  Surface critical behavior (SCB) refers to the singularities of physical
quantities on the surface at the bulk phase transition. It is closely related
to and even richer than the bulk critical behavior. In this work, we show that
three types of SCB universality are realized in the dimerized Heisenberg models
at the (2+1)-dimensional O(3) quantum critical points by engineering the
surface configurations. The ordinary transition happens if the surface is
gapped in the bulk disordered phase, while the gapless surface state generally
leads to the multicritical special transition, even though the latter is
precluded in classical phase transitions because the surface is in the lower
critical dimension. An extraordinary transition is induced by the ferrimagnetic
order on the surface of the staggered Heisenberg model, in which the surface
critical exponents violate the results of the scaling theory and thus seriously
challenge our current understanding of extraordinary transitions.
