Interacting electrons in one dimension beyond the Luttinger-liquid limit

Over the past several decades, Luttinger-liquid theory has a framework for interacting electrons in one dimension. However, the validity of the theory is strictly limited to low-energy excitations where the electron dispersion is linear. Interacting electrons in one-dimension beyond the Luttinger-liquid limit, where the underlying dispersion of electrons is no longer linear, exhibit intriguing manifestations of the interactions, which have direct implications on many experimental systems. For example, consider the energy relaxation of particles or holes, the unoccupied states in a Fermi sea. Whereas in Luttinger-liquid theory such energy relaxation is strictly forbidden, in a nonlinearly dispersing one-dimensional electron system energy relaxation is allowed but very different for particles and holes. Here, we use momentum-resolved tunnelling to selectively inject energetic particles and holes into a quantum wire and study their relaxation processes. Our measurements confirm that energetic particles undergo fast relaxation to a thermalized distribution and holes retain their original injection energy, thereby providing a clear demonstration of electron dynamics beyond the Luttinger limit. A model of thermalization derived in the limit of weak interactions shows quantitative agreement with the experimental findings.