  Anomalous proximity effect is a remarkable phenomenon which arises due to the
cooperative interplay oftopologically-protected quasiparticles at zero-energy
and odd-frequency Cooper pairs. So far, a spin-triplet superconductor, which is
difficult to find in nature, has been supposed to be essential for the
anomalous proximity effect, and therefore the experimental observation of this
effect has been considered to be a challenging issue. In this paper, we
demonstrate that the anomalous proximity effect also occurs in a
semiconductor/high-$T_c$ cuprate superconductor hybrid devices in which two
potentials coexist: a spin-singlet $d$-wave pair potential and a spin-orbit
coupling potential sustaining persistent spin-helix. By utilizing the
spin-singlet superconductors, we propose an alternative and promising route to
observe the anomalous proximity effect.
