Unusual superconducting isotope effect in the presence of a quantum criticality

The isotope effect in superconductivity (SC) is used to make a concrete connection to a quantum critical point (QCP) that is tunable by isotopic mass substitution. We find a distinct contribution to the isotope exponent in SC and derive an explicit relation to the critical exponent of a QCP. The relation between the two exponents is general and can be used as an experimental signature for the connection between SC and a QCP. We demonstrate it in a scenario where the SC pairing is due to modes related to a structural instability. Within this model the isotope exponent is derived in terms of microscopic parameters.