Entanglement Renormalization of Thermofield Double States

Entanglement renormalization is a method for "coarse graining" a quantum state in real space, with the multiscale entanglement renormalization ansatz as a notable example. We obtain an entanglement renormalization scheme for finite-temperature (Gibbs) states by applying the multiscale entanglement renormalization ansatz to their canonical purification, the thermofield double state. As an example, we find an analytically exact renormalization circuit for a finite-temperature two-dimensional toric code that 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.