Charge Sharpening via Emergent Gauge Theory

In this work, we study the dynamics of $d$-dimensional random circuits possessing a generic global symmetry $G$ composed of local gates and symmetric measurements. Such structured circuits can be decomposed into two parts - a symmetric layer responsible for the propagation of charged degrees of freedom and a non-symmetric layer responsible for the propagation of uncharged degrees of freedom. Interestingly the symmetric layer yields a $G$-symmetric wavefunction of a lattice gauge theory. For instance, in the case of a $\mathbb{Z}_n$ symmetric unitary circuit, the wavefunction can be chosen to be the deconfined fixed point state of the $\mathbb{Z}_{n}$ lattice gauge theory. The inclusion of measurements leads to the creation of fluxes in the wavefunction causing it to eventually undergo a confinement transition. We argue that this phase transition precisely corresponds to the charge-sharpening transition of the quantum circuit. Our methodology is general and can be adapted to any abelian or non-abelian symmetry groups which reproduces and extends known results.