Bose-Einstein condensations of magnons in quantum magnets with spin-orbit coupling in a Zeeman field

We study the response of a quantum magnet with spin-orbit coupling (SOC) to a Zeeman field by constructing effective actions and performing Renormalization Group (RG) analysis. There are several novel classes of quantum phase transitions at a low $h_{c1}$ and an upper critical field $h_{c2}$ driven by magnon condensations at commensurate (C-) or in-commensurate (IC-) momenta $0<k_0<\pi$. The intermediate IC- Skyrmion crystal (IC-SkX) phase is controlled by a line of fixed points in the RG flows labeled by $k_0$. We derive the relations between the quantum spin and the order parameters of the effective actions which determine the spin-orbital structures of the IC-SkX phase. We also analyze the operator contents near $h_{c1}$ and $h_{c2}$ which determine the exotic excitation spectra inside the IC-SkX. The intrinsic differences between the magnon condensations at the C- and IC- momenta are explored. The two critical fields $h_{c1}<h_{c2}$ and the intermediate IC-SkX phase could be a generic feature to any quantum magnets with SOC in a Zeeman field. Experimental implications to some materials or cold atom systems with SOC in a Zeeman field are presented.