Quantum Field Theory of X-Cube Fracton Topological Order and Robust Degeneracy from Geometry

TL;DR

This work derives a quantum field theory description for the X-cube fracton model by mapping lattice operators to continuum gauge fields and currents, yielding a Lagrangian that is not TQFT-like but subconformally invariant. It demonstrates how fracton and dimension-1 particle dynamics, constrained mobility, and nontrivial braiding are captured through generalized current conservation and membrane-based processes, and it shows how the theory couples to matter while preserving mobility constraints. The authors compute ground-state degeneracies both in the lattice model and in the field theory, revealing exponential scaling with system size and, crucially, a curvature-induced degeneracy on geometries with trivial topology. They also discuss minimal coupling to matter, and explore extensions and limitations, including the potential for curvature-based degeneracy in other non-liquid fracton phases. Overall, the paper provides a general framework to construct fracton field theories from a charge-density input and highlights geometry as a robust source of degeneracy beyond topology.

Abstract

We propose a quantum field theory description of the X-cube model of fracton topological order. The field theory is not (and cannot be) a topological quantum field theory (TQFT), since unlike the X-cube model, TQFTs are invariant (i.e. symmetric) under continuous spacetime transformations. However, the theory is instead invariant under a certain subgroup of the conformal group. We describe how braiding statistics and ground state degeneracy are reproduced by the field theory, and how the the X-cube Hamiltonian and field theory can be minimally coupled to matter fields. We also show that even on a manifold with trivial topology, spatial curvature can induce a ground state degeneracy that is stable to arbitrary local perturbations! Our formalism may allow for the description of other fracton field theories, where the only necessary input is an equation of motion for a charge density.

Figures (11)

• Figure 1: Fracton operator $\hat{\mathcal{O}}$ and dimension-1 particle $\hat{\mathcal{A}}^{(a)}$ operators of the X-cube model (Eq. (\ref{['eq:Xcube H']})). $\hat{\mathcal{O}}$ is a product of twelve $\hat{Z}$ operators on the links bordering a cube. $\hat{\mathcal{A}}^{(a)}$ operators are a product of four $\hat{X}$ operators on the four links neighboring a vertex which are orthogonal to the $x^a$ direction. $\hat{X}$ and $\hat{Z}$ are defined in Eq. (\ref{['eq:commutator']}).
• Figure 2: In order for the fractons (labeled a and b) to move to the right, they must exchange a fracton dipole (circled in black). Generating this fracton configuration from the vacuum requires a nonzero fracton current $I^{12}$ in the blue region. Thus, $I^{cd}$ can be interpreted as a fracton dipole current since it describes a combination of 1) fracton dipoles oriented in the $x^c$-direction moving in the $x^d$-direction and 2) fracton dipoles oriented in the $x^d$-direction moving in the $x^c$-direction. (Recall that $I^{cd} = I^{dc}$.) The dipole exchange results in a fracton flow (green, Eq. (\ref{['eq:fracton flow']})), most of which will be canceled out by an additional dipole exchange.
• Figure 3: Dimension-1 particles are "braided" around a fracton, resulting in a $-2\pi/N$ phase. (red) Dimension-1 particle current $J^a$ around the corners of a cube. (blue) Rectangular membrane where $A_3$ is nonzero. A single fracton is located at the corner of the membrane, which is inside of the cube.
• Figure 4: (a) A pair of oppositely charged fractons are "braided" around an $x$-axis dimension-1 particle by exchanging fracton dipoles in the blue shaded region (Fig. \ref{['fig:fractonCurrent']}), resulting in a $+2\pi/N$ phase. The fractons are moved parallel to each other; the green arrows are used to indicate the sign of the fracton flow $\tilde{I}^\alpha$(b) Another fracton current $I$ configuration (blue) which produces the same fracton flow $\tilde{I}$ (green). The blue arrows specify the sign of $B_{ab}$. However, no phase results from this configuration. (red line) Region where $B_{23}$ is nonzero. (red point) An X-axis dimension-1 particle. (blue) Membrane where the fracton current $I^{ab}$ is nonzero and where $\hat{X}$ operators are placed in Eq. (\ref{['eq:F']}). (green) Fracton flow $\tilde{I}$ (Eq. (\ref{['eq:fracton flow']})).
• Figure 5: $\hat{\tau}^x_\mathbf{x} \hat{\mathcal{O}}_\mathbf{x}$, $\hat{\sigma}^x_\mathbf{x} \hat{\mathcal{A}}^{(a)}_\mathbf{x}$, $\hat{\mathcal{C}}_\mathbf{x}^{(a)}$, and $\hat{\mathcal{F}}_\mathbf{x}^{(a)}$ operators of the X-cube model (Eq. (\ref{['eq:Xcube H matter']})) after coupling to $\hat{\sigma}^\mu$ and $\hat{\tau}^\mu$ matter. $\hat{\sigma}^\mu_{\mathbf{x},a}$ are centered on the vertices, while $\hat{\tau}^\mu_\mathbf{x}$ are centered on the cubes.