Renormalization Group Circuits for Weakly Interacting Continuum Field Theories

TL;DR

The paper introduces quantum circuit perturbation theory to construct local unitaries linking scale-invariant product states to ground states of weakly interacting continuum field theories, enabling systematic, perturbative cMERA constructions. It develops a continuum BD-based operator framework and generalizes cMERA to interacting fields, delivering a 1-loop cMERA circuit for φ^4 theory where the disentangler implements momentum-space Wilsonian RG in position space. The work demonstrates exponential locality of the perturbative entangler kernels and articulates a precise bridge between momentum-space RG and position-space entanglement renormalization, with practical implications for variational numerics in weakly interacting QFTs. Collectively, it provides analytic tools and perturbative templates for crafting non-Gaussian cMERA circuits and suggests avenues for extending to fermions, gauge theories, and holographic contexts, as well as guiding efficient numerical variational strategies.

Abstract

We develop techniques to systematically construct local unitaries which map scale-invariant, product state wavefunctionals to the ground states of weakly interacting, continuum quantum field theories. More broadly, we devise a "quantum circuit perturbation theory" to construct local unitaries which map between any pair of wavefunctionals which are each Gaussian with arbitrary perturbative corrections. Further, we generalize cMERA to interacting continuum field theories, which requires reworking the existing formalism which is tailored to non-interacting examples. Our methods enable the systematic perturbative calculation of cMERA circuits for weakly interacting theories, and as a demonstration we compute the 1-loop cMERA circuit for scalar $\varphi^4$ theory and analyze its properties. In this case, we show that Wilsonian renormalization of the spatial momentum modes is equivalent to a local position space cMERA circuit. This example provides new insights into the connection between position space and momentum space renormalization group methods in quantum field theory. The form of cMERA circuits derived from perturbation theory suggests useful ansatzes for numerical variational calculations.

Figures (2)

• Figure 1: The kernels comprising the entangler in position space at different values of $u$, with a soft cutoff (i.e., a sigmoid instead of a Heaviside step function). Here we work in $(1+1)$ dimensions. The upper left kernel is the free kernel $f_{2,0}(x,0 \, ; u)$ and the upper right kernel is the quadratic kernel $f_{2,1}(x,0 \,; u)$. The momentum space equations corresponding to the upper plots were given in \ref{['eq:f2i']}. The lower left and lower right plots are the quartic kernels $f_4^{(1)}(x,0,0,0 \,; u)$ and $f_4^{(3)}(x,0,0,0 \,; u)$, respectively. The corresponding momentum space equations were given in \ref{['f41sol2']}. In all plots we have set $m=1$ and $\Lambda=100$.
• Figure 2: Branch cuts in the complex $z$--plane.