Spin Quadrupolar Order in d-wave Unconventional Magnetism

Unconventional magnetism represents a class of metallic states whose Fermi surfaces exhibit spin-dependent splittings under the nontrivial representations of the rotation group. The d-wave α-phase of unconventional magnetism provides a continuum realization of the nonrelativistic spin-splitting physics described in crystals within the framework of altermagnetism; the two share the same symmetry under rotation. In this work, we investigate how the associated momentum-space spin quadrupole manifests itself in the real-space spin structure of a crystal. Within linear-response theory, we calculate the spin-charge cross-susceptibility of the d-wave state in a weak nonmagnetic periodic scalar potential. The combined symmetry of the d-wave α-phase requires the response to change sign between two symmetry-related reciprocal-lattice vectors, producing an intra-unit-cell d-wave spin-density modulation without enlarging the unit cell. The resulting real-space spin modulation is induced by the scalar lattice potential rather than being a consequence of spontaneous symmetry breaking. It thus provides a real-space linear-response signature of the even-parity momentum-space spin multipole.