Simulating a catalyst induced quantum dynamical phase transition of a Heyrovsky reaction with different models for the environment

We show analytically that the molecular dissociation occurring in a Heyrovsky reaction can be interpreted as a Quantum Dynamical Phase Transition, i.e. an analytical discontinuity in the molecular energy spectrum. Through appropriate election of the molecular orbital basis, it is shown that the metallic substrate plays the role of an environment that produces an energy uncertainty that induces the critical behavior not possible in a quantum closed system. This, together with perturbation theory allows us to give analytical estimates for the critical parameters. Within suitable approximations we find that dissociation occurs when the bonding to the surface is twice the bonding of the molecule. However simple, this conclusion involves high order perturbative solution of the model and the substrate. The model is further simplified to discuss how this critical phenomenon can be evaluated through an idealized perturbative tunneling microscopy setting. In this case, the energy uncertainties in one or both atoms are either Lorentzian or Gaussian, where the former could be identified with a voltage probe described in the Fermi Golden Rule approximation. In order to generate a Gaussian energy uncertainty we introduce and solve a particular model that can be assimilated to a spin bath. The partially coherent tunneling current is obtained with the Keldysh formalism in its linearized version: the Generalized Landauer-B\"uttiker Equations. The results clarify some of the subtleties involved in the thermodynamic limit and in the use of the simpler steady state energy representation instead of the more natural, but complex, time domain representation.