Transitions between different topologically ordered phases have been studied by artificially creating boundaries between these gapped phases and thus studying their effects relating to condensation and tunneling of particles from one phase to the other. In this work we introduce exactly solvable models which are similar to the quantum double models (QDM) of Kitaev where such domain walls are dynamically generated making them a part of the spectrum. These systems have a local symmetry and may or may not have a global symmetry leading to the possibility of a ground state degeneracy arising from both, global symmetry breaking and a topological degeneracy. The domain wall states now separate the different topologically ordered phases belonging to the different sectors controlled by the global symmetry, when present. They have interesting properties including fusion with the deconfined anyons of the topologically ordered phase, to either create new domain walls, or their annihilation. They can also act like a scatterer permuting anyons of the topologically ordered phases on either side of the domain wall. Thus these domain wall states can be thought of as a synthetic scatterer which can be created in any part of the lattice. We show these effects for the simplest case of the $\mathbb{Z}_2$ toric code phase decorated with a global $\mathbb{Z}_2$ symmetry and then make remarks about the case when we have the QDM based on two arbitrary groups $G_1$ and $G_2$ in which case we may no longer have a global symmetry.Transitions between different topologically ordered phases have been studied by artificially creating boundaries between these gapped phases and thus studying their effects relating to condensation and tunneling of particles from one phase to the other. In this work we introduce exactly solvable models which are similar to the quantum double models (QDM) of Kitaev where such domain walls are dynamically generated making them a part of the spectrum. These systems have a local symmetry and may or may not have a global symmetry leading to the possibility of a ground state degeneracy arising from both, global symmetry breaking and a topological degeneracy. The domain wall states now separate the different topologically ordered phases belonging to the different sectors controlled by the global symmetry, when present. They have interesting properties including fusion with the deconfined anyons of the topologically ordered phase, to either create new domain walls, or their annihilation. They can also act like a scatterer permuting anyons of the topologically ordered phases on either side of the domain wall. Thus these domain wall states can be thought of as a synthetic scatterer which can be created in any part of the lattice. We show these effects for the simplest case of the $\mathbb{Z}_2$ toric code phase decorated with a global $\mathbb{Z}_2$ symmetry and then make remarks about the case when we have the QDM based on two arbitrary groups $G_1$ and $G_2$ in which case we may no longer have a global symmetry.

arXiv:1510.06627Oct 22, 20151 citation

Lattice model constructions for gapless domain walls between topological phases

Chenfeng Bao, Shuo Yang, Chenjie Wang +1 more

Domain walls between different topological phases are one of the most interesting phenomena that reveal the nontrivial bulk properties of topological phases. Very recently, gapped domain walls between different topological phases have been intensively studied. In this paper, we systematically construct a large class of lattice models for gapless domain walls between twisted and untwisted gauge theories with arbitrary finite group $G$ . As simple examples, we numerically study several finite groups (including both Abelian and non-Abelian finite group such as ${S}_{3}$ ) in $2\mathrm{D}$ using the state-of-the-art loop optimization of tensor network renormalization algorithm. We also propose a physical mechanism for understanding the gapless nature of these particular domain wall models. Finally, by taking advantage of the classification and construction of twisted gauge theories using group cohomology theory, we generalize such constructions into arbitrary dimensions, which might provide us a systematical way to understand gapless domain walls and topological quantum phase transitions.Domain walls between different topological phases are one of the most interesting phenomena that reveal the nontrivial bulk properties of topological phases. Very recently, gapped domain walls between different topological phases have been intensively studied. In this paper, we systematically construct a large class of lattice models for gapless domain walls between twisted and untwisted gauge theories with arbitrary finite group $G$. As simple examples, we numerically study several finite groups (including both Abelian and non-Abelian finite group such as ${S}_{3}$) in $2\mathrm{D}$ using the state-of-the-art loop optimization of tensor network renormalization algorithm. We also propose a physical mechanism for understanding the gapless nature of these particular domain wall models. Finally, by taking advantage of the classification and construction of twisted gauge theories using group cohomology theory, we generalize such constructions into arbitrary dimensions, which might provide us a systematical way to understand gapless domain walls and topological quantum phase transitions.

arXiv:1801.00719Physical Review ResearchApr 14, 202230 citations

Models for Gapped Boundaries and Domain Walls

Alexei Kitaev, Liang Kong

arXiv:1104.5047Communications in Mathematical PhysicsJun 6, 2012443 citations

Gapped Domain Walls, Gapped Boundaries, and Topological Degeneracy

Tian Lan, Juven Wang, Xiao-Gang Wen

Gapped domain walls, as topological line defects between (2+1)D topologically ordered states, are examined. We provide simple criteria to determine the existence of gapped domain walls, which apply to both Abelian and non-Abelian topological orders. Our criteria also determine which (2+1)D topological orders must have gapless edge modes, namely, which (1+1)D global gravitational anomalies ensure gaplessness. Furthermore, we introduce a new mathematical object, the tunneling matrix W, whose entries are the fusion-space dimensions W(ia), to label different types of gapped domain walls. By studying many examples, we find evidence that the tunneling matrices are powerful quantities to classify different types of gapped domain walls. Since a gapped boundary is a gapped domain wall between a bulk topological order and the vacuum, regarded as the trivial topological order, our theory of gapped domain walls inclusively contains the theory of gapped boundaries. In addition, we derive a topological ground state degeneracy formula, applied to arbitrary orientable spatial 2-manifolds with gapped domain walls, including closed 2-manifolds and open 2-manifolds with gapped boundaries.Gapped domain walls, as topological line defects between (2+1)D topologically ordered states, are examined. We provide simple criteria to determine the existence of gapped domain walls, which apply to both Abelian and non-Abelian topological orders. Our criteria also determine which (2+1)D topological orders must have gapless edge modes, namely, which (1+1)D global gravitational anomalies ensure gaplessness. Furthermore, we introduce a new mathematical object, the tunneling matrix W, whose entries are the fusion-space dimensions W(ia), to label different types of gapped domain walls. By studying many examples, we find evidence that the tunneling matrices are powerful quantities to classify different types of gapped domain walls. Since a gapped boundary is a gapped domain wall between a bulk topological order and the vacuum, regarded as the trivial topological order, our theory of gapped domain walls inclusively contains the theory of gapped boundaries. In addition, we derive a topological ground state degeneracy formula, applied to arbitrary orientable spatial 2-manifolds with gapped domain walls, including closed 2-manifolds and open 2-manifolds with gapped boundaries.

arXiv:1408.6514Physical Review LettersFeb 18, 2015155 citations

\mathbb{Z}_2

toric code phase decorated with a global

\mathbb{Z}_2

symmetry and then make remarks about the case when we have the QDM based on two arbitrary groups

G_1

and

G_2

in which case we may no longer have a global symmetry.

G

. As simple examples, we numerically study several finite groups (including both Abelian and non-Abelian finite group such as

{S}_{3}

) in

2\mathrm{D}

using the state-of-the-art loop optimization of tensor network renormalization algorithm. We also propose a physical mechanism for understanding the gapless nature of these particular domain wall models. Finally, by taking advantage of the classification and construction of twisted gauge theories using group cohomology theory, we generalize such constructions into arbitrary dimensions, which might provide us a systematical way to understand gapless domain walls and topological quantum phase transitions.