SciPost Phys. 15, 001 (2023) ·
published 6 July 2023

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2+1d topological phases are well characterized by the fusion rules and braiding/exchange statistics of fractional point excitations. In 4+1d, some topological phases contain only fractional loop excitations. What kind of loop statistics exist? We study the 4+1d gauge theory with 2form $\mathbb{Z}_2$ gauge field (the looponly toric code) and find that while braiding statistics between two different types of loops can be nontrivial, the self "exchange" statistics are all trivial. In particular, we show that the electric, magnetic, and dyonic loop excitations in the 4+1d toric code are not distinguished by their selfstatistics. They tunnel into each other across 3+1d invertible domain walls which in turn give explicit unitary circuits that map the loop excitations into each other. The SL(2,$\mathbb{Z}_2$) symmetry that permutes the loops, however, cannot be consistently gauged and we discuss the associated obstruction in the process. Moreover, we discuss a gapless boundary condition dubbed the "fractional Maxwell theory" and show how it can be Higgsed into gapped boundary conditions. We also discuss the generalization of these results from the $\mathbb{Z}_2$ gauge group to $\mathbb{Z}_N$.
Nandagopal Manoj, Kevin Slagle, Wilbur Shirley, Xie Chen
SciPost Phys. 10, 094 (2021) ·
published 29 April 2021

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The Xcube model, a prototypical gapped fracton model, has been shown to have a foliation structure. That is, inside the 3+1D model, there are hidden layers of 2+1D gapped topological states. A screw dislocation in a 3+1D lattice can often reveal nontrivial features associated with a layered structure. In this paper, we study the Xcube model on lattices with screw dislocations. In particular, we find that a screw dislocation results in a finite change in the logarithm of the ground state degeneracy of the model. Part of the change can be traced back to the effect of screw dislocations in a simple stack of 2+1D topological states, hence corroborating the foliation structure in the model. The other part of the change comes from the induced motion of fractons or subdimensional excitations along the dislocation, a feature absent in the stack of 2+1D layers.
SciPost Phys. 6, 041 (2019) ·
published 2 April 2019

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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 submanifold 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 selfduality similar to gauging global symmetries in 1D.
SciPost Phys. 6, 015 (2019) ·
published 31 January 2019

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Fracton models exhibit a variety of exotic properties and lie beyond the conventional framework of gapped topological order. In a previous work, we generalized the notion of gapped phase to one of foliated fracton phase by allowing the addition of layers of gapped twodimensional resources in the adiabatic evolution between gapped threedimensional models. Moreover, we showed that the Xcube model is a fixed point of one such phase. In this paper, according to this definition, we look for universal properties of such phases which remain invariant throughout the entire phase. We propose multipartite entanglement quantities, generalizing the proposal of topological entanglement entropy designed for conventional topological phases. We present arguments for the universality of these quantities and show that they attain nonzero constant value in nontrivial foliated fracton phases.