SciPost Phys. 15, 231 (2023) ·
published 7 December 2023

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We study the boundary states of the archetypal three dimensional topological order, i.e. the three dimensional $\mathbb{Z}_2$ toric code. There are three distinct elementary types of boundary states that we will consider in this work. In the phase diagram that includes the three elementary boundaries there may exist a multicritical point, which is captured by the socalled deconfined quantum critical point (DQCP) with an "easyaxis" anisotropy. Moreover, there is an emergent $\mathbb{Z}_{2,d}$ symmetry that swaps two of the boundary types, and it becomes part of the global symmetry of the DQCP. The emergent $\mathbb{Z}_{2,d}$ (where d represents "defect") symmetry on the boundary is originated from a type of surface defect in the bulk. We further find a gapped boundary with a surface topological order that is invariant under the emergent symmetry.
SciPost Phys. 15, 150 (2023) ·
published 10 October 2023

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We present a general approach to the bulkboundary correspondence of noninvertible topological phases, including both topological and fracton orders. This is achieved by a novel bulk construction protocol where solvable $(d+1)$dimensional bulk models with noninvertible topology are constructed from the socalled generalized Ising (GI) models in $d$ dimensions. The GI models can then terminate on the boundaries of the bulk models. The construction generates abundant examples, including not only prototype ones such as $\mathbb{Z}_2$ toric code models in any dimensions no less than two, and the Xcube fracton model, but also more diverse ones such as the $\mathbb{Z}_2× \mathbb{Z}_2$ topological order, the 4d $\mathbb{Z}_2$ topological order with pureloop excitations, etc. The boundary of the solvable model is potentially anomalous and corresponds to precisely only sectors of the GI model that host certain total symmetry charges and/or satisfy certain boundary conditions. We derive a concrete condition for such bulkboundary correspondence. The condition is violated only when the bulk model is either trivial or fracton ordered. A generalized notion of KramersWannier duality plays an important role in the construction. Also, utilizing the duality, we find an example where a single anomalous theory can be realized on the boundaries of two distinct bulk fracton models, a phenomenon not expected in the case of topological orders. More generally, topological orders may also be generated starting with lattice models beyond the GI models, such as those with symmetry protected topological orders, through a variant bulk construction, which we provide in an appendix.
SciPost Phys. 12, 052 (2022) ·
published 3 February 2022

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We investigate a family of invertible phases of matter with higherdimensional exotic excitations in even spacetime dimensions, which includes and generalizes the Kitaev's chain in 1+1d. The excitation has $\mathbb{Z}_2$ higherform symmetry that mixes with the spacetime Lorentz symmetry to form a higher group spacetime symmetry. We focus on the invertible exotic loop topological phase in 3+1d. This invertible phase is protected by the $\mathbb{Z}_2$ oneform symmetry and the timereversal symmetry, and has surface thermal Hall conductance not realized in conventional timereversal symmetric ordinary bosonic systems without local fermion particles and the exotic loops. We describe a UV realization of the invertible exotic loop topological order using the $SO(3)_$ gauge theory with unit discrete theta parameter, which enjoys the same spacetime twogroup symmetry. We discuss several applications including the analogue of ``fermionization'' for ordinary bosonic theories with $\mathbb{Z}_2$ nonanomalous internal higherform symmetry and timereversal symmetry.
Dr Ji: "Thank you for the positive com..."
in Submissions  report on Boundary states of Three Dimensional Topological Order and the Deconfined Quantum Critical Point