# Spacetime symmetries and conformal data in the continuous multi-scale entanglement renormalization ansatz

## TL;DR

This paper studies how continuous multi-scale entanglement renormalization ansatz (cMERA) can capture spacetime symmetries and conformal data in free quantum field theories, showing that symmetries are preserved and key data can be extracted quasi-locally.

## Contribution

It demonstrates that in Gaussian cMERA for free theories, spacetime symmetries and conformal data are preserved and can be extracted, extending understanding of cMERA's capabilities.

## Key findings

- Spacetime symmetries are mapped into cMERA states via a quasi-local unitary.
- cMERA stress-energy tensor is quasi-local, not strictly local.
- Conformal data such as scaling dimensions and OPE coefficients are preserved in cMERA.

## Abstract

The generalization of the multi-scale 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 paper we investigate, in the simpler context of Gaussian cMERA for free theories, the extent to which the cMERA state $|\Psi^\Lambda\rangle$ with finite UV cut-off $\Lambda$ can capture the spacetime symmetries of the ground state $|\Psi\rangle$. For a free boson conformal field theory (CFT) in 1+1 dimensions as a concrete example, we build a quasi-local unitary transformation $V$ that maps $|\Psi\rangle$ into $|\Psi^\Lambda\rangle$ and show two main results. (i) Any spacetime symmetry of the ground state $|\Psi\rangle$ is also mapped by $V$ into a spacetime symmetry of the cMERA $|\Psi^\Lambda\rangle$. 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, the corresponding cMERA stress-energy tensor $T_{\mu\nu}^{\Lambda}(x) = V T_{\mu\nu}(x) V^{\dagger}$ is quasi-local. (ii) From the cMERA, we can extract quasi-local scaling operators $O^{\Lambda}_{\alpha}(x)$ 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.