π excitation of thet−Jmodel

In this paper, we present analytical and numerical calculations of the π resonance in the $t-J$ model. We show in detail how the $\pi$ resonance in the particle-particle channel couples to the dynamical spin correlation function in a superconducting state. The contribution of the π resonance to the spin excitation spectrum can be estimated from general model-independent sum rules, and it agrees with our detailed calculations. The results are in overall agreement with the exact diagonalization studies of the $t-J$ model. Earlier calculations predicted the correct doping dependence of the neutron resonance peak in the ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+x}$ superconductor, and in this paper detailed energy and momentum dependence of the spin correlation function is presented. The microscopic equations of motion obtained within current formalism agree with that of the SO(5) nonlinear σ model, where the π resonance is interpreted as a pseudo-Goldstone mode of the spontaneous SO(5) symmetry breaking.