Entanglement Renormalization for Quantum Fields in Real Space

We show how to construct renormalization group (RG) flows of quantum field theories in real space, as opposed to the usual Wilsonian approach in momentum space. This is achieved by generalizing the multiscale entanglement renormalization ansatz to continuum theories. The variational class of wave functions arising from this RG flow are translation invariant and exhibits an entropy-area law. We illustrate the construction for a free nonrelativistic boson model, and argue that the full power of the construction should emerge in the case of interacting theories.

Figure 1

The momentum occupation $n(p)$ for the cMERA approximation of the ground state of the Hamiltonian in Eq. (8) with $\mu /2\nu =1.1$ ($\Delta /2\nu \approx 0.46$) for different values of the integration limit $s_{\xi }$. Note that the error becomes negligible if $s_{\xi }$ is sufficiently larger than $-1/2\log ({\Lambda }^2/\Delta )$ and that the UV error (vertical offset between exact and approximate results at large momentum) is completely invisible even for the moderate choice of the UV cutoff $\Lambda =3\sqrt{2\nu }$.