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The isotope effect in the superconducting transition temperature is anomalousif the isotope coefficient $\alpha<0$ or $\alpha>1/2$. In this work, we showthat such anomalous behaviors can naturally arise within theBardeen-Cooper-Schrieffer framework if both phonon and non-phonon modescoexist. Different from the case of the standard Eliashberg theory (with onlyphonon) in which $\alpha\le1/2$, the isotope coefficient can now take arbitraryvalues in the simultaneous presence of phonon and the other non-phonon mode. Inparticular, most strikingly, a pair-breaking phonon can give rise to largeisotope coefficient $\alpha>1/2$ if the unconventional superconductivity ismediated by the lower frequency non-phonon boson mode. Based on our studies,implications on several families of superconductors are discussed.

Anomalous isotope effect in BCS superconductors with two boson modes