Magnetic susceptibility in the pseudogap phase of cuprate perovskites

We calculate the magnetic susceptibility in the pseudogap phase of cuprates using the two-dimensional $t-J$ model and the Mori projection operator technique. In this phase, the Fermi arcs lead to a quasi-elastic incommensurate magnetic response for low temperatures. The theory accounts for the reorientation of the susceptibility maxima from the axial to the diagonal direction occurring at small hole concentrations in La$_{2-x}$Sr$_x$CuO$_4$. A small cusp of the hole dispersion near the Fermi level, which is connected with the spin-polaron band, affords the growth of the maxima with decreasing frequency. As in the superconducting phase, the susceptibility maxima have an hourglass dispersion. The assumption of the phase separation into regions of superconducting and pseudogap phases in underdoped cuprates explains the lack of the superconducting gap in their susceptibility. The observed rapid change of widths of the susceptibility maxima with frequency is related to the contribution of the pseudogap phase. This contribution may also explain the strengthening of the quasi-elastic response by modest magnetic fields and by small impurity concentrations.