Fractonic Chern-Simons and BF theories

TL;DR

The paper develops a general framework for higher-rank, abelian Chern-Simons and BF–type field theories in 3D that bind charge to flux via a generalized constraint, aiming to capture fracton physics. A detailed analysis of a rank-2, $C_3$-symmetric fractonic CS theory reveals subdimensional lineon excitations, a cage-net structure of gauge-invariant operators, and a ground-state degeneracy that depends on both topology and geometry, which on the lattice maps to a $\,\mathbb{Z}_s$ generalization of the Chamon code. The work also discusses a continuum boundary structure with gapless dipole modes, a lattice regularization avoiding some CS subtleties, and a gapless three-field variant that can circumvent confinement. Overall, the results establish a concrete field-theoretic path to fracton order, illustrate its rich operator algebra and lattice correspondence, and open questions about continuum realizations and broader generalizations. The study thus provides a principled connection between fracton lattice models and higher-rank gauge theories with potential implications for quantum information and strongly correlated systems.

Abstract

Fracton order is an intriguing new type of order which shares many common features with topological order, such as topology-dependent ground state degeneracies, and excitations with mutual statistics. However, it also has several distinctive geometrical aspects, such as excitations with restricted mobility, which naturally lead to effective descriptions in terms of higher rank gauge fields. In this paper, we investigate possible effective field theories for 3D fracton order, by presenting a general philosophy whereby topological-like actions for such higher-rank gauge fields can be constructed. Our approach draws inspiration from Chern-Simons and BF theories in 2+1 dimensions, and imposes constraints binding higher-rank gauge charge to higher-rank gauge flux. We show that the resulting fractonic Chern-Simons and BF theories reproduce many of the interesting features of their familiar 2D cousins. We analyze one example of the resulting fractonic Chern-Simons theory in detail, and show that upon quantization it realizes a gapped fracton order with quasiparticle excitations that are mobile only along a sub-set of 1-dimensional lines, and display a form of fractional self-statistics. The ground state degeneracy of this theory is both topology- and geometry- dependent, scaling exponentially with the linear system size when the model is placed on a 3-dimensional torus. By studying the resulting quantum theory on the lattice, we show that it describes a $\mathbb{Z}_s$ generalization of the Chamon code.

Figures (13)

• Figure 1: Illustration of the $x,y,z$ and $u,v,w$ coordinates. The red plane is the 111 plane and the three boundary lines of the red triangle denote the $u,v,w$ direction. The $C_3$ rotation on the 111 plane rotates the red triangle by $2\pi/3$ and permutes the axis along $x,y,z$.
• Figure 2: Gauge invariant cage nets of the symmetric rank 2 theory. With differential operators given in Eq. (\ref{['Gtrans']}), the cages can have edges along the $x,y,z$ or $u,v,w$ directions.
• Figure 3: Left: The red/yellow/green sheets represent thick ribbon operators. An isolated ribbon must harbor a charge at each corner; however, with appropriately oriented gauge fields these corner charges cancel in the configuration shown. Right: By lengthening and thinning the ribbons, we obtain a Wilson ribbon cage-net. Note that the Wilson ribbon cage nets are gauge invariant only if we choose the width of ribbons along the $u,v,w$ directions to be longer by a factor of $\sqrt{2}$ than those along the $x,y,z$ directions.
• Figure 4: Lineon excitation configurations: Each end of an open Wilson ribbon extended along the $x$ direction must terminate on a dipole (or anti-dipole) with dipole moment oriented along $u$. Likewise, a Wilson ribbon extended along the $u$ direction hosts a dipole oriented along $x$ at each end-point. Since Wilson ribbons cannot turn, these dipoles are mobile only along the direction of the ribbon, and are hence 1-dimensional lineon excitations. Here star represent gauge fields $A_1, A_2$ while dot represent charge $\rho$.
• Figure 5: By tuning $\alpha,\beta$ in Eq. (\ref{['Eq:Dgens']}), we are changing the height of the tetrahedron in our cage-net configuration. There are an infinite number of equivalent cage-net configurations related by this deformation. c) A rotated view of the cage-net tetrahedron. The dimensions of the base of the tetrahedron (indicated with a shaded red triangle) is fixed, while its height along the $(111)$ direction (represented by the green dotted line) varies depending on $\arctan{(\alpha/\beta)}$. a) Examples of equivalent cage net tetrahedra. The purple dot corresponds to the case where $\alpha,\beta=1$. The yellow dot corresponds to $\alpha=2,\beta=1$. b) When $\beta=0$, the height of the tetrahedron goes to infinity and the cage-net configurations becomes a prism.