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 J1-J2-J3 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 χ(q, ω) of the CuO2 cuprate plane, the behavior of the spin spectrum in the two-plane case, and the occurrence of an incommensurable χ(q, ω) peak. In the case of the complete J1-J2-J3 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 J1-J2-J3 model and agreement with the neutron scattering data for the J1-J2 model of cuprates. the occurrence of a spin gap Δ at an arbitrarily low temperature T = 0 into account and describing the long-range order (LRO) at T = 0. It is known that as the doping level increases in two-dimensional AFMs, the spin correlation length decreases. Introducing frustration leads to a similar behavior. It is obvious that there is no direct correspondence between doping and frustration. Nevertheless, an increase in frustration can be qualitatively treated as an increase in x. It is believed that the CuO 2 cuprate plane corresponds to a nonvanishing frustration even at x = 0 [10], and the purely spin model has been used many times to analyze the spin response of the doped CuO 2 plane (see, e.g., [11]).Here, in a spherically symmetric self-consistent approach (SSSA) [7], [8], [12], [13], we study the J 1 -J 2 -J 3 model in its different versions: the J 1 model (J 2 = J 3 = 0); the J 1 -J 2 model (J 3 = 0), which is the case most studied in cluster computations; and the J 1 -J 2 -J 3 model-the complete model, which exhibits a series of phases. Our treatment is based on the two-time temperature spin Green's function G q (ω) [14].In contrast to spin-wave two-sublattice approaches, the SSSA states both with and without LRO are singlet, the Marshall and Mermin-Wagner theorems are satisfied automatically, and the spin SU (2) and translation symmetries are preserved (we do not consider the so-called box and columnar states). In particular, this implies that the averaged spin operator at a site is equal to zero, S α n = 0, α = x, y, z, and that the spin correlators C r = S α n+r S α n are independent of the position of the site n. The method allows finding the microscopic characteristics that cannot be extracted from cluster computations, such as the spin-excitation spectra ω q , the dependence Δ(T ) and its position, and the explicit form of the function χ(q, ω, T ). The main difference between our method and the standard realization of the SSSA is that we go beyond the mean field approximation by introducing damping into the expression for the s...
