Phase shifts, band geometry, and responses in triple-Q charge and spin density waves

Triple-Q density waves are commonly found in various materials, such as charge density waves in transition metal dichalcogenides and spin density waves (skyrmion crystals) in B20 compounds. Compared to single-Q density waves, triple-Q density waves possess an additional internal degree of freedom, a phase shift arising from the phase of the order parameters, in addition to the translations of the density waves. In this study, we systematically investigate the significant effects stemming from both triple-Q charge density waves and spin density waves order parameters, with particular emphasis on potential phase shifts. We demonstrate that these phase shifts play a crucial role in influencing the interference effects of triple-Q density waves on electronic states. Due to such interference, the band geometry in the momentum space becomes nontrivial at the hot spots, where multiband Dirac-type fermions are induced near the Fermi energy. Furthermore, we explicitly establish that the nontrivial band geometry, combined with symmetry breaking induced by phase shifts, leads to a variety of intriguing linear and nonlinear responses.