Entanglement renormalization for gauge invariant quantum fields

TL;DR

This paper extends the cMERA variational wavefunctional framework to gauge invariant quantum fields, specifically U(1) gauge theories, enabling accurate long-distance descriptions while maintaining gauge invariance and local Hamiltonian properties.

Contribution

It introduces a gauge invariant cMERA construction for U(1) gauge theories, addressing gauge constraints compatibility with the entanglement structure.

Findings

Constructed a gauge invariant cMERA wavefunctional for U(1) gauge theory.

Showed the wavefunctional reproduces long-distance properties of the ground state.

Derived a local Hamiltonian with the same low-energy physics as the original.

Abstract

The continuous multiscale entaglement renormalization ansatz (cMERA) [Haegeman et al., Phys. Rev. Lett., 110, 100402 (2013)] is a variational wavefunctional for ground states of quantum field theories. So far, only scalar bosons and fermions have been considered. In this paper we explain how to generalize the cMERA framework to gauge invariant quantum fields. The fundamental difficulty to be addressed is how to make the gauge constraints (local linear constraints in the Hilbert space) compatible with the UV structure of the cMERA wavefunctional (which is generated by a quasi-local entangler). For simplicity, we consider $U(1)$ gauge theory in $d+1$ spacetime dimensions, a non-interacting theory with massless Hamiltonian $H_{U(1)}$ and Gaussian scale invariant ground state $|\Psi_{U(1)}\rangle$. We propose a gauge invariant cMERA wavefunctional $|\Psi^{\Lambda}_{U(1)}\rangle$ that, by construction, accurately reproduces the long distance properties of $|\Psi_{U(1)}\rangle$ while remaining somewhat unentangled at short distances. Moreover, $|\Psi^{\Lambda}_{U(1)}\rangle$ is the exact ground state of a gauge invariant, local Hamiltonian $H^{\Lambda}_{U(1)}$ whose low energy properties coincide with those of $H_{U(1)}$. Our construction also extends the cMERA formalism to massive (non-gauge invariant) vector boson quantum fields.