Theory of spin response in underdoped cuprates as strongly fluctuating<i>d</i>-wave superconductors

We study the spin dynamics in underdoped cuprates at low temperatures by considering them as quasi-two-dimensional d-wave superconductors with strong quantum phase fluctuations. An effective theory of spin degrees of freedom of nodal quasiparticles coupled to vortex defects in the phase of the superconducting order parameter is formulated. It represents the minimal extension of the three-dimensional QED theory of the pseudogap phase into the superconducting region. The theory predicts a single superconductor-spin-density wave (SC-SDW) phase transition, without coexistence between the two phases. At the transition, which may be fluctuation-induced first order, vortices condense (and SC is lost) and the chiral symmetry for fermions simultaneously breaks (and SDW is formed). We compute the spin-spin correlation function in the fluctuating superconducting state and explain the evolution of the spin response with energy in underdoped ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{6+x}$ observed in neutron-scattering experiments. In particular, we predict that at very low frequencies $\ensuremath{\omega}\ensuremath{\sim}(1/10){\ensuremath{\omega}}_{\mathrm{res}},$ with ${\ensuremath{\omega}}_{\mathrm{res}}$ being the energy of the ``resonance'' at $\stackrel{\ensuremath{\rightarrow}}{Q}=(\ensuremath{\pi},\ensuremath{\pi}),$ (weak) spin response should become narrowly peaked at four diagonally incommensurate wave vectors that span between the nodes of the superconducting order parameter. These peaks represent the inherent collective mode of the phase fluctuating d-wave superconductor, the condensation of which would bring about the SDW order in the pseudogap phase. Our interpretation of the resonance suggests that it should become more elusive in the superconductors with lower ${T}_{c},$ in accord with its conspicuous absence in ${\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}{\mathrm{CuO}}_{4}.$

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