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arXiv:2212.09754 · SciPost Physics

Boundary states of three dimensional topological order and the deconfined quantum critical point

Wenjie Ji, Nathanan Tantivasadakarn, Cenke Xu

Published Dec 7, 2023• 17 citations
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Abstract

We study the boundary states of the archetypal three dimensional topological order, i.e. the three dimensional \mathbb{Z}_2 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 multi-critical point, which is captured by the so-called deconfined quantum critical point (DQCP) with an “easy-axis” anisotropy. Moreover, there is an emergent \mathbb{Z}_{2,d} 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} 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.

DOI
10.21468/scipostphys.15.6.231

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We investigate the quantum phase diagram of the KKK-layer Ising toric code corresponding to KKK layers of two-dimensional toric codes coupled by Ising interactions. While for small Ising interactions the system displays Z2K{\mathbb{Z}}_{2}^{K}Z2K​ topological order originating from the toric codes in each layer, the system shows Z2{\mathbb{Z}}_{2}Z2​ topological order in the high-Ising limit. The latter is demonstrated for general KKK by deriving an effective low-energy model in KthK\mathrm{th}Kth-order degenerate perturbation theory, which is given as an effective anisotropic single-layer toric code in terms of collective pseudospins 1/2 referring to the two ground states of isolated Ising chain segments. For the specific cases K=3K=3K=3 and K=4K=4K=4 we apply high-order series expansions to determine the gap series in the low- and high-Ising limit. Extrapolation of the elementary energy gaps gives convincing evidence that the ground-state phase diagram consists of a single quantum critical point and our findings suggest a quantum phase transition in the 3D Ising* universality class for both KKK separating both types of topological order, which is consistent with former findings for the bilayer Ising toric code.
Bulk topology and criticality can both lead to nontrivial boundary effects. Topological orders are often characterized by their robust edge states, while bulk critical points can have different boundary scalings governed by boundary conditions. The interplay between these two different boundary effects is an intriguing problem. The boundary of the deconfined quantum critical point (DQCP) is the ideal platform for the interplay of the two boundary effects, as the DQCP is also an intrinsically gapless symmetry protected topological (igSPT) state. In this work we discuss the boundary of several analogues of the DQCP. We demonstrate that the fluctuation of the bulk order parameters and their various boundary conditions lead to a rich possibility of the edge states, including a "pseudogap" (or super power-law decay) behavior. We also discuss the quantum information perspective of our work, i.e. the implication of our results on DQCP under weak-measurement. Weak-measurement followed by post-selection can change the boundary condition at the temporal boundary in the path-integral representation of a density matrix, which will lead to different behaviors of the "strange correlator".
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Deconfined criticality and gapless topological states have recently attracted growing attention, as both phenomena go beyond the traditional Landau paradigm. However, the deep connection between these two critical states, particularly in lattice realization, remains insufficiently explored. In this Letter, we reveal that certain deconfined criticality can be regarded as an intrinsically gapless topological state without gapped counterparts in a one dimensional lattice model. Using a combination of field-theoretic arguments and large-scale numerical simulations, we establish the global phase diagram of the model, which features deconfined critical lines separating two distinct spontaneous symmetry breaking ordered phases. More importantly, we unambiguously demonstrate that the mixed anomaly inherent to deconfined criticality enforces topologically robust edge modes near the boundary, providing a general mechanism by which deconfined criticality manifests as a gapless topological state. Our findings not only offer a new perspective on deconfined criticality but also deepen our understanding of gapless topological phases of matter.