Higgs mechanism of rank-2 gauge theory and fracton topological order

Fractons are quasiparticle excitations in d = 3 gapped topological ordered phases whose motion is constrained. Their physical mechanism for generating them is still missing. It has been noticed that in the higher rank U(1) gauge theories, charges cannot move freely due to the extra conservation laws. To make connection to the gapped fracton models, we study what happens when the higher rank gauge theories are Higgsed. Surprisingly, all the higher rank Z_N gauge theories obtained through Higgsing do not contain fractons. In particular, we show that the rank-2 Z_2 gauge theories are equivalent to several copies of Toric Codes while fracton topological order appears when certain degrees of freedom of the gauge fields are frozen. We demonstrate this result by identifying the conservation law under Higgsing and also by explicitly analyzing the model after Higgsing.