  We discuss a tensor network method for constructing the adiabatic gauge
potential -- the generator of adiabatic transformations -- as a matrix product
operator, which allows us to adiabatically transport matrix product states.
Adiabatic evolution of tensor networks offers a wide range of applications, of
which two are explored in this paper: improving tensor network optimization and
scanning phase diagrams. By efficiently transporting eigenstates to quantum
criticality and performing intermediary density matrix renormalization group
(DMRG) optimizations along the way, we demonstrate that we can compute ground
and low-lying excited states faster and more reliably than a standard DMRG
method at or near quantum criticality. We demonstrate a simple automated step
size adjustment and detection of the critical point based on the norm of the
adiabatic gauge potential. Remarkably, we are able to reliably transport states
through the critical point of the models we study.
