Anti-theft tags contain a tiny mechanical oscillator that is activated by a magnetic field. The detectors at a shop's doorway can detect the oscillation magnetically. The tag can be activated and deactivated by magnetizing or demagnetizing it, respectively.
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Sources:
http://www.metglas.com/products/magnetic_materials/2826mb.asp
https://www.google.com/patents/WO1999013442A1?cl=en&dq=magnetostrictive+marker&hl=en&sa=X&ved=0ahUKEwjT8bCCtLLJAhVP9GMKHYThCY8Q6AEIMjAD
https://patents.google.com/patent/WO2008032274A2/en?q=magnetostrictive&q=amorphous&q=theft
https://www.google.com/patents/US4510489?dq=4,510,489&hl=en&sa=X&ved=0ahUKEwi0x_SosbLJAhVJyWMKHbl_AI4Q6AEIHDAA
https://www.google.com/patents/US4298862
https://en.wikipedia.org/wiki/Magnetostriction
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