  Rydberg atoms in an optical tweezer array have been used as a quantum
simulator of the spin-$1/2$ antiferromagnetic Ising model with longitudinal and
transverse fields. We suggest how to implement the next-nearest-neighbor (NNN)
interaction whose sign is opposite to that of the nearest neighbor one in the
Rydberg atom systems. We show that this can be achieved by weakly coupling one
Rydberg state with another Rydberg state. We further study the surface
criticality associated with the first-order quantum phase transition between
the antiferromagnetic and paramagnetic phases, which emerges due to the
sign-inverted NNN interaction. From the microscopic model, we derive a
Ginzburg-Landau (GL) equation, which describes static and dynamic properties of
the antiferromagnetic order parameter near the transition. Using both
analytical GL theory and numerical method based on a mean-field theory, we
calculate the order parameter in the proximity of a boundary of the system in
order to show that the healing length of the order parameter logarithmically
diverges, signaling the surface criticality.
