Generalized $U(1)$ Gauge Field Theories and Fractal Dynamics

TL;DR

This paper develops a theoretical framework for generalized $U(1)$ gauge theories with fractal and subdimensional charge dynamics, extending higher-rank gauge theories to broken rotational symmetries and describing fractal operators.

Contribution

It introduces a continuum effective field theory for fractal and subdimensional charges, generalizing higher-rank $U(1)$ gauge theories to broken rotational symmetries and fractal configurations.

Findings

Describes charge configurations at fractal operator corners.

Provides a continuum theory for Haah's code and Sierpinski prism models.

Introduces a 3+1D $U(1)$ theory without a $ ext{Z}_p$ counterpart.

Abstract

We present a theoretical framework for a class of generalized $U(1)$ gauge effective field theories. These theories are defined by specifying geometric patterns of charge configurations that can be created by local operators, which then lead to a class of generalized Gauss law constraints. The charge and magnetic excitations in these theories have restricted, subdimensional dynamics, providing a generalization of recently studied higher-rank symmetric $U(1)$ gauge theories to the case where arbitrary spatial rotational symmetries are broken. These theories can describe situations where charges exist at the corners of fractal operators, thus providing a continuum effective field theoretic description of Haah's code and Yoshida's Sierpinski prism model. We also present a $3+1$-dimensional $U(1)$ theory that does not have a non-trivial discrete $\mathbb{Z}_p$ counterpart.