Spontaneous spin-orbit coupling induced by quantum phonon dynamics

Spin-orbit coupling is a important focus of condensed matter physics as well as electron-phonon interaction. Traditionally spin-orbit coupling is regarded as a single-body effect arising from relativity, and electron-phonon interaction is often considered spin-independent. In this Letter, we bridge spin-orbit coupling and electron-phonon interaction, and propose a novel mechanism to dynamically generate spin-orbit coupling. Based on symmetry analysis, a spin-dependent electron-phonon coupling model is constructed, and is solved by sign-problem-free quantum Monte Carlo simulations. The phase diagram versus phonon frequency ω and coupling constant λ is fully investigated. The spin-orbit coupling emerges as an order in the ground state for any λ in the adiabatic limit, accompanied by a breathing mode of lattice distortion and a staggered loop spin current. This phase dominates in the entire range of ω with λ<λ_{∞}, a critical value in the ω→∞ limit. At λ>λ_{∞}, the emergent spin-orbit coupling is suppressed as increasing ω, and a phase transition occurs leading to charge-density-wave ordering degenerate with superconductivity. Our work opens up the possibility of hidden spin-orbit coupling in materials where it is otherwise forbidden by lattice symmetry and paves the way to explore possible materials for spintronics.