It is said that wake turbulence occurs when an aircraft generates lift and there is no wake turbulence after touch down or before take off. That's how we avoid wake turbulence. However, I've thought we have lift while on landing roll or take off roll because there is the angle of incidence and at least a certain amount of speed, just not enough lift to be airborne. Based on my idea, I don't really understand why does wake turbulence suddenly disappear like a magic when we land. Can someone explain this please?
Well there may be vortex formation on the ground under certain conditions, but it doesn't stick around long enough to really be measured or cause trouble.
Any time a wing is producing lift it will generate a wake. This happens even if the wing generates just enough lift to make the aircraft "lighter" but not enough to get it off the ground: Vortices will still start to form at the wingtips, but as they begin their rotation (downward and outward) they will quickly hit the ground and be dispersed.
If the runway were a narrow strip with the wingtips hanging out in free air you would still see vortex formation in these edge cases, but pilots generally don't like to take off on a tightrope.
Airplanes also usually don't stay in that kind of edge case situation where the wing is generating lift but the wheels are still on the ground for very long: Either they're taking off (soon after the wing starts producing some lift it's producing enough lift and the airplane begins flying), or they're landing (within a few seconds after you touch down you've slowed to a point where the wing's lift production, and thus vortex formation, is negligible). Thus for practical purposes of wake turbulence avoidance the wake begins when the airplane rotates, and ends where it touches down.
It's also worth noting that if you happen to be following close enough for the incipient wingtip vortices to cause trouble on the ground you'll probably be experiencing other issues (jet blast, prop blast, etc.) which will remind you that it's prudent to maintain a safe separation distance from other aircraft on the ground.
Wake turbulence is related to the amount of lift produced. This in turn is proportional to the angle of attack of the wing. Before take-off rotation the wing produces little lift, so the effect of wake turbulence is small. Ground effect will weaken the wake further, but it would technically be wrong to say that no wake turbulence is produced.
The strength of the wake turbulence of a flying aircraft is proportional to the mass of the aircraft and inverse with airspeed squared. This means that the turbulence gets stronger while the aircraft descends and decelerates, and wake turbulence reaches a peak right before touchdown. With the wing close to the ground, ground effect will weaken the turbulence, so when the wheels touch the ground the intensity of the wake is maybe two thirds of its free flight strength at the same speed. Only when the aircraft derotates (the process when pitch attitude is reduced until the nose wheel touches the ground), the wake turbulence will be significantly reduced.
After derotation the lower zero-lift angle due to the flap deflection for landing will let the wing produce more lift at the same speed than in case of the aircraft with flaps set for take-off. Again, ground effect and extending spoilers or wing-mounted speed brakes reduces the wake (with flaps down the ground effect is even stronger), but here it would be even more incorrect to say that there is no wake turbulence. It is only much weaker than in flight.