Subsystem symmetry enriched topological order in three dimensions

Abstract

We introduce a model of three-dimensional (3D) topological order enriched by planar subsystem symmetries. The model is constructed starting from the 3D toric code, whose ground state can be viewed as an equal-weight superposition of two-dimensional (2D) membrane coverings. We then decorate those membranes with 2D cluster states possessing symmetry-protected topological order under linelike subsystem symmetries. This endows the decorated model with planar subsystem symmetries under which the looplike excitations of the toric code fractionalize, resulting in an extensive degeneracy per unit length of the excitation. We also show that the value of the topological entanglement entropy is larger than that of the toric code for certain bipartitions due to the subsystem symmetry enrichment. Our model can be obtained by gauging the global symmetry of a short-range entangled model which has symmetry-protected topological order coming from an interplay of global and subsystem symmetries. We study the nontrivial action of the symmetries on boundary of this model, uncovering a mixed boundary anomaly between global and subsystem symmetries. To further study this interplay, we consider gauging several different subgroups of the total symmetry. The resulting network of models, which includes models with fracton topological order, showcases more of the possible types of subsystem symmetry enrichment that can occur in 3D.

Long-range entanglement from measuring symmetry-protected topological phases

“…Second, the origin of LRE under measurement is tied to a specific anomaly involving the symmetries-related to the anomaly living at the boundary of the original SPT phase-thereby constraining the nature of the resulting LRE. Third, this allows for the preparation of states that are not realized by stabilizer codes, such as topological order described by twisted gauge theories or non-Abelian fracton orders [60][61][62][63][64][65][66][67][68][69]. Fourth, we achieve a new perspective on Kramers-Wannier (KW) [18,[70][71][72][73][74][75][76][77] and Jordan-Wigner (JW) [78][79][80][81][82][83][84][85][86] transformations.…”

“…Note that this already contains certain non-Abelian phases, e.g., D 4 topological order arises upon gauging the Z 3 2 symmetry of an SPT phase with type-III cocycle [96,97]. (For obtaining non-Abelian topological order associated to any solvable group, see Section III C.) Similarly, our procedure allows for the creation of twisted fracton phases by gauging 3D subsystem SPT phases [65,83,84,98,99]. Thus a much wider class of states can be obtained from local unitary circuits and LOCC (local operations and classical communications) [54] than previously established.…”

Bifurcating subsystem symmetric entanglement renormalization in two dimensions

“…In two dimensions, we expect that all nontrivial bifurcating symmetric ERG flows in gapped phases originate due to SSPTs. This is because in 2D only conventional topological phases, which contain unique fixed points, are possible [36,37], and there are no nontrivial SSET phases beyond stacking an SSPT with a decoupled topological order [63]. We remark that in three or higher dimensions, it is possible to have nontrivial gapped fracton topological phases that contain bifurcating ERG fixed points [30][31][32].…”

“…We remark that in three or higher dimensions, it is possible to have nontrivial gapped fracton topological phases that contain bifurcating ERG fixed points [30][31][32]. It is also possible to enrich conventional topological phases to form nontrivial SSET phases [63]. While we have focused on the simplest nontrivial setting of two-dimensional phases in this paper, the bifurcating ERG of subsystem symmetry-enriched phases presents an interesting avenue for future study.…”

Non-Abelian Hybrid Fracton Orders

“…In this general setting, it is generally not possible to gauge the full non-Abelian subsystem symmetry group; only an Abelian subgroup of the symmetry group can be gauged, thus giving rise to an Abelian fracton order 1 . Other examples of non-Abelian fracton orders have been recently proposed, by strongly coupling nonabelian topological orders in 2D and 3D [20][21][22][23][24][25], by gauging the abelian subsystem symmetries in certain symmetryprotected topological phases [26,27], or gauging a global symmetry which permutes fracton excitations in an abelian 1 Suppose the system of interest has three intersecting planar symmetries, each of which transforms a given plane under a non-abelian group G. The group commutator of two intersecting planar symmetries acts only on the intersection line. Therefore, the system has additional line symmetries given by the commutator subgroup [G, G].…”