  There is a well-known mapping between energy normal (super-) diffusion and
normal (anomalous) heat conduction in one-dimensional (1D) nonlinear lattices.
The momentum conserving nonlinear lattices exhibit energy super-diffusion
behavior with the only exception of coupled rotator model. Yet, for all other
1D momentum nonconserving nonlinear lattices studied so far, the energy
diffusion or heat conduction is normal. Here we propose a 1D nonlinear lattice
model with negative couplings, which is momentum non-conserving due to the
translational symmetry breaking. Our numerical results show that energy
super-diffusion instead of normal diffusion can be found for this model, which
indicates that neither momentum non-conservation is a sufficient condition for
energy normal diffusion nor momentum conservation is a necessary condition for
energy super-diffusion. Zero frequency phonon mode at Brillouin zone boundary
induces a new conserved momentum parity, which is the key for the energy
super-diffusion and anomalous heat conduction. Removing the zero frequency
mode, such as by on-site potential, is a sufficient condition for normal heat
conduction in 1D nonlinear lattices.
