  Triple-Q density waves are often realized in many materials such as charge
density waves in transition metal dichalcogenides (TMDC) and spin density
waves, such as Skyrmion crystal, in B20 compounds. Compared with the single-Q
density waves, triple-Q density waves have an internal degree of freedom (phase
shift) made from the phase of the order parameters in addition to the
translations of density waves. Although this phase shift has been studied in
Ginzburg-Landau theory or in spin Hamiltonian, its impact on the electronic
states has not yet been explored from the microscopic point of view. Here, we
systematically investigate the interference of different Q density waves
revealing its pivotal effects depending on the phase shift. We show that the
phase shifts play a crucial role in affecting the interference effects of
triple-Q density waves on electronic states. Due to such interference, the band
geometry in the momentum space becomes nontrivial, where multiband Dirac-like
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.
