Abstract
It is theoretically shown that, in the fourfold symmetric $d$-wave superconducting phase, a paramagnetic pair-breaking (PPB) enhanced sufficiently by increasing the applied magnetic field induces not only the Fulde--Ferrell--Larkin--Ovchinnikov (FFLO) superconducting state but also an incommensurate antiferromagnetic (AFM) order with the $\mathbf{Q}$ vector parallel to a gap node. This AFM ordering tends to occur only below ${H}_{c2}$ at low temperatures, i.e., in the presence of a nonvanishing superconducting energy gap $\ensuremath{\Delta}$ rather than in the normal phase. Through a detailed study on the resulting AFM order and its interplay with the FFLO spatial modulation of $\ensuremath{\Delta}$, it is argued that the strange high-field, low-temperature (HFLT) superconducting phase of CeCoIn${}_{5}$ is a coexisting phase of the FFLO and incommensurate AFM orders, and that this PPB mechanism of an AFM ordering is also the origin of the AFM quantum critical fluctuation which has occurred close to ${H}_{c2}(0)$ in several unconventional superconductors including CeCoIn${}_{5}$.
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