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

Abstract: Recently, there has been growing interest in the impacts of phase shifts within the triple-Q spin density wave (SDW) order parameters. Concurrently, it is widely recognized that incommensurate triple-Q charge density waves (CDW) are also prevalent in low-dimensional materials, where the phase degrees of freedom in the order parameters are generally allowed. In this study, we systematically investigate the pivotal effects arising from both triple-Q CDW and SDW order parameters, with particular consideration given to possible phase shifts. We show that the phase shifts play a crucial role in determining the real-space topology of triple-Q density waves. More importantly, we show that the triple-Q CDW and SDW order parameters would influence the band geometry in the momentum space, where multiband Dirac-like fermions are induced by the triple-Q density wave order parameters near the Fermi energy. Furthermore, we explicitly establish that such nontrivial band geometry, combined with symmetry-breaking induced by phase shifts, leads to a variety of intriguing linear and nonlinear responses.