The Two-Dimensional Rashba-Holstein Model: A Quantum Monte Carlo Approach

TL;DR

This work examines how Rashba spin-orbit coupling (RSOC) interacts with Holstein electron-phonon coupling on a half-filled 2D square lattice using unbiased finite-temperature determinant Quantum Monte Carlo. The authors map the ground-state order as a function of RSOC strength $\alpha$ and phonon frequency $\omega_0$, revealing that RSOC destabilizes a Rashba metal via particle-hole instabilities and drives a CDW for any $\alpha$, while in the pure Rashba limit four Weyl cones appear at half-filling and a finite-$\lambda$ quantum critical point (Gross-Neveu Ising universality) separates a semimetal from CDW. In the antiadiabatic limit the model exhibits an emergent symmetry that unifies SC and CDW order; at finite $\omega_0$ there is robust coexistence of CDW and SC in certain regimes, with SC long-range order emerging only for large $\omega_0$. These results advance understanding of competing CDW and SC phases in systems with spin-orbit coupling and electron-phonon interactions, with potential relevance to related materials.

Abstract

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 a 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 semi-metal and CDW emerges at finite phonon frequency dependent coupling $λ_c$. 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.

Figures (9)

• Figure 1: (a) Sketch of the Holstein model: vibrations of lattice ions, represented by local dispersionless phonons, mediate an indirect electron-electron interaction. (b) Illustration of spin splitting due to Rashba spin-orbit coupling in a 2D electron gas, producing spin textures with opposite chiralities. (c) The electronic dispersion relation along the high-symmetry points in a square lattice with finite Rashba spin-orbit coupling. (d) Pictorial representation of four Weyl cones, with two located at the Fermi level (indicated by the red line). (e) Same as in panel (c), but in the pure Rashba hopping limit where $t/\alpha=0$. (f) Same as in panel (d), but in the pure Rashba hopping limit where $t/\alpha=0$.
• Figure 2: The change-density wave structure factor, $S_{CDW}$(Q), as a function of the inverse of temperature ($\beta$) for different values system sizes and fixed (a) $\alpha=0.2$, (b) $\alpha=0.6$, (c) $\alpha=0.8$ and (d) $\alpha=1.0$. The phonon frequency is $\omega_{0}/t=1$ and the electron-phonon coupling is given by $\lambda/t=2$. Here, and in all subsequent figures, when not shown, error bars are smaller than symbol size.
• Figure 3: (a) Finite-size scaling of the ground-state staggered CDW structure factor for different RSOC, $\alpha$, and fixed $\omega_{0}=1$ and $\lambda/t=2$. (b) The CDW order parameter, $\delta_{CDW}$, as a function of $\alpha$, for the same parameters of the previous panel. The red curve is just a guide to the eye.
• Figure 4: The ground state phase diagram of the Rashba-Holstein model for fixed $\omega_{0}=1$. The solid red line is just a crossover between a robust CDW (magenta region) and a weak CDW (white region) one.
• Figure 5: The CDW correlation ratio $R_{CDW}(\beta)$ as a function of the inverse temperature ($\beta$=1/T), for fixed (a) $\alpha= 0.2$ and (b) $\alpha= 1.2$, and different lattice sizes. (c) The extrapolation of the crossings of the $R_{CDW}(\beta)$, between $L$ and $L-\Delta L$ system sizes, $T_{c}(L, L - \Delta L)$. Here, we fixed $\Delta L =4$. (d) The extrapolated critical temperatures of the Rashba-Holstein model for fixed $\omega_0 /t =1$. The red curve is just a guide to the eye.