  We show that a large class of gapless states are renormalization group fixed
points in the sense that they can be grown scale by scale using local
unitaries. This class of examples includes some theories with dynamical
exponent different from one, but does not include conformal field theories. The
key property of the states we consider is that the ground state wavefunction is
related to the statistical weight of a local statistical model. We give several
examples of our construction in the context of Ising magnetism.
