  In this work, we propose a class of ferroelectrics (which we denote
"double-path" ferroelectrics), characterized by two competing polarization
switching paths for which the change in polarization is different and in fact
of opposite sign. Depending on which path is favorable under given conditions,
this leads to different identification of up- and down-polarized states. Since
the sign of piezoelectric response depends on the assignment of up- or
down-polarized state for a specific structure, this means that the material can
exhibit different signs of the piezoelectric response under different
conditions. We focus on HfO$_2$ as a key example. Our first-principles
calculations show that there are two competing paths in HfO$_2$, resulting from
different displacements of the atoms from the initial to the final structures,
and the change in polarization along these two paths is of opposite sign. These
results provide a natural explanation for the recently observed discrepancy in
the signs of piezoelectric responses in HfO$_2$ between theoretical
first-principles calculations and experimental observation. Further, this
allows predictions of how to favor one path over another by changes in
conditions and compositional tuning. This family of materials also includes
other candidates, such as CuInP$_2$S$_6$ and theoretically proposed
LaVO$_3$-SrVO$_3$ superlattice. We finally note that double-path ferroelectrics
possess novel electromechanical properties since the signs of their
piezoelectric responses can be switched.
