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 $|{\mathrm{\Psi }}^{\mathrm{\Lambda }}⟩$ with finite UV cutoff $\mathrm{\Lambda }$ can capture the spacetime symmetries of the ground state $|\mathrm{\Psi }⟩$. 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 $|\mathrm{\Psi }⟩$ into $|{\mathrm{\Psi }}^{\mathrm{\Lambda }}⟩$ and show two main results. (i) Any spacetime symmetry of the ground state $|\mathrm{\Psi }⟩$ is also mapped by $V$ into a spacetime symmetry of the cMERA $|{\mathrm{\Psi }}^{\mathrm{\Lambda }}⟩$. 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 $T_{\mu \nu }^{\mathrm{\Lambda }}(x)=V{T}_{\mu \nu }(x)V^{†}$ is quasilocal. (ii) From the cMERA, we can extract quasilocal scaling operators $O_{\alpha }^{\mathrm{\Lambda }}(x)$ characterized by the exact same scaling dimensions ${\mathrm{\Delta }}_{\alpha }$, 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.

Figure 1

Left: Function $g(k)=\exp [-e^{-\gamma }(k/\mathrm{\Lambda }{)}^2]/2$ for the optimized Gaussian cMERA of Ref. [18], where $\gamma \approx 0.57722$ is Euler’s constant [23]. For comparison, the CFT dilation operator $D$ such that $D|\mathrm{\Psi }⟩=0$ corresponds to choosing $g(k)=1/2$ and the nonrelativistic dilation operator $L$ such that $L|\mathrm{\Lambda }⟩=0$ corresponds to $g(k)=0$). Right: Function $\alpha (k)$ for $g(k)$ interpolating between the linear dependence $k$ for the CFT ground state $|\mathrm{\Psi }⟩$ at small $k$ and the constant value $\mathrm{\Lambda }$ for the product state $|\mathrm{\Lambda }⟩$ at large $k$.