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

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.

Figures (5)

• Figure 1: Charge configurations created in our $U(1)$ "Haah's Code" gauge theory. Positive charges are orange, negative charges are blue, and black circles indicate the location of a local operator acts. (a) and (b): Generating charge configurations, created by the local action of $e^{iA_1}$ and $e^{iA_2}$ respectively. (c): Typical charge configuration with an isolated $+1$ charge created with repeated application of $e^{iA_1}$ (we have set $a_0=1$ for legibility). Isolating one charge requires a "sheet" of charge of linear size $d$ to be created at distance $d$.
• Figure 2: Hamiltonian for our chosen $\mathbb{Z}_p$ generalization of Haah's code, also used in Ref. HaahU1Code. There is one of each term per elementary cube of the cubic lattice, two $N$-component spins per site, and each term is a product of five generalized Pauli operators.
• Figure 3: Hamiltonian for our chosen $\mathbb{Z}_N$ generalization of Yoshida's Sierpinski prism model. There is one of each term per elementary cube of the cubic lattice, two $N$-component spins per site, and each term is a product of five generalized Pauli operators.
• Figure 4: Charge configurations created in our $U(1)$ Sierpinski prism gauge theory. Color coding is the same as Fig. \ref{['fig:HaahCharges']}. (a) and (b): Generating charge configurations, created by the local action of $e^{iA_1}$ and $e^{iA_2}$ respectively. (c): Typical charge configuration with an isolated $+1$ charge created with repeated application of $e^{iA_2}$ (we have set $a_0=1$ for legibility). Isolating one charge in the $xy$-plane requires a line of charges of length $d$ to be created at distance $d$.
• Figure 5: Charge configurations created by local operators in our simple $U(1)$ model with Gauss law in Eq. \ref{['eqn:NewU1Gauss']}.