Field digitization scaling in a $\mathbb{Z}_N \subset U(1)$ symmetric model

The simulation of quantum field theories, both classical and quantum, requires regularization of infinitely many degrees of freedom. However, in the context of field digitization (FD) -- a truncation of the local fields to $N$ discrete values -- a comprehensive framework to obtain continuum results is currently missing. Here, we propose to analyze FD by interpreting the parameter $N$ as a coupling in the renormalization group (RG) sense. As a first example, we investigate the two-dimensional classical $N$-state clock model as a $\mathbb{Z}_N$ FD of the $U(1)$-symmetric $XY$-model. Using effective field theory, we employ the RG to derive generalized scaling hypotheses involving the FD parameter $N$, which allows us to relate data obtained for different $N$-regularized models in a procedure that we term $\textit{field digitization scaling}$ (FDS). Using numerical tensor-network calculations at finite bond dimension $\chi$, we further uncover an unconventional universal crossover around a low-temperature phase transition induced by finite $N$, demonstrating that FDS can be extended to describe the interplay of $\chi$ and $N$. Finally, we analytically prove that our calculations for the 2D classical-statistical $\mathbb{Z}_N$ clock model are directly related to the quantum physics in the ground state of a (2+1)D $\mathbb{Z}_N$ lattice gauge theory which serves as a FD of compact quantum electrodynamics. Our study thus paves the way for applications of FDS to quantum simulations of more complex models in higher spatial dimensions, where it could serve as a tool to analyze the continuum limit of digitized quantum field theories.