Anomalous isotopic effect near the charge-ordering quantum criticality

Within the Hubbard-Holstein model, we evaluate the various crossover lines marking the opening of pseudogaps in the cuprates, which, in our scenario, are ruled by the proximity to a charge-ordering quantum criticality (stripe formation). We provide also an analysis of their isotopic dependencies, as produced by critical fluctuations. We find no isotopic shift of the temperature $T^0$ marked as a reduction of the quasiparticle density of states in various experiments, and a substantial positive shift of the pseudogap-formation temperature $T^*$. We infer that the superconducting critical temperature $T_c$ has almost no shift in the optimally- and overdoped regimes while it has a small negative isotopic shift in the underdoped, which increses upon underdoping. We account also for the possible dynamical nature of the charge-ordering transition, and explain in this way the spread of the values of $T^*$ and its of isotopic shift, obtained with experimental probes having different characteristic timescales.