Fracton topological order from Higgs and partial confinement mechanisms of rank-two gauge theory

TL;DR

This work shows that fracton topological order can be generated from rank-2 ${\rm U}(1)$ gauge theories through Higgsing and partial confinement. It provides two concrete routes to the X-cube model: (i) Higgsing to a ${\mathbb Z}_2$ scalar-charge theory followed by selective flux-loop condensation, and (ii) condensing monopoles to obtain a hollow ${\rm U}(1)$ theory, then Higgsing to ${\mathbb Z}_2$ and mapping to X-cube. It also demonstrates that the ${\mathbb Z}_2$ scalar-charge theory is equivalent to four copies of the $d=3$ toric code, while the checkerboard fracton model arises from a rank-2 ${\mathbb Z}_2$ theory with alternating Gauss laws. Together, these results provide a unified framework linking proximate gauge theories to gapped fracton orders and suggest broader avenues for exploring higher-rank fracton phases via symmetry-breaking and confinement mechanisms.

Abstract

Fractons are gapped point-like excitations in $d=3$ topological ordered phases whose motion is constrained. They have been discovered in several gapped models but a unifying physical mechanism for generating them is still missing. It has been noticed that in symmetric-tensor ${\rm U}(1)$ gauge theories, charges are fractons and cannot move freely due to, for example, the conservation of not only the charge but also the dipole moment. To connect these theories with fully gapped fracton models, we study Higgs and partial confinement mechanisms in rank-2 symmetric-tensor gauge theories, where charges or magnetic excitations, respectively, are condensed. Specifically, we describe two different routes from the rank-2 ${\rm U}(1)$ scalar charge theory to the X-cube fracton topological order, finding that a combination of Higgs and partial confinement mechanisms is necessary to obtain the fully gapped fracton model. On the other hand, the rank-2 $\mathbb{Z}_2$ scalar charge theory, which is obtained from the former theory upon condensing charge-2 matter, is equivalent to four copies of the $d=3$ toric code and does not support fracton excitations. We also explain how the checkerboard fracton model can be viewed as a rank-2 $\mathbb{Z}_2$ gauge theory with two different Gauss' law constraints on different lattice sites.

Figures (14)

• Figure 1: Two paths from the rank-2 ${\rm U}(1)$ scalar charge theory to the X-cube model.
• Figure 2: Rank-2 ${\rm U}(1)$ gauge theory defined on the cubic lattice. The off-diagonal elements of $E_{\mu \nu}$ and $A_{\mu \nu}$ live on plaquettes, while diagonal elements reside on sites. Gauge charges $n_{\bf r}$, with conjugate phase $\theta_{\bf r}$, also reside on sites, which are labeled by ${\bf r}$.
• Figure 3: Two possible electric charge configurations in the scalar charge theory are shown in (a) and (b). Any configurations related to these by cubic symmetry can also appear.
• Figure 4: Two elements of the ${\mathbb{Z}}_2$ magnetic flux tensor $F_{\mu \nu}$ are shown as products of $\mathbb{Z}_2$ gauge field operators $Z_{\mu\nu}$.
• Figure 5: The octahedron term $G_{\bf r}$ at ${\bf r}$ involves eighteen Pauli operators denoted as solid dots. Six of them are at sites (red). The other ones are at plaquettes adjacent to ${\bf r}$ in $xy$ (yellow), $yz$ (blue) and $xz$ (green) planes. The open circle shows the location of the charge created by violating this $G_{\bf r}$ term.