  Magnetic spins and charges interact strongly in high-temperature
superconductors. New physics emerges as layers of copper oxide are tuned
towards the boundary of the superconducting phase. As the pseudogap increases
the characteristic spin excitation energy decreases. We show that our
well-annealed YBa2Cu3O6+x (YBCO6+x) single crystals are orthorhombic and
superconducting but not antiferromagnetically ordered. Near the critical
concentration for superconductivity for x = 0.35 the spins fluctuate on two
energy scales, one a relaxational spin response at ~2 meV and the other a slow
central mode that is resolution-limited in energy (<0.08 meV) but broad in
momentum. The gradual formation on cooling of a central mode over a range of
momenta suggests that the spin ground state from which coherent superconducting
pairing emerges may be quantum disordered. We show that YBCO6.35 adopts a
homogeneous state that consists of highly-organized frozen sub-critical
three-dimensional spin correlations. The continuous spin evolution indicates
that a single quantum state occurs in contrast to claims from site-based probes
that lightly doped YBCO undergoes a transition to antiferromagnetic Bragg order
followed by a sharp transition to a cluster glass phase. For x = 0.35, where Tc
= 18 K is reduced to 1/5 of Tcmax, the spin ground state is reached without a
sharp transition and consists of short correlations extending over only 8
Angstrom between cells and 42 Angstrom within the planes. Polarized neutrons
show the angular spin distribution to be isotropic unlike the AF insulator.
Since moment is conserved we interpret this as evidence for hole-induced spin
rotations rather than decay.
