Field-tunable quadruple-Q states driven by momentum-space frustration

Multiple-Q magnetism in itinerant electron systems enables complex spin crystals and noncoplanar textures even in centrosymmetric settings. We study a minimal momentum-space spin model on a square lattice with four symmetry-related ordering wave vectors, including bilinear and biquadratic interactions under an out-ofplane magnetic field. Using simulated annealing, we obtain the field-dependent phase diagram and identify successive transitions among single-Q, double-Q, and multiple inequivalent quadruple-Q states. The quadruple-Q manifold exhibits rich internal structures: the states sharing the same wave vectors differ in phase locking, amplitude distribution, and noncoplanarity, leading to distinct real-space textures and scalar spin chirality patterns. Our results demonstrate that momentum-space frustration and biquadratic coupling provide an efficient route to stabilizing diverse quadruple-Q spin crystals, offering a general framework for higher-order spin textures in centrosymmetric itinerant magnes.