Foliated fracton order from gauging subsystem symmetries

TL;DR

This work develops a general, abelian subsystem symmetry gauging framework that maps ungauged models with planar or linear subsystem symmetries to 3D foliated fracton gauge theories, with the resulting fracton content determined by the symmetry charges in the ungauged phase.By applying a uniform procedure that places gauge degrees of freedom at centers of minimal coupling terms and enforces Gauss laws, the authors connect planar symmetry charges to planon, lineon, and fracton excitations, recovering X-cube and related models in multiple lattice geometries.The paper also reveals a self-duality for gauging linear subsystem symmetries in 2D/3D, paralleling the 1D global-case dualities, and shows how symmetry breaking and SSPT phases map under gauging, offering a unifying perspective on foliated fracton order and its signatures.Overall, the results provide a concrete toolkit for constructing and classifying foliated fracton phases from subsystem-symmetric building blocks and highlight the deep link between ungauged symmetry content and gauged topological mobility constraints.

Abstract

Based on several previous examples, we summarize explicitly the general procedure to gauge models with subsystem symmetries, which are symmetries with generators that have support within a sub-manifold of the system. The gauging process can be applied to any local quantum model on a lattice that is invariant under the subsystem symmetry. We focus primarily on simple 3D paramagnetic states with planar symmetries. For these systems, the gauged theory may exhibit foliated fracton order and we find that the species of symmetry charges in the paramagnet directly determine the resulting foliated fracton order. Moreover, we find that gauging linear subsystem symmetries in 2D or 3D models results in a self-duality similar to gauging global symmetries in 1D.

Figures (9)

• Figure 1: Correspondence of foliation structure in 3D systems with planar subsystem symmetry and 3D foliated fracton models.
• Figure 2: Gauging global $Z_2$ symmetry in 2D. (a) Vertex $A_v = \sigma_v^x \prod_{e \ni v} \tau_e^x$ and plaquette $B_p = \prod_{e\in p} \tau_e^z$ terms that appear in Eq. (\ref{['eq:gGlobal']}). (b-c) String operators.
• Figure 3: Gauging planar symmetry on a cubic lattice. (a) The minimal symmetric coupling term: a product of four $\sigma^z$ around a plaquette (of the cubic lattice of blue spheres). A gauge field $\tau$ (green sphere) is therefore placed at the center of the plaquette, and all other plaquettes. The gray lines form the dual lattice. (b) The red vertex is involved in the twelve minimal coupling terms highlighted by red squares. The gauge symmetry term is a product of a $\sigma^x$ at the red sphere and twelve $\tau^x$ on the green spheres. (c) The product of four minimal coupling terms around the four blue plaquettes is the identity. The corresponding flux term is a product of four $\tau^z$ on the green spheres.
• Figure 4: Gauging planar symmetry on FCC lattice. (a) A minimal symmetric coupling term: a product of four $\sigma^z$ at the corners of the red tetrahedron. A gauge field $\tau$ (green sphere) is placed at the center of tetrahedron. Within the above cube, there are eight tetrahedra (one for each corner of the cube) and gauge fields. The gray lines form the dual lattice. (b) The red vertex is involved in the eight minimal coupling tetrahedron terms centered at the green spheres. The gauge symmetry term is thus a product of a $\sigma^x$ at the red sphere and eight $\tau^x$ on the green spheres. The product of the eight minimal coupling tetrahedron terms is the identity. The corresponding flux term is a product of eight $\tau^z$ on the green spheres.
• Figure 5: Gauging planar symmetry on BCC lattice. (a) A minimal symmetric coupling term: a product of two $\sigma_0^z$ and one $\sigma_\mathrm{c}^z$. The black lines form the cubic lattice, while the gray lines form the dual lattice. There are 12 minimal coupling terms within the shown dual-lattice cube: one for each gray edge of the cube. For the other terms, the $\sigma_\mathrm{c}^z$ at the center becomes a $\sigma_\mathrm{a}^z$ or $\sigma_\mathrm{b}^z$ when the two $\sigma_0^z$ are displaced in the $x$ or $y$ direction, respectively. (b) The body-center is involved in $4 \times 6$ minimal coupling terms, which are centered at the green spheres, which lie on the faces of the black cube. (The orange lines are guides for the eye.) The gauge symmetry term is therefore a product of a $\sigma^x$ at the center and 24 $\tau^x$ on the green spheres. (c) The $\sigma_\mathrm{c}^z$ operator in the center is involved in 4 minimal coupling terms (highlighted in red). The gauge symmetry term is therefore a product of a $\sigma_\mathrm{c}^x$ at the center and four $\tau^x$ on the green spheres. (d) The product of the four minimal coupling triangular terms is the identity. The corresponding flux term is a product of four $\tau^z$ on the four green spheres.