Frustrated quantum two-dimensional $J_1$-$J_2$-$J_3$ antiferromagnet in a spherically symmetric self-consistent approach

In the framework of a spherically symmetric self-consistent approach to two-time retarded spin–spin Green's functions, we develop the theory of a two-dimensional frustrated $J_1$-$J_2$-$J_3$ quantum $S=1/2$ antiferromagnet. We show that taking the damping of spin fluctuations into account is decisive in forming both the spin-liquid state and the state with long-range order. In particular, the existence of damping allows explaining the scaling behavior of the susceptibility $\chi(\mathbf{q},\omega)$ of the CuO$_2$ cuprate plane, the behavior of the spin spectrum in the two-plane case, and the occurrence of an incommensurable $\chi(\mathbf{q},\omega)$ peak. In the case of the complete $J_1$-$J_2$-$J_3$ model, in a single analytic approach, we find continuous transitions between three phases with long-range order (“checkerboard”, stripe, and helical $(q,q)$ phases) through the spin-liquid state. We obtain good agreement with cluster computations for the $J_1$-$J_2$-$J_3$ model and agreement with the neutron scattering data for the $J_1$-$J_2$ model of cuprates.