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Continuous phase transitions exhibit richer critical phenomena on the surfacethan in the bulk, because distinct surface universality classes can be realizedat the same bulk critical point by tuning the surface interactions. Theexploration of surface critical behavior provides a window looking intohigher-dimensional boundary conformal field theories. In this work, we studythe surface critical behavior of a two-dimensional (2D) quantum criticalHeisenberg model by tuning the surface coupling strength, and discover a directspecial transition on the surface from the ordinary phase into an extraordinaryphase. The extraordinary phase has a long-range antiferromagnetic order on thesurface, in sharp contrast to the logarithmic decaying spin correlations in the3D classical O(3) model. The special transition point has a new set of criticalexponents, $y_{s}=0.86(4)$ and $\eta_{\parallel}=-0.33(1)$, which are distinctfrom the special transition of the classical O(3) model and indicate a newsurface universality class of the 3D O(3) Wilson-Fisher theory.

Special Transition and Extraordinary Phase on the Surface of a Two-Dimensional Quantum Heisenberg Antiferromagnet