  Tensor networks are often used to accurately represent ground states of
quantum spin chains. Two popular choices of such tensor network representations
can be seen to implement linear maps that correspond, respectively, to
euclidean time evolution and to global scale transformations. In this paper, by
exploiting the local structure of the tensor networks, we explain how to also
implement local or non-uniform versions of both euclidean time evolution and
scale transformations. We demonstrate our proposal with a critical quantum spin
chain on a finite circle, where the low energy physics is described by a
conformal field theory (CFT), and where non-uniform euclidean time evolution
and local scale transformations are conformal transformations acting on the
Hilbert space of the CFT. We numerically show, for the critical quantum Ising
chain, that the proposed tensor networks indeed transform the low energy states
of the periodic spin chain in the same way as the corresponding conformal
transformations do in the CFT.
