Fluctuating Domain Wall Wavefunctions for Symmetry Protected Topological Phases

Xie Chen, Yuan-Ming Lu, Ashvin Vishwanath

Ryohei Kobayashi

We discuss a way to construct a commuting projector Hamiltonian model for a ($3+1$)-dimensional topological superconductor in class DIII. The wave function is given by a sort of string net of the Kitaev wire, decorated on the time-reversal ($T$) domain wall. Our Hamiltonian is provided on a generic three-dimensional (3D) manifold equipped with a discrete form of the spin structure. We will see how the 3D spin structure induces a 2D spin structure (called a ``Kasteleyn'' direction on a 2D lattice) on $T$ domain walls, which makes it possible to define fluctuating Kitaev wires on them. Upon breaking the $T$ symmetry in our model, we find the unbroken remnant of the symmetry, which is defined on the time-reversal domain wall. The domain wall supports the 2D nontrivial symmetry-protected topological (SPT) phase protected by the unbroken symmetry, which allows us to determine the SPT classification of our model, based on the recent quantum field theory argument by Hason, Komargodski, and Thorngren.

Hong-Hao Song, Chen Peng, Rui-Zhen Huang +1 more

Lukasz Fidkowski, Jeongwan Haah, Matthew B. Hastings

We construct an exactly solvable commuting projector model for a ($4+1$)-dimensional ${\mathbb{Z}}_{2}$-symmetry-protected topological phase (SPT) which is outside the cohomology classification of SPTs. The model is described by a decorated domain wall construction, with ``three-fermion'' Walker-Wang phases on the domain walls. We describe the anomalous nature of the phase in several ways. One interesting feature is that, in contrast to in-cohomology phases, the effective ${\mathbb{Z}}_{2}$ symmetry on a ($3+1$)-dimensional boundary cannot be described by a quantum circuit and instead is a nontrivial quantum cellular automaton. A related property is that a codimension-two defect (for example, the termination of a ${\mathbb{Z}}_{2}$ domain wall at a trivial boundary) will carry nontrivial chiral central charge 4 mod 8. We also construct a gapped symmetric topologically ordered boundary state for our model, which constitutes an anomalous symmetry-enriched topological phase outside of the classification of Chen and Hermele, and define a corresponding anomaly indicator.

Bei Zeng, Xie Chen, D. L. Zhou +1 more

Yabo Li, Zijian Song, Aleksander Kubica +1 more

We introduce a systematic method for constructing gapped domain walls of topologically ordered systems by gauging a lower-dimensional symmetry-protected topological (SPT) order. Based on our construction, we propose a correspondence between 1d SPT phases with a non-invertible $G\times \text{Rep}(G)\times G$ symmetry and invertible domain walls in the quantum double associated with the group $G$. We prove this correspondence when $G$ is Abelian and provide evidence for the general case by studying the quantum double model for $G=S_3$. We also use our method to construct \emph{anchoring domain walls}, which are novel exotic domain walls in the 3d toric code that transform point-like excitations to semi-loop-like excitations anchored on these domain walls.