The Xperia Z2 is a no-nonsense Android phone that cuts down on all the gimmicks found on the Galaxy S5 and the LG G3. Sony, instead, has added only those features that are genuinely useful in day to day life. Once such feature is Double tap to wake, which makes it extremely easy to wake up the device from sleep by simply tapping on the display twice.
However, for some strange reason, the feature is disabled by default on the Z2 and Sony does not even advertise it to make the general consumers aware of its presence. Since the feature is buried in the Settings menu, chances are most Z2 owners would never even come across it.
On your Xperia Z2, head over to Settings -> Display and enable the Tap to wake up option present near the bottom of the list. Keep in mind that enabling this option will increase the standby drain on the handset. However, the Z2 already has the best battery life amongst these year flagship devices so this should not really be much of an issue.
