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 line-like subsystem symmetries. This endows the decorated model with planar subsystem symmetries under which the loop-like 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 non-trivial 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.
Highlights
A new paradigm in the classification of gapped phases of matter has recently begun thanks to the discovery of models with novel subdimensional physics
Our SSET model can be understood as a decorated 3D toric code model [43,44]
This toric code consists of qubits on the faces of a cubic lattice, and the ground states can be visualized as equal-weight superpositions over all basis states where the faces in the state |1 form unions of closed 2D membranes
Summary
A new paradigm in the classification of gapped phases of matter has recently begun thanks to the discovery of models with novel subdimensional physics This includes the fracton topological phases [1,2,3,4,5,6,7,8,9,10,11,12], in which topological quasiparticles are either immobile or confined to move only within subsystem such as lines or planes, as well as models with subsystem symmetries, which are symmetries that act nontrivially only on rigid subsystems of the entire system [7,8,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]. In anticipation of the main model of this paper, consider a 3D system with planar subsystem symmetries, and suppose there are two pointlike excitations living on a single plane It may seem as though it is possible for each excitation to carry a fractional charge, as above. We find models containing both pointlike excitations with restricted mobility, as well as looplike excitations, and a nontrivial symmetry action that couples the two types
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