Imaginary spin-orbital coupling in parity-time symmetric systems with momentum-dependent gain and loss

Spin-orbital coupling (SOC) and parity-time $(\mathcal{P}\mathcal{T})$ symmetry both have attracted paramount research interest in condensed matter physics, cold atom physics, optics and acoustics to develop spintronics, quantum computation, precise sensors and novel functionalities. Natural SOC is an intrinsic relativistic effect. However, there is an increasing interest in synthesized SOC nowadays. Here, we show that in a $\mathcal{P}\mathcal{T}$ -symmetric pseudo spin-1/2 system (in other words, two-state system), the momentum-dependent balanced gain and loss can synthesize a new type of SOC, which we call imaginary SOC. The imaginary SOC can substantially change the energy spectrum of the system. Firstly, we show that it can generate a pure real energy spectrum with a double-valleys structure. Therefore, it has the ability to generate supersolid stripe states. Especially, the imaginary SOC stripe state can have a high contrast of one. Moreover, the imaginary SOC can also generate a spectrum with tunable complex energy band, in which the waves are either amplifying or decaying. Thus, the imaginary SOC would also find applications in the engineering of $\mathcal{P}\mathcal{T}$ -symmetry-based coherent wave amplifiers/absorbers. Potential experimental realizations of imaginary SOC are proposed in cold atomic gases and systems of coupled waveguides constituted of nonlocal gain and loss.