Special Transition and Extraordinary Phase on the Surface of a (2+1)-Dimensional Quantum Heisenberg Antiferromagnet

Continuous phase transitions exhibit richer critical phenomena on the surface than in the bulk, because distinct surface universality classes can be realized for the same bulk critical point by tuning the surface interactions. The exploration of surface critical behavior provides a window looking into the higher-dimensional boundary conformal field theories. In this work, we discover a direct special transition from the ordinary phase into an extraordinary phase on the surface of a two-dimensional (2D) quantum critical Heisenberg model by tuning the surface coupling strength. The extraordinary phase has a long-range antiferromagnetic order on the surface, in sharp contrast to the logarithmically decaying spin correlations in the 3D classical O(2) and O(3) models. The anomalous dimension of the surface antiferromagnetic order in the extraordinary phase $\eta_{\parallel}=-0.29(1)$ is independent of the surface coupling strength. The special transition point also has a new set of critical exponents, $y_{s}=0.82(4)$ and $\eta_{\parallel}=-0.304(5)$, which are distinct from the special transition of the classical O(3) model and indicates a new surface universality class of the 3D O(3) Wilson-Fisher theory.