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Spin-orbital coupling (SOC) and parity-time ($\mathcal{PT}$) symmetry bothhave attracted paramount research interest in condensed matter physics, coldatom physics, optics and acoustics to develop spintronics, quantum computation,precise sensors and novel functionalities. Natural SOC is an intrinsicrelativistic effect. However, there is an increasing interest in synthesizedSOC nowadays. Here, we show that in a $\mathcal{PT}$-symmetric spin-1/2 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 theenergy spectrum of the system. Firstly, we show that it can generate a purereal energy spectrum with a double-valleys structure. Therefore, it has theability to generate supersolid stripe states. Especially, the imaginary SOCstripe state can have a high contrast of one. Moreover, the imaginary SOC canalso generate a spectrum with tunable complex energy band, in which the wavesare either amplifying or decaying. Thus, the imaginary SOC would also findapplications in the engineering of $\mathcal{PT}$-symmetry-based coherent waveamplifiers/absorbers. Potential experimental realizations of imaginary SOC areproposed in cold atomic gases and systems of coupled waveguides constituted ofnonlocal gain and loss.

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