How to split a Sine wave to two positive and negative parts by real circuit?
It all depends how accurately you want to preserve the shape of each half of the waveform. If accuracy isn't a big concern then using a positive half wave rectifier and a negative half wave rectifier gives you what you need: -



To get a negative rectifier just reverse the diode connection. Be aware that there is a small voltage drop across the diode and this means that the peak of the signal output is somewhat lower than the original sinewave. If the operating frequency is a few hundred Hz or above you'll need to choose a diode that has low reverse recovery time: -



If you need more accuracy you can use an op-amp precision rectifier: -