  We propose quenched disorders could bring novel quantum excitations and
models to certain quantum magnets. Motivated by the recent experiments on the
quantum Ising magnet TmMgGaO$_4$, we explore the effects of the quenched
disorder and the interlayer coupling in this triangular lattice Ising
antiferromagnet. It is pointed out that the weak quenched (non-magnetic)
disorder would convert the emergent 2D Berezinskii-Kosterlitz-Thouless (BKT)
phase and the critical region into a gauge glass. There will be an emergent
Halperin-Saslow mode associated with this gauge glass. Using the Imry-Ma
argument, we further explain the fate of the finite-field $C_3$ symmetry
breaking transition at the low temperatures. The ferromagnetic interlayer
coupling would suppress the BKT phase and generate a tiny ferromagnetism. With
the quenched disorders, this interlayer coupling changes the 2D gauge glass
into a 3D gauge glass, and the Halperin-Saslow mode persists. This work merely
focuses on addressing a phase regime in terms of emergent U(1) gauge glass
behaviors and hope to inspire future works and thoughts in weakly disordered
frustrated magnets in general.
