Magnetic scaling in cuprate superconductors

We determine the magnetic phase diagram for the ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_{6+\mathit{x}}$ and ${\mathrm{La}}_{2\mathrm{-}\mathit{x}}$ ${\mathrm{Sr}}_{\mathit{x}}$ ${\mathrm{CuO}}_4$ systems from various NMR experiments. We discuss the possible interpretation of NMR and neutron scattering experiments in these systems in terms of both the nonlinear σ 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 wave vector (π,π), such as ${}_{}{}^{63}{}_{}{}^{}$ ${\mathit{T}}_1$, the ${}_{}{}^{63}{}_{}{}^{}\mathrm{Cu}$ spin-lattice relaxation time, both display a crossover at a temperature ${\mathit{T}}_{\mathrm{cr}}$, which increases linearly with decreasing hole concentration, from a nonuniversal regime to a z=1 scaling regime characterized by spin pseudogap behavior. We pursue the consequences of the ansatz that ${\mathit{T}}_{\mathrm{cr}}$ corresponds to a fixed value of the antiferromagnetic correlation length ξ and show how this enables one to extract the magnitude and temperature dependence of ξ from measurements of ${\mathit{T}}_1$ alone. We show that like ${\mathit{T}}_{\mathrm{cr}}$, the temperature ${\mathit{T}}_{\mathrm{*}}$ 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.