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

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.
Comment: 6 pages, 3 figures, plus Supplementary Material