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Register a new userNon-Abelian fracton order from gauging a mixture of subsystem and global symmetries
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We demonstrate 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 non-trivial extension of them. We demonstrate this gauging procedure on a cubic lattice in three dimensions with four examples: $G=mathbb{Z}_3^{text{sub}} rtimes mathbb{Z}_2^{text{glo}}$, $G=(mathbb{Z}_2^{text{sub}} times mathbb{Z}_2^{text{sub}}) rtimes mathbb{Z}_2^{text{glo}}$, $1to mathbb {Z}_2^text {sub}to Gto mathbb {Z}_2^text {glo}to 1$, and $1to mathbb {Z}_2^text {sub}to Gto K_4^text {glo}to 1$. The former two cases and the last one produce the non-Abelian fracton orders. Our construction of the gauging procedure 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.
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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.
We discuss the procedure for gauging on-site $mathbb{Z}_2$ 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 Haahs cubic code, and the checkerboard model. Where the former two models possess an on-site $mathbb{Z}_2$ 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.
We introduce lattice gauge theories which describe three-dimensional, gapped quantum phases exhibiting the phenomenology of both conventional three-dimensional topological orders and fracton orders, starting from a finite group $G$, a choice of an Abelian normal subgroup $N$, and a choice of foliation structure. These hybrid fracton orders -- examples of which were introduced in arXiv:2102.09555 -- can also host immobile, point-like excitations that are non-Abelian, and therefore give rise to a protected degeneracy. We construct solvable lattice models for these orders which interpolate between a conventional, three-dimensional $G$ gauge theory and a pure fracton order, by varying the choice of normal subgroup $N$. We demonstrate that certain universal data of the topological excitations and their mobilities are directly related to the choice of $G$ and $N$, and also present complementary perspectives on these orders: certain orders may be obtained by gauging a global symmetry which enriches a particular fracton order, by either fractionalizing on or permuting the excitations with restricted mobility, while certain hybrid orders can be obtained by condensing excitations in a stack of initially decoupled, two-dimensional topological orders.
We consider a 2D quantum spin model with ring-exchange interaction that has subsystem symmetries associated to conserved magnetization along rows and columns of a square lattice, which implies the conservation of the global dipole moment. In a certain regime, the model is non-integrable, but violates the eigenstate thermalization hypothesis through an extensive Hilbert space fragmentation, including an exponential number of inert subsectors with trivial dynamics, arising from kinetic constraints. While subsystem symmetries are quite restrictive for the dynamics, we show that they alone cannot account for such a number of inert states, even with infinite-range interactions. We present a procedure for constructing shielding structures that can separate and disentangle dynamically active regions from each other. Notably, subsystem symmetries allow the thickness of the shields to be dependent only on the interaction range rather than on the size of the active regions, unlike in the case of generic dipole-conserving systems.
In this work, we show that the checkerboard model exhibits the phenomenon of foliated fracton order. We introduce a renormalization group transformation for the model that utilizes toric code bilayers as an entanglement resource, and show how to extend the model to general three-dimensional manifolds. Furthermore, we use universal properties distilled from the structure of fractional excitations and ground-state entanglement to characterize the foliated fracton phase and find that it is the same as two copies of the X-cube model. Indeed, we demonstrate that the checkerboard model can be transformed into two copies of the X-cube model via an adiabatic deformation.
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