Spacetime Symmetries and Conformal Data in the Continuous Multiscale Entanglement Renormalization Ansatz

The generalization of the multiscale entanglement renormalization ansatz (MERA) to continuous systems, or cMERA [Haegeman et al., Phys. Rev. Lett. 110, 100402 (2013)], is expected to become a powerful variational ansatz for the ground state of strongly interacting quantum field theories. In this Letter, we investigate, in the simpler context of Gaussian cMERA for free theories, the extent to which the cMERA state vertical bar Psi<^>> with finite UV cutoff <^> can capture the spacetime symmetries of the ground state vertical bar Psi >. For a free boson conformal field theory (CFT) in 1 + 1 dimensions, as a concrete example, we build a quasilocal unitary transformation V that maps vertical bar Psi > into vertical bar Psi<^>> and show two main results. (i) Any spacetime symmetry of the ground state vertical bar Psi > is also mapped by V into a spacetime symmetry of the cMERA vertical bar Psi<^>>. However, while in the CFT, the stress-energy tensor T-mu nu (x) (in terms of which all the spacetime symmetry generators are expressed) is local, and the corresponding cMERA stress-energy tensor T-mu nu(<^>)(x) = VT mu nu(x)V-dagger is quasilocal. (ii) Fromthe cMERA, we can extract quasilocal scaling operators O-alpha(<^>)(x) characterized by the exact same scaling dimensions Delta(alpha), conformal spins s(alpha), operator product expansion coefficients C-alpha beta gamma, and central charge c as the original CFT. Finally, we argue that these results should also apply to interacting theories.