  A recent study shows that an emergent evolution dimension, in addition to
time, can be induced if the quantum system depends on a continuous parameter.
The evolution in the emergent evolution dimension is described by a parallel
transport, the closest fiber bundle analog to a geodesic in (pseudo-)Riemannian
geometry. Nevertheless, the evolution in the emergent dimension often exhibits
some singular behaviors at critical points. In this work, we demonstrate that
these singularities can be indications of quantum phase transitions. We then
show that these singularities, like those at the black hole event horizon, can
be removed locally.
