  We propose a general method of cooling -- periodic driving generated by
spatially deformed Hamiltonians -- and study it in general one-dimensional
quantum critical systems described by a conformal field theory. Our protocol is
able to efficiently cool a finite-temperature Gibbs (mixed) state down to zero
temperature at prescribed sub-regions exponentially rapidly in Floquet time
cycles. At the same time, entropy and energy are transferred and localized to
the complementary regions that shrink with time. We derive these conclusions
through an exact analytic solution of the full time evolution of reduced
density matrices. We also use numerics in free-fermion lattice models as a
benchmark and find remarkable agreement. Such conformal Floquet refrigerators
open a promising new route to cooling synthetic quantum systems.
