Looking for boundary sensor
A few suggestions:

Lets assume that there is only a wire/ metal strip buried beneath the ground (not very deep) as the 'invisible' boundary.

(1) The wire/strip has a direct current passing through it.

You could detect the magnetic field around the wire with a Hall sensor.

(2) The wire/strip has a low frequency alternating current.

Hall sensor or pick up coil (detecting AC 'hum')

(3) The wire/strip has a high frequency alternating current 

Tuned LC circuit - probably with a ferrite rod to localize signal.

(4) The wire/strip is carrying a pulse signal (e.g. output of a square wave generator)

as per solution (3) but doesn't need to be tuned (detects wideband 'noise') - similar to electric cable detector in walls.

(5) The wire/strip is not connected to anything

Some form of metal detector.