A smartphone has two entirely separate computers in it, the one that runs Android or iOS, and the baseband controller. The latter is built to be very power-efficient while waiting for a radio signal. It will wake up the dormant application OS computer when a push notification comes in.

The same likely can be built on top of an RPi, using, say, wake-on-LAN signaling, or some GPIO as an interrupt source. You'll have to suspend your OS while idle for a prolonged time though.