Entanglement renormalization for weakly interacting fields

We adapt the techniques of entanglement renormalization tensor networks to weakly interacting quantum field theories in the continuum. A key tool is “quantum circuit perturbation theory,” which enables us to systematically construct unitaries that map between wave functionals which are Gaussian with arbitrary perturbative corrections. As an application, we construct a local continuous multiscale entanglement renormalization ansatz (cMERA) circuit that maps an unentangled scale-invariant state to the ground state of ${\phi }^4$ theory to one loop. Our local cMERA circuit corresponds exactly to one-loop Wilsonian renormalization group (RG) flow on the spatial momentum modes. In other words, we establish that perturbative Wilsonian RG on spatial momentum modes can be equivalently recast as a local cMERA circuit in ${\phi }^4$ theory and argue that this correspondence holds more generally. Our analysis also suggests useful numerical ansätze for cMERA in the nonperturbative regime.

Figure 1

The kernels comprising the entangler in position space at different values of $u$. Here we have set $m=1$ and $\mathrm{\Lambda }=100$.