  The quantum phase transition between $\mathbb{Z}_{2}$ plaquette valence bound
solid (PVBS) and superfluid (SF) phases on the planar pyrochlore lattice
(square ice) is under debate, because the conventional deconfined theory does
not support continuous one, but the numerical evidence is still not solid.
Here, we propose the system can be effectively described by a 2+1 dimensional
Abelian-Higgs model which may host the mixed 't Hooft anomaly, so that the
deconfinement can exist on the domain walls at the transition point. To verify
it, we study the spin excitation spectra by combining stochastic analytic
continuation and quantum Monte Carlo simulation. In both PVBS and SF phases, a
flat band with bow-tie structure is observed and can be explained by group
theoretic analysis. At the transition point, the spectra turn to be continuous
and gapless, which indicates the topological excitations. From the snapshot of
the spin configuration in real space, we found the existence of the domain wall
in which the symmetries satisfy the anomaly. Meanwhile, a Luttinger liquid like
continuum implies additional domain walls (point-defect) can emerge in the
domain walls (line-defect) and take the role of deconfinement at transition
point, as prediction of mixed 't Hooft anomaly. Our work can build a new bridge
between the topological gauge field theory and a strongly correlated system.
