Exotic surface behaviors induced by geometrical settings of two-dimensional dimerized quantum XXZ model

We study the surface behavior of the two-dimensional columnar dimerized quantum antiferromagnetic XXZ model with easy-plane anisotropy, putting special attention on the surface critical behaviors of the (2+1)-dimensional quantum critical points of the model, belonging to the classical three-dimensional O(2) universality class, for both $S=1/2$ and $S=1$ spins using quantum Monte Carlo simulations. We found utterly different surface behaviors at two different surfaces of geometrical settings: one surface is the dangling-ladder surface, which is exposed by cutting a row of weak bonds, and the other is the dangling-chain surface, which is formed by cutting a row of strong bonds along the direction perpendicular to the strong bonds of a periodic system. We find an ordinary transition on the dangling-ladder surface for both $S=1$ and $S=1/2$ spin systems, similar to what found in the Heisenberg limit. However, the dangling-chain surface shows much richer surface behaviors than in the Heisenberg limit. For the $S=1/2$ easy-plane model, we find the surface undergoes an unexpected phase transition from a Tomonaga-Luttinger liquid to an unknown critical state in the gapped bulk state as the dimerization strength is weaken. We denote this state as a bulk induced surface critical state (BISC). At the bulk critical point, we show evidences supporting an extraordinary surface transition with a long-range order established by effective long-range interactions due to bulk critical fluctuations. The possibility that the state is an extra-ordinary-log state seems unlikely. For the $S=1$ system, we find surface behaviors similar to that of the three-dimensional classical XY model with surface coupling enhanced strong enough, suggesting an extra-ordinary-log state at the bulk critical point.