A remark on gapped domain walls between topological phases
Yasuyuki Kawahigashi
TL;DR
The paper formalizes gapped domain walls and gapped boundaries between topological phases as irreducible local Lagrangian $Q$-systems in $\mathcal{C}\boxtimes\mathcal{D}^{opp}$ and develops a matrix-constraint framework for the tunneling matrix $Z$. It proves that Witt equivalence of $\mathcal{C}$ and $\mathcal{D}$ is necessary for the existence of a gapped domain wall, and derives key conditions $Z_{\lambda\mu}\in\mathbb{N}$, $S^{\mathcal{C}} Z = Z S^{\mathcal{D}}$, and $T^{\mathcal{C}} Z = Z T^{\mathcal{D}}$, along with a fusion-compatibility inequality. However, the three matrix conditions are not sufficient, as shown by a counterexample (D2) where the charge-conjugation modular invariant of a quantum double satisfies them but does not correspond to any $Q$-system; thus $Z$ does not uniquely determine the gapped domain wall. The results connect condensation and Longo-Rehren subfactor theory to the domain-wall problem and clarify limitations of existing conjectures, informing the classification of domain walls in topological phases.
Abstract
We give a mathematical definition of a gapped domain wall between topological phases and a gapped boundary of a topological phase. We then provide answers to some recent questions studied by Lan, Wang and Wen in condensed matter physics based on works of Davydov, Müger, Nikshych and Ostrik. In particular, we identify their tunneling matrix and a coupling matrix of Rehren, and show that their conjecture does not hold.