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

TL;DR

This paper introduces a framework called field digitization scaling (FDS) for analyzing the continuum limit of digitized quantum field theories, using RG and tensor networks, with applications to classical and quantum models.

Contribution

It develops a renormalization group approach to field digitization, relating different discretizations, and demonstrates its effectiveness through tensor network simulations and analytical proofs.

Findings

Uncovers a universal crossover near phase transitions induced by finite N.

Establishes a direct relation between classical Z_N clock models and quantum Z_N lattice gauge theories.

Proposes FDS as a tool for continuum analysis in digitized quantum simulations.

Abstract

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.