In this work, we perform a detailed study of heat transport in one dimensional long-ranged anharmonic oscillator systems, such as the long-ranged Fermi-Pasta-Ulam-Tsingou model. For these systems, the long-ranged anharmonic potential decays with distance as a power-law, controlled by an exponent $δ\geq 0$. For such a non-integrable model, one of the recent results that has captured quite some attention is the puzzling ballistic-like transport observed for $δ= 2$, reminiscent of integrable systems. Here, we first employ the reverse nonequilibrium molecular dynamics simulations to look closely at the $δ= 2$ transport in three long-ranged models, and point out a few problematic issues with this simulation method. Next, we examine the process of energy relaxation, and find that relaxation can be appreciably slow for $δ= 2$ in some situations. We invoke the concept of nonlinear localized modes of excitation, also known as discrete breathers, and demonstrate that the slow relaxation and the ballistic-like transport properties can be consistently explained in terms of a novel depinning of the discrete breathers that makes them highly mobile at $δ= 2$. Finally, in the presence of quartic pinning potentials we find that the long-ranged model exhibits Fourier (diffusive) transport at $δ= 2$, as one would expect from short-ranged interacting systems with broken momentum conservation. Such a diffusive regime is not observed for harmonic pinning.
