Twisted Fracton Models in Three Dimensions

TL;DR

Twisted Fracton Models in Three Dimensions introduces a systematic method to generate 3D fracton phases by twisting gauge symmetries with 3-cocycles ω∈Z^3(G,U(1)), extending known type-I fracton models to yield non-Abelian excitations and geometry-dependent ground-state degeneracy. The authors construct twisted X-cube and twisted checkerboard models based on finite Abelian groups G, derive their ground-state degeneracies on the three-torus via DW weights, and develop a categorical framework for fusion and braiding using twisted quantum doubles D^ω(G). They demonstrate the existence of inextricably non-Abelian fractons and 1d mobile quasiparticles, show how GSD depends on system size, and provide explicit results for G=Z_2 and G=Z_2^3, highlighting new topological content beyond conventional 2D DW theories. The work establishes a rigorous 3D, intrinsically fracton-based toolkit for describing quasiparticle types, mobility constraints, and braiding in non-Abelian fracton phases, with potential implications for fault-tolerant quantum computation and connections to 3D TQFT-like structures. These twisted fracton models thus extend the landscape of 3D topological orders and offer new avenues for realizing robust quantum information storage and manipulation in higher dimensions.

Abstract

We study novel three-dimensional gapped quantum phases of matter which support quasiparticles with restricted mobility, including immobile "fracton" excitations. So far, most existing fracton models may be instructively viewed as generalized Abelian lattice gauge theories. Here, by analogy with Dijkgraaf-Witten topological gauge theories, we discover a natural generalization of fracton models, obtained by twisting the gauge symmetries. Introducing generalized gauge transformation operators carrying an extra phase factor depending on local configurations, we construct a plethora of exactly solvable three-dimensional models, which we dub "twisted fracton models." A key result of our approach is to demonstrate the existence of rich non-Abelian fracton phases of distinct varieties in a three-dimensional system with finite-range interactions. For an accurate characterization of these novel phases, the notion of being inextricably non-Abelian is introduced for fractons and quasiparticles with one-dimensional mobility, referring to their new behavior of displaying braiding statistics that is, and remains, non-Abelian regardless of which quasiparticles with higher mobility are added to or removed from them. We also analyze these models by embedding them on a three-torus and computing their ground state degeneracies, which exhibit a surprising and novel dependence on the system size in the non-Abelian fracton phases. Moreover, as an important advance in the study of fracton order, we develop a general mathematical framework which systematically captures the fusion and braiding properties of fractons and other quasiparticles with restricted mobility.

Figures (25)

• Figure 1: Lattice model of gauge theory in 2+1 dimensions. (a) The physical degrees of freedom are on the black oriented edges of the square lattice. Auxiliary grey edges are added to give a complete triangulation. (b) $P_{v}^{g}$ for $v=3$ is presented by a triangulated pyramid with $\left[33'\right]$ colored by $g\in G$, where the edge orientations are picked according to the local ordering of vertices $0<1<2<3<3'<4<5<6$. For each tetrahedron, $\text{sgn}\left(\left[v_{0}v_{1}v_{2}v_{3}\right]\right)$ equals the sign of the triple product $\overrightarrow{v_{0}v_{1}}\cdot\left(\overrightarrow{v_{0}v_{2}}\times\overrightarrow{v_{0}v_{3}}\right)$. For example, $\text{sgn}\left(\left[0133'\right]\right)=+1$ and $\text{sgn}\left(\left[233'5\right]\right)=-1$.
• Figure 2: Stacking two pyramids over a vertex as a graphic representation of $P_{v}^{g}P_{v}^{h}$.
• Figure 3: Two orders of stacking two pyramids over two adjacent vertices, with (a) and (b) presenting $P_{v_{0}}^{g}P_{v_{1}}^{h}$ and $P_{v_{1}}^{h}P_{v_{0}}^{g}$ respectively.
• Figure 4: Default triangulation of a prism over $\tau\times \mathtt{I}$, where $\tau=\left[\tau_{0}\tau_{1}\tau_{2}\right]$ and $\mathtt{I}=\left[0,1\right]$. The Dijkgraaf-Witten weight on this prism is $\left[\tau_{0}\tau_{1}\tau_{2}\tau_{2}^{\prime}\right]\left[\tau_{0}\tau_{0}^{\prime}\tau_{1}^{\prime}\tau_{2}^{\prime}\right]/\left[\tau_{0}\tau_{1}\tau_{1}^{\prime}\tau_{2}^{\prime}\right]$, where each tetrahedron $\left[v_{0}v_{1}v_{2}v_{3}\right]$ stands for $\omega\left(\left[v_{0}v_{1}\right],\left[v_{1}v_{2}\right],\left[v_{2}v_{3}\right]\right)$ with edges short for the group elements assigned by the coloring.
• Figure 5: A triangulation of a cube. The eight vertices are ordered as $0<0'<1<1'<2<2'<3<3'$; their ordering assigns orientations to edges, triangles and tetrahedrons. Gluing the three pairs of opposite faces of the cube gives a triangulated three-dimensional torus $\mathtt{T}^{3}$.