  We show that the asymmetric inter-particle interactions can induce rapid
decay of the heat current correlation in one-dimensional momentum conserving
lattices. When the asymmetry degree is appropriate, even exponential decay may
arise. This fact suggests that the power-law decay predicted by the
hydrodynamics may not be applied to the lattices with asymmetric inter-particle
interactions, and as a result, the Green-Kubo formula may instead lead to a
convergent heat conductivity in the thermodynamic limit. The mechanism of the
rapid decay is traced back to the fact that the heat current has to drive a
mass current additionally in the presence of the asymmetric inter-particle
interactions.
