We construct a general gauging procedure of a pure matter theory on a lattice with a mixture of subsystem and global symmetries. This mixed symmetry can be either a semidirect product of a subsystem symmetry and a global symmetry, or a nontrivial extension of them. We demonstrate this gauging procedure on a cubic lattice in three dimensions with four examples: $G={\mathbb{Z}}_{3}^{\text{sub}}\ensuremath{\rtimes}{\mathbb{Z}}_{2}^{\text{glo}}, G=({\mathbb{Z}}_{2}^{\text{sub}}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{\text{sub}})\ensuremath{\rtimes}{\mathbb{Z}}_{2}^{\text{glo}}, 1\ensuremath{\rightarrow}{\mathbb{Z}}_{2}^{\text{sub}}\ensuremath{\rightarrow}G\ensuremath{\rightarrow}{\mathbb{Z}}_{2}^{\text{glo}}\ensuremath{\rightarrow}1$, and $1\ensuremath{\rightarrow}{\mathbb{Z}}_{2}^{\text{sub}}\ensuremath{\rightarrow}G\ensuremath{\rightarrow}{K}_{4}^{\text{glo}}\ensuremath{\rightarrow}1$. The former two cases and the last one produce the non-Abelian fracton orders. Our construction provides an identification of the electric charges of these fracton orders with irreducible representations of the symmetry. Furthermore, by constraining the local Hilbert space, the magnetic fluxes with different geometry (tube-like and plaquette-like) satisfy a subalgebra of the quantum double models (QDMs). This algebraic structure leads to an identification of the magnetic fluxes to the conjugacy classes of the symmetry.
We discuss the procedure for gauging on-site Z_2 Z 2 global symmetries of three-dimensional lattice Hamiltonians that permute quasi-particles and provide general arguments demonstrating the non-Abelian character of the resultant gauged theories. We then apply this general procedure to lattice models of several well known fracton phases: two copies of the X-Cube model, two copies of Haah’s cubic code, and the checkerboard model. Where the former two models possess an on-site Z_2 Z 2 layer exchange symmetry, that of the latter is generated by the Hadamard gate. For each of these models, upon gauging, we find non-Abelian subdimensional excitations, including non-Abelian fractons, as well as non-Abelian looplike excitations and Abelian fully mobile pointlike excitations. By showing that the looplike excitations braid non-trivially with the subdimensional excitations, we thus discover a novel gapped quantum order in 3D, which we term a ‘panoptic" fracton order. This points to the existence of parent states in 3D from which both topological quantum field theories and fracton states may descend via quasi-particle condensation. The gauged cubic code model represents the first example of a gapped 3D phase supporting (inextricably) non-Abelian fractons that are created at the corners of fractal operators.
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.
Fracton theories possess exponentially degenerate ground states, excitations with restricted mobility, and nontopological higher-form symmetries. This paper shows that such theories can be defined on arbitrary spatial lattices in three dimensions. The key element of this construction is a generalization of higher-form gauge theories to so-called
Fp gauge theories, in which gauge transformations of rank-
k fields are specified by rank-
(k−p) gauge parameters. The
Z2 rank-two theory of type
F2, placed on a cubic lattice and coupled to scalar matter, is shown to have a topological phase exactly dual to the well-known X-cube model. Generalizations of this example yield novel fracton theories. In the continuum, the
U(1) rank-two theory of type
F2 is shown to have a perturbatively gapless fracton regime that cannot be consistently interpreted as a tensor gauge theory of any kind. The compact scalar fields that naturally couple to this
F2 theory also show gapless fracton behavior; on a cubic lattice they have a conserved
U(1) charge and dipole moment, but these particular charges are not necessarily conserved on more general lattices. The construction straightforwardly generalizes to
F2 theories of nonabelian rank-two gauge fields, giving first examples of pure nonabelian higher-rank theories.
We broaden the scope of quantum field theory by introducing a general class of discrete gauge theories that realize either topological order or fracton behavior across dimensions. We start from translation-invariant systems endowed with unconventional charge-conservation laws, which we term \textit{exponential polynomial symmetries}. Gauging these symmetries yields
ZN gauge theories in 2D that exhibit topological order whose quasiparticles have constrained mobility and whose ground-state degeneracy shows ultraviolet (UV) dependence. These features are reminiscent of spatial symmetry-enriched topological order, wherein quasiparticle excitations transform nontrivially under lattice translations. We further propose a Chern-Simons variant that produces non-CSS stabilizer codes and outline a framework for exponentially symmetric subsystem SPT phases. Finally, we extend this gauging procedure to 3D, obtaining new variants of fracton topological order.
Understanding quantum phases and phase transitions in the presence of symmetries is a central objective of quantum many-body physics. A powerful modern paradigm for investigating this problem is topological holography, which relates symmetries in
k dimensions to "bulk" topological orders in
(k+1) dimensions. While conceptually profound, most existing bulk construction methods rely on sophisticated mathematical formalisms and can be difficult to apply to certain symmetry types. In this work, we propose a physically intuitive and versatile method, termed the layered gauging construction, to systematically generate
(k+1)-dimensional (liquid or fracton) topological orders from
k-dimensional generalized symmetries. Roughly speaking, the prescription is to stack many layers of
k-dimensional quantum systems with certain symmetries into a
(k+1)-dimensional pile, and then sequentially gauge a diagonal symmetry acting on each nearest-neighbor pair of layers. The detailed procedure depends on the specific symmetry types. We have successfully implemented the method in a number of examples in different spatial dimensions, with symmetries that are conventional, higher-form, subsystem, anomalous, nonabelian, or noninvertible. We hence conjecture the method to be very general. For example, from the subsystem symmetry of the
2d plaquette Ising model, we derive the X-cube model and also an anisotropic fracton topological order. Additionally, starting from an anomalous
Z2 symmetry in
1d, we construct a new square lattice model realizing the double semion topological order.