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.