Ion heating by Alfvén waves has been considered for long as the mechanism explaining why the solar corona has a temperature several orders of magnitude higher than the photosphere. Unfortunately, as the measured wave frequencies are much smaller than the ion cyclotron frequency, particles were expected to behave adiabatically, impeding a direct wave-particle energy transfer to take place, except through decorrelating stochastic mechanisms related to broadband wave spectra. This paper proposes a new paradigm for this mechanism by showing it is actually much simpler, more general, and very efficient. Indeed, for measured wave amplitudes in the quiet corona, ion orbits are shown to cross quasi-periodically one or several slowly pulsating separatrices in phase space. Now, a separatrix is an orbit with an infinite period, thus much longer than the pulsation one. Therefore, each separatrix crossing cancels adiabatic invariance, and yields a very strong energy transfer from the wave, and thus particle heating. This occurs whatever be the wave spectrum, even a monochromatic one. The proposed mechanism is so efficient that it might lead to a self-organized picture of coronal heating: all Alfvén waves exceeding a threshold are immediately quenched and transfer their energy to the ions.