  Entanglement renormalization is a method for coarse-graining a quantum state
in real space, with the multi-scale entanglement renormalization ansatz (MERA)
as a notable example. We obtain an entanglement renormalization scheme for
finite-temperature (Gibbs) states by applying MERA to their canonical
purification, the thermofield double state. As an example, we find an
analytically exact renormalization circuit for finite temperature
two-dimensional toric code which maps it to a coarse-grained system with a
renormalized higher temperature, thus explicitly demonstrating its lack of
topological order. Furthermore, we apply this scheme to one-dimensional free
boson models at a finite temperature and find that the thermofield double
corresponding to the critical thermal state is described by a Lifshitz theory.
We numerically demonstrate the relevance and irrelevance of various
perturbations under real space renormalization.
