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We calculate the magnetic susceptibility in the pseudogap phase of cupratesusing the two-dimensional $t-J$ model and the Mori projection operatortechnique. In this phase, the Fermi arcs lead to a quasi-elastic incommensuratemagnetic response for low temperatures. The theory accounts for thereorientation of the susceptibility maxima from the axial to the diagonaldirection occurring at small hole concentrations in La$_{2-x}$Sr$_x$CuO$_4$. Asmall cusp of the hole dispersion near the Fermi level, which is connected withthe spin-polaron band, affords the growth of the maxima with decreasingfrequency. As in the superconducting phase, the susceptibility maxima have anhourglass dispersion. The assumption of the phase separation into regions ofsuperconducting and pseudogap phases in underdoped cuprates explains the lackof the superconducting gap in their susceptibility. The observed rapid changeof widths of the susceptibility maxima with frequency is related to thecontribution of the pseudogap phase. This contribution may also explain thestrengthening of the quasi-elastic response by modest magnetic fields and bysmall impurity concentrations.

Magnetic susceptibility in the pseudogap phase of cuprate perovskites