Anomalous Isotopic Effect Near the Charge-Ordering Quantum Criticality

Within the Hubbard-Holstein model, we evaluate the 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 find that their isotopic dependence, due to critical fluctuations, implies a substantial positive shift of the pseudogap-formation temperature $T^{*}$. We infer that the isotopic shift of the superconducting $T_c$ is nearly absent in the optimally and overdoped regimes and is negative and increasing upon underdoping. The dynamical nature of the charge-ordering transition may explain the spread of the experimental values of $T^{*}$.