Commensurate to incommensurate transition of three-dimensional charge density waves

The charge density wave (CDW) is a widely regarded emergent phenomenon in condensed matter physics. To establish a systematic understanding of the CDW, we develop a diagrammatic self-consistent field approach for a cubic Holstein model employing the fluctuation exchange approximation, and explore the emergence and transition of three-dimensional CDWs. A commensurate CDW (c-CDW) locked at $(\ensuremath{\pi},\ensuremath{\pi},\ensuremath{\pi})$ is favored near half filling, and the transition temperature is predicted around half of the nearest-neighbor hopping. Large hole doping leads to a suppression of the CDW transition temperature and the emergence of incommensurate CDW (i-CDW), which is evidenced by a drifting of the ordering vector away from $(\ensuremath{\pi},\ensuremath{\pi},\ensuremath{\pi})$ towards $(\ensuremath{\pi},\ensuremath{\pi},0)$. The phonon frequency significantly impacts the transition temperature and the phase boundary between c-CDW and i-CDW, and the optimal frequency for enlarging the CDW regime is also predicted near half of the nearest-neighbor hopping. These theoretical results provide a systematic understanding of CDW and a fresh perspective on emergent phenomena dominated by the electron-phonon interaction.