Immersed figure-8 annuli and anyons

TL;DR

This work investigates immersion of topological regions in entanglement bootstrap, focusing on the figure-8 annulus $X_8$ in 2D and the question of Abelian sectors on immersed regions. By developing the information-convex-set framework and an Abelian-area tunneling construction, it proves that an Abelian sector on $X_8$ enforces a 2D strong isomorphism: the information-convex sets of any two homeomorphic immersed regions become isomorphic, even when the regions are not smoothly deformable. The results are established rigorously for Abelian anyon theories, and the analysis connects immersed annuli to anyon transportation in the presence of defects, suggesting a route to extract universal topological data from single-wavefunction states. The work also introduces a transportation thought experiment and discusses how topological defects influence the information-convex structure, highlighting open problems for non-Abelian theories and higher dimensions with potential implications for classifying topological order via immersion.

Abstract

Immersion (i.e., local embedding) is relevant to the physics of topologically ordered phases through the entanglement bootstrap. An annulus can be immersed in a disk or a sphere as a "figure-8", which cannot be smoothly deformed to an embedded annulus. We investigate a simple problem: is there an Abelian state on the immersed figure-8 annulus, locally indistinguishable from the ground state of the background physical system? We show that if the answer is affirmative, a strong sense of isomorphism must hold: two homeomorphic immersed regions must have isomorphic information convex sets, even if they cannot smoothly deform to each other on the background physical system. We explain why to care about strong isomorphism in physical systems with anyons and give proof in the context of Abelian anyon theory. We further discuss a connection between immersed annuli and anyon transportation in the presence of topological defects. In the appendices, we discuss related problems in broader contexts.

Figures (18)

• Figure 1: An immersed "figure-8" annulus ($X_8$) on a background 2-dimensional physical system. The background physical system, often taken to be a ball or a sphere, is equipped with a reference state $\sigma$. For the reference state, entanglement bootstrap axioms A0 and A1 are imposed on bounded radius balls such as the yellow ball illustrated. Entropy combinations $\Delta(B,C)$ and $\Delta(B,C,D)$ are defined in Eq. \ref{['eq:Delta-def']}.
• Figure 2: Regions used in the definition of information convex set. (a) $\Omega$ is an annulus embedded in the plane. $\Omega_+ \supset \Omega$ is its thickening. $b\subset \Omega_+$ is a bounded radius disk. (b) An immersed region $\Omega$, which is homeomorphic to a punctured torus.
• Figure 3: An illustration related to elementary steps. $A$ can be large and has an arbitrary topology, and it can be immersed. $BCD$ is a partition of a bounded radius (embedded) disk that we impose axiom A1.
• Figure 4: Examples of regions that detect fusion spaces. (a) A region that detects $\mathbb{V}_{ab}^c$, and (b) a region that detects $\mathbb{V}_{\mu a}^{\nu}$. Here, $a,b,c\in {\mathcal{C}}$ and $\mu,\nu \in {\mathcal{C}}_8$.
• Figure 5: The partition of an immersed annulus $Z=BCD$ used in defining quantum dimension (Definition \ref{['def:d_h-2D']}). We first map the immersed annulus to a topological annulus by a homeomorphism indicated by the arrow. We then partition the topological annulus as illustrated.

Theorems & Definitions (29)

• Remark
• Remark
• Definition 1: Information convex set for embedded region shi2020fusion
• Definition 2: Immersed region
• Definition 3: Information convex set for immersed region knots-paper
• Remark
• Definition 4: Quantum dimension
• Definition 5: Abelian sector
• Definition 6: Abelian anyon theory
• Lemma 7: Abelian sector criterion