  Two-dimensional (2D) silicon carbide is an emergent direct band-gap
semiconductor, recently synthesized, with potential applications in electronic
devices and optoelectronics. Here, we study nuclear quantum effects in this 2D
material by means of path-integral molecular dynamics (PIMD) simulations in the
temperature range from 25 to 1500~K. Interatomic interactions are modeled by a
tight-binding Hamiltonian fitted to density-functional calculations. Quantum
atomic delocalization combined with anharmonicity of the vibrational modes
cause changes in structural and thermal properties of 2D SiC, which we quantify
by comparison of PIMD results with those derived from classical molecular
dynamics simulations, as well as with those given by a quantum harmonic
approximation. Nuclear quantum effects are found to be appreciable in
structural properties such as the layer area and interatomic distances.
Moreover, we consider a {\em real} area for the SiC sheet, which takes into
account bending and rippling at finite temperatures. Differences between this
area and the in-plane area are discussed in the context of quantum atomic
dynamics. The bending constant ($\kappa = 1.0$ eV) and the 2D modulus of
hydrostatic compression ($B_{xy}$ = 5.5 eV/\AA$^2$) are clearly lower than the
corresponding values for graphene. This study paves the way for a deeper
understanding of the elastic and mechanical properties of 2D SiC.
