  Isotope effect with a large coefficient $\alpha=-\partial \ln T_c/\partial
\ln M$ is usually taken as an 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 up with decreasing $T_c$ upon doping, even after exceeding the
BCS value $1/2$. Such an anomalous isotope effect seems to challenge the
non-phonon picture and still leave room for the phonon-dominated mechanism. In
this work, 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 up with decreasing $T_c$ and tends to diverge as
$T_c\to0$, 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.
