  Heat transport in one-dimensional (1D) momentum-conserving lattices is
generally assumed to be anomalous, thus yielding a power-law divergence of
thermal conductivity with system length. However, whether heat transport in
two-dimensional (2D) system is anomalous or not is still on debate because of
the difficulties involved in experimental measurements or due to the
insufficiently large simulation size. Here, we simulate energy and momentum
diffusion in the 2D nonlinear lattices using the method of fluctuation
correlation functions. Our simulations confirm that energy diffusion in the 2D
momentum-conserving lattices is anomalous and can be well described by the
L\'{e}vy-stable distribution. We also find that the disappear of side peaks of
heat mode may suggest a weak coupling between heat mode and sound mode in the
2D nonlinear system. It is also observed that the harmonic interactions in the
2D nonlinear lattices can accelerate the energy diffusion. Contrary to the
hypothesis of 1D system, we clarify that anomalous heat transport in the 2D
momentum-conserving system cannot be corroborated by the momentum
superdiffusion any more. Moreover, as is expected, lattices with a nonlinear
on-site potential exhibit normal energy diffusion, independent of the
dimension. Our findings offer some valuable insights into the mechanism of
thermal transport in 2D system.
