Coexistence of d-wave superconductivity and antiferromagnetism induced by paramagnetic depairing

It is shown theoretically that, in the superconducting state with $d_{x^2-y^2}$-pairing, a strong Pauli paramagnetic depairing (PD) induces not only the modulated Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconducting state but also an incommensurate antiferromagnetic (AFM) (or spin-density wave) ordering with ${\bf Q}$-vector nearly paralle to a gap node. This intrinsic mechanism of field-induced coexistence of the $d$-wave superconducting and AFM orders does not need to assume a pair-density wave but is enhanced by the FFLO ordering. It is argued that this is the common origin of {\it both} the coexistent FFLO and AFM phases of CeCoIn$_5$ {\it and} the AFM quantum critical behavior around the superconducting $H_{c2}(0)$ seen in several heavy fermion superconductors and overdoped cuprates.