Spin gaps and bilayer coupling in YBa2Cu3O7- delta and YBa2Cu4O8.

We investigate the relevance to the physics of underdoped ${\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{6+\mathit{x}}$ and ${\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{4}$${\mathrm{O}}_{8}$ of the quantum critical point which occurs in a model of two antiferromagnetically coupled planes of antiferromagnetically correlated spins. We use a Schwinger boson mean-field theory and a scaling analysis to obtain the phase diagram of the model and the temperature and frequency dependence of various susceptibilities and relaxation rates. We distinguish between a low \ensuremath{\omega},T coupled-planes regime in which the optic spin excitations are frozen out and a high \ensuremath{\omega},T decoupled-planes regime in which the two planes fluctuate independently. In the coupled-planes regime the yttrium nuclear relaxation rate at low temperatures is larger relative to the copper and oxygen rates than would be naively expected in a model of uncorrelated planes. Available data suggest that in ${\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{4}$${\mathrm{O}}_{8}$ the crossover from the coupled to the decoupled planes regime occurs at Tg700 K or T\ensuremath{\sim}200 K. The predicted correlation length is of order six lattice constants at T=200 K. Experimental data related to the antiferromagnetic susceptibility of ${\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{4}$${\mathrm{O}}_{8}$ may be made consistent with the theory, but available data for the uniform susceptibility are inconsistent with the theory.