In this work, we investigate the impact of Rashba spin-orbit coupling (RSOC) on the formation of charge-density-wave (CDW) and superconducting (SC) phases in the Holstein model on a half-filled square lattice. Using unbiased finite-temperature quantum Monte Carlo simulations, we go beyond mean-field approaches to determine the ground-state order parameter as a function of RSOC and phonon frequency. Our results reveal that the Rashba metal is unstable due to particle-hole instabilities, favoring the emergence of a CDW phase for any RSOC value. In the limit of pure Rashba hopping, the model exhibits a distinct behavior with the appearance of four Weyl cones at half-filling, where quantum phase transitions are expected to occur at strong interactions. Indeed, a quantum phase transition, belonging to the Gross-Neveu Ising universality class between a semimetal and CDW, emerges at finite phonon frequency-dependent coupling ${\lambda }_c$. In the antiadiabatic limit, we observe an enhanced symmetry in the infrared that unifies SC and CDW orders. These results advance our understanding of competing CDW and SC phases in systems with spin-orbit coupling, providing insights that may help clarify the behavior of related materials.

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• Charge density waves
• Electron-phonon coupling
• Peierls transition
• Weyl semimetal
• Lattice models in condensed matter
• Quantum Monte Carlo