  Hafnium dioxide (HfO2) is a promising ferroelectric (FE) material for
achieving high-density nonvolatile memory and neuromorphic computing, due to
its compatibility with the mainstream integrated circuit technology and the
surprisingly enhanced ferroelectricity by reduced thickness. The FE switching
dynamics is essential to the device performance, but the complexity of HfO2
atomic structure causes unknown of various FE switching paths and domain wall
configurations. Here, we demonstrate that its low-barrier paths and domain
walls can be comprehensively found and understood from a perspective of
topological symmetry. By discussing pseudo-chirality and equivalent
transformation relations in crystal with first principles and lattice modes, we
classify and analyze 4 low-barrier FE switching paths and 93 irreducible
topology domain wall configurations in HfO2. Anisotropic switching mechanism is
found based on the mobility investigation for 12 types of 180{\deg} side domain
walls. This methodology is expected to be generally applicable to displacive
ferroelectrics with low unit cell point group symmetries, and lay a foundation
for mechanism study of the switching dynamics.
