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We study the spin dynamics in underdoped cuprates at low temperatures byconsidering them as quasi two dimensional d-wave superconductors (dSC) withstrong phase fluctuations. An effective theory of spin degrees of freedom ofnodal quasiparticles coupled to vortex defects in the phase of thesuperconducting order parameter is formulated. It represents the minimalextension of the QED3 theory of the pseudogap phase into the superconductingregion. The theory predicts a single dSC-SDW transition, which may befluctuation induced first order, without the region of coexistence between thetwo ordered phases. We compute the spin-spin correlator in the dSC, and explainthe evolution of the spin response with frequency as observed in neutronscattering experiments on YBCO. We predict that at the frequencies much belowthe resonance the (weak) spin response should become narrowly peaked at four`diagonally' incommensurate wave vectors that span between the nodes of thesupercopnducting order parameter. These peaks represent an inherent collectivemode of the phase fluctuating dSC, the condensation of which would bring aboutthe SDW order in the pseudogap phase. Our interpretation of the resonancesuggests it should become more elusive in the superconductors with lower Tc, inaccord with its' conspicuous absence in LSCO.Comment: 10 pages, 7 figures, Revtex, the extended version of cond-mat/0211418  with a more detailed discussion of spin respons

Theory of spin response in underdoped cuprates as strongly fluctuatingd-wave superconductors