Dynamics in the planar pyrochlore lattice: flat band, domain wall and anomaly

The quantum phase transition between Z₂ plaquette valence bound solid (PVBS) and superfluid (SF) phases on the planar pyrochlore lattice (square ice) is under debate. To gain a further insight, here, we focus on the dynamical features of the hard-core Bose-Hubbard model on this lattice and study the 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 certain symmetries. At the transition point, the spectra turn to be continuous and gapless. A 2+1 dimensional Abelian-Higgs model with mixed’t Hooft anomaly is proposed to describe the transition, where the anomaly matching predicts that the deconfinement can exist on the domain walls. From the snapshot of the spin configuration in real space, we found the existence of the domain wall. We also found that the spectrum along a specific path in momentum space from PVBS phase to the transition point can be well described by a XXZ spin chain, and the critical theory of XXZ spin chain matches the anomaly. The two-spinon continuum along this specific path implies additional domain walls (point defect) can emerge in the domain walls (line defect) and take the role of deconfinement at transition point.