Failure of the Generalized Eigenstate Thermalization Hypothesis in integrable models with multiple particle species

It has been recently observed for a particular quantum quench in the XXZ spin chain that local observables do not equilibrate to the predictions of the Generalized Gibbs Ensemble (GGE). In this work we argue that the breakdown of the GGE can be attributed to the failure of the Generalized Eigenstate Thermalization Hypothesis (GETH), which has been the main candidate to explain the validity of the GGE. We provide explicit counterexamples to the GETH and argue that generally it does not hold in models with multiple particle species. Therefore there is no reason to assume that the GGE should describe the long time limit of observables in these integrable models.