Entanglement renormalization of thermofield double states

We generalize the multi-scale entanglement renormalization ansatz [G. Vidal, Phys. Rev. Lett. 99, 220405(2007)] to thermofield double (TFD) states, which are canonical purifications of finite-temperature Gibbs states. This provides both an efficient preparation scheme for TFD states and a way to remove short-range quantum correlations from a Gibbs state, thus unveiling its long-range entanglement structure. As proof of concept, we apply this technique to the two-dimensional toric code model at finite temperature and find an exact renormalization group circuit which connects the finite-temperature toric code model to the infinite temperature state, thus demonstrating the absence of finite temperature topological order for 2d toric code. We also apply it to free boson models at finite temperature and find results consistent with the conventional momentum-space renormalization group procedure in these models.