[Paper Review] Generalized $U(1)$ Gauge Field Theories and Fractal Dynamics

Daniel Bulmash, Maissam Barkeshli|arXiv (Cornell University)|Jun 5, 2018

Theoretical and Computational Physics32 references19 citations

TL;DR

This paper introduces a generalized framework for $U(1)$ gauge field theories with subdimensional dynamics by defining Gauss law constraints through geometric patterns of charge configurations created by local operators. It establishes a continuum effective field theory for fracton models like Haah’s code and Yoshida’s Sierpinski prism model, demonstrating that non-trivial fracton phases can exist without a $π$-flux or $π$-flux counterpart, and proves conditions under which a non-degenerate magnetic field can be consistently defined.

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.

Motivation & Objective

• To develop a continuum effective field theory for generalized $U(1)$ gauge theories with restricted, subdimensional dynamics of charges and magnetic excitations.
• To provide a field-theoretic description of fracton models such as Haah’s code and Yoshida’s Sierpinski prism model, which previously lacked such a description.
• To identify conditions under which a non-degenerate, Maxwell-type gauge theory can be constructed from a set of allowed charge configurations at the cutoff scale.
• To demonstrate the existence of $U(1)$ theories without non-trivial $π$-flux or $π$-flux counterparts, indicating novel topological phases.
• To clarify the role of local operators in determining the mobility of charges and the structure of gauge-invariant observables.

Proposed method

• Define generalized Gauss laws via differential operators $D_i$ that map electric fields $E_i$ to charge density $\rho$, based on the geometric patterns of charge configurations created by local operators at the cutoff scale.
• Use the condition that a charge is mobile in a direction if and only if a dipole oriented along that direction can be created by a local operator, linking mobility to operator algebra.
• Construct gauge-invariant magnetic fields as $\tilde{G}_i A_j - \tilde{G}_j A_i$ for $i \neq j$, where $\tilde{G}_i$ are differential operators derived from the $D_i$.
• Prove that a non-degenerate magnetic field exists if and only if the operators $\tilde{D}_i^\lambda$ admit a common factor $\tilde{G}_i$, ensuring consistency and non-degeneracy.
• Use lattice regularization in both cubic and rhombohedral coordinates to connect the continuum theory to discrete models, showing that the theory reduces to standard $U(1)$ gauge theory on a rhombohedral lattice under appropriate coordinate transformation.
• Derive effective Hamiltonians and conserved quantities from the generalized Gauss law and gauge transformation rules, ensuring consistency with quantum constraints.

Experimental results

Research questions

• RQ1 Under what conditions can a continuum effective field theory be constructed from a specified set of cutoff-scale charge configurations?
• RQ2 How can a non-degenerate magnetic field be consistently defined in generalized $U(1)$ gauge theories with subdimensional dynamics?
• RQ3 Can $U(1)$ gauge theories exist that lack a non-trivial $\mathbb{Z}_p$ counterpart, even when their $\mathbb{Z}_p$ versions have fully mobile anyons?
• RQ4 What determines the mobility of charges in such generalized gauge theories—specifically, when can a charge be moved freely in a given direction?
• RQ5 How do the algebraic properties of the differential operators $D_i$ relate to the existence of gauge-invariant observables and the structure of the gauge group?

Key findings

• A generalized $U(1)$ gauge theory can be constructed from any set of allowed charge configurations created by local operators at the cutoff scale, with the dynamics of charges fully determined by these configurations.
• The theory provides the first continuum effective field theory description of Haah’s code and Yoshida’s Sierpinski prism model, where isolated charges require exponentially large energy to create due to fractal operator structure.
• For $M=2$ flavors of $U(1)$ gauge fields, a non-degenerate magnetic field exists if and only if the differential operators $\tilde{D}_i^\lambda$ share a common factor $\tilde{G}_i$, ensuring non-degeneracy and gauge-invariance.
• The paper proves that $N > M$ is sufficient for the existence of a non-trivial magnetic field in $M=1$ and $M=2$ cases, with explicit constructions provided for $M=2$, $N=3$.
• It is shown that there exist $U(1)$ theories without a non-trivial $\mathbb{Z}_p$ counterpart, indicating the existence of novel topological phases not captured by conventional anyon statistics.
• Lattice regularization in rhombohedral coordinates yields a standard $U(1)$ one-form gauge theory, confirming consistency with known discrete models under appropriate coordinate transformations.