Spin gap and magnetic excitations in the cuprate superconductors

We analyze the spectrum of magnetic excitations as observed by neutron diffraction and NMR experiments in ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_{6+\mathit{x}}$, in the framework of the single-band t-t’-J model in which the next-nearest-neighbor hopping term has been introduced in order to fit the shape of the Fermi surface revealed by photoemission. Within the slave-boson approach, we have as well examined the d-wave superconducting state, and the singlet-resonating-valence-bond phase appropriate to describe the normal state of heavily doped systems. Our calculations show a smooth evolution of the spectrum from one phase to the other, with the existence of a spin gap in the frequency dependence of χ’ ’(Q,ω). The value of the threshold of excitations ${\mathit{E}}_{\mathit{G}}$ is found to increase with doping, while the characteristic temperature scale ${\mathit{T}}_{\mathit{m}}$ at which the spin gap opens exhibits a regular decrease, reaching ${\mathit{T}}_{\mathit{c}}$ only in the overdoped regime. This very atypical combined variation of ${\mathit{E}}_{\mathit{G}}$ and ${\mathit{T}}_{\mathit{m}}$ with doping results from strong-correlation effects in the presence of the realistic band structure considered here. We point out that the presence of a resonance in the spectrum χ’ ’(Q,ω) is in good agreement with the neutron-diffraction results obtained at x=0.92 and 1.0. This resonance is analyzed as a Kohn anomaly of the second kind in the Cooper channel. Finally, we examine the evolution of the Knight shift and of χ’ ’(q,ω) at any q, allowing one to study the magnetic correlation length ξ as a function of doping, frequency, and temperature.