The isotope effect with a large coefficient $\alpha =-\partial ln{T}_c/\partial lnM$ is usually taken as evidence of phonon-mediated superconductors in the Bardeen-Cooper-Schrieffer (BCS) theory. However, in cuprates which are now widely believed to be strong correlation-induced $d$-wave superconductors, $\alpha$ is experimentally observed to be quite small at optimal doping, but keeps growing with decreasing $T_c$ upon doping, even after exceeding the BCS value $1/2$. Such an anomalous isotope effect seems to challenge the nonphonon picture and still leave room for the phonon-dominated mechanism. In this Letter, we show that the anomalous dependence of $\alpha$ on $T_c$ can actually be obtained in spin-fluctuation-induced $d$-wave superconductors by studying the Hubbard model on square lattices with functional renormalization group. We have considered two types of electron-phonon couplings (EPCs). The first type couples to electron densities, including the Holstein, breathing, and buckling phonons, called Holstein-like. For all these EPCs, $\alpha$ is negative and drops down towards $-\infty$ with decreasing $T_c$ upon doping. On the opposite, for the other type of Peierls-like EPC coupling to electron hoppings on the nearest bonds, also called Su-Schrieffer-Heeger phonon, $\alpha$ is positive, grows with decreasing $T_c$, and tends to diverge as $T_c\to 0$, in qualitative agreement with the experiments. The difference between these two types of EPCs can be understood by their isotope effects on spin fluctuations. From this study, we conclude that the SSH phonon can explain the anomalous isotope effect in cuprates, although it is not the leading pairing mechanism.

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• Isotope effect
• d-wave
• Unconventional superconductors
• Functional renormalization group