Double-Q Checkerboard Bubble Crystal in Centrosymmetric Tetragonal Magnets

We report our numerical studies on the emergence of a double-Q checkerboard bubble crystal in centrosymmetric tetragonal magnets. The double-Q checkerboard bubble crystal is characterized by a fourfold-symmetric collinear spin configuration consisting of a superposition of two sinusoidal waves with the out-of-plane spin modulations along the [110] and [1¯10] directions. The numerical calculations based on the simulated annealing for an effective spin model with the momentum-resolved easy-axis exchange interactions reveal that the double-Q checkerboard bubble crystal is energetically degenerate with the single-Q collinear state when the ordering wave vector lies on the quarter of the reciprocal lattice vector along the ⟨110⟩ direction. We show that such a degeneracy is lifted by considering the biquadratic interaction. We also find that the double-Q checkerboard bubble crystal turns into another double-Q state characterized by the in-plane spin modulations by increasing an external magnetic field.