\xi$ from measurements of $T_1$ alone. We show that like $T_{cr}$, the temperature $T_*$ which marks a crossover at low temperatures from the $z=1$ scaling regime to a quantum disordered regime, exhibits the same dependence on doping for the 2:1:4 and 1:2:3 systems, and so arrive at a unified description of magnetic behavior in the cuprates, in which the determining factor is the planar hole concentration. We apply our quantitative results for YBa$_2$Cu$_3$O$_7$ to the recent neutron scattering experiments of Fong {\em et al}, and show that the spin excitation near $40 meV$ measured by them corresponds to a spin gap excitation, which is overdamped in the normal state, but becomes visible in the superconducting state.