Chiral Edge States Emerging on Anyon-Net

TL;DR

This work introduces a symmetry-driven lattice construction for chiral non-Abelian topological phases using a 2+1D anyon-chain framework built from modular tensor categories, enforcing exact microscopic MTC symmetry. By mapping the 2+1D problem to a long-range 1D chain and employing tensor-network techniques, the authors provide numerical evidence of chiral edge modes in Ising and Fibonacci anyon systems, with entanglement spectra matching boundary CFT predictions. The approach yields a flexible platform to realize a broad class of chiral topological orders via input MTCs, without relying on solvable models, and points toward exploring bulk properties and higher-dimensional generalizations. The findings extend the toolkit for studying strongly coupled 2+1D systems and pave the way for microscopic realizations of a wide range of topological phases grounded in MTC symmetry.

Abstract

We propose a symmetry-based approach to constructing lattice models for chiral topological phases, focusing on non-Abelian anyons. Using a 2+1D version of anyon chains and modular tensor categories(MTCs), we ensure exact MTC symmetry at the microscopic level. Numerical simulations using tensor networks demonstrate chiral edge modes for topological phases with Ising and Fibonacci anyons. Our method contrasts with conventional solvability approaches, providing a new theoretical avenue to explore strongly coupled 2+1D systems, revealing chiral edge states in non-Abelian anyonic systems.

Figures (5)

• Figure 1: Real-space configuration of $(a)$ left-chiral, $(b)$ non-chiral, and $(c)$ right-chiral phases. The heatmaps represent the local excitation energy density $\delta E_{\text{loc}}$ normalized by the energy gap $\Delta$ for $\theta/\pi = (a) ~ 0.1$, $(b) ~ 0.5$, and $(c) ~ 0.9$ of the Fibonacci-type model. The excitation is localized on the edge for $\theta/\pi = 0.1$ and $\theta/\pi = 0.9$, while it spreads into the bulk for $\theta/\pi = 0.5$. The arrows represent the local chirality relative to the ground state, with their color representing the chirality and their size scaling according to the amplitude.
• Figure 2: The above anyon diagram represents a state on a square lattice with $W = H = 4$. Here, the vertical lines are labeled by the chosen object $\rho$.
• Figure 3: The correlation length of the ground states for $(a)$ the Ising-type$(H=4)$ and $(c)$ Fibonacci-type$(H=5)$ models. The correlation length was calculated from the second/third leading eigenvalues of the MPS transfer matrix with $\chi=24\sim60$/$24\sim140$, respectively. The data points from larger $\chi$ are indicated with darker black. The entanglement entropy of the half strip grows as $S_A\sim\frac{c}{6}\ln(\xi)$ in the gapless regions of $(b)$ the Ising-type and $(d)$ Fibonacci-type model. The insets show $H=6$ results with larger $\chi\sim270$, indicating the gapless regions extend as phases. Meanwhile, $\theta/\pi =0.3$ and $0.4$ exhibit saturation of $S_A$, indicating finite correlation lengths.
• Figure 4: Entanglement spectrum of $(a)$ the Ising-type with $\theta=\frac{\pi}{8}(H=4)$ and the Fibonacci-type anyon model at $(b)$$\theta=0.2\pi(H=6)$ and $(c)$$\theta=0.9\pi(H=6)$ in the infinite strip geometry. All spectra are normalized so that the first excitations in the $\mathds{1}$ sector match the theoretical prediction. The theoretical boundary CFT spectrum and multiplicity are presented with crosses.
• Figure 5: The energy spectrum of the $4\times4$$(a)$ Ising-type and $(b)$ Fibonacci-type models obtained by exact diagonalization. The horizontal axis $k$ is the “quasi-momentum" of the edge states. Chiral energy-momentum dispersion is observed in the same region as discussed in the main text (denoted with red points).