Identifying chiral topological order in microscopic spin models by modular commutator

TL;DR

The paper tackles the challenge of extracting the chiral central charge $c_-$ from microscopic spin models, including non-Abelian chiral topological orders, by employing the modular commutator formalism. It computes $J(A,B,C) = i \mathrm{Tr}(\rho_{ABC}[K_{AB},K_{BC}])$ from a single ground-state wavefunction and uses $J(A,B,C) = \frac{\pi}{3} c_-$ to obtain $c_-$, with the topological entanglement entropy $\\gamma$ serving as a cross-check. Applying the method to two non-integrable models—the Zeeman-Kitaev honeycomb model (Ising topological order) and the kagome Heisenberg model with a scalar spin-chirality term (semionic CSL)—yields $c_- = \tfrac{1}{2}$ and $\\gamma = \ln 2$ for the Kitaev case, and $c_- \approx 1$, $\\gamma \approx \tfrac{1}{2}\\ln 2$ for the kagome CSL, concordant with Ising TQFT and $U(1)_2$ Chern–Simons theory. Finite-size scaling shows rapid convergence toward these universal values, establishing modular commutators as a robust bulk diagnostic for chiral topological order in strongly correlated quantum magnets.

Abstract

The chiral central charge $c_-$ is a key topological invariant of the edge characterizing the bulk two-dimensional chiral topological order, but its direct evaluation in microscopic spin models has long been a challenge, especially for non-abelian topological order. Building on the recently developed modular commutator formalism, we numerically obtain $c_-$ directly from single ground-state wave functions of two-dimensional interacting spin models that have chiral topological order. This provides a geometry-independent and bulk diagnostic of chirality. We study two nonintegrable systems -- the Zeeman-Kitaev honeycomb model and the kagome antiferromagnet -- both subjected to scalar spin chirality perturbations. We find that the modular commutator yields results consistent with the expected topological quantum field theories. We also compute the topological entanglement entropy which provides an independent diagnostic of the topological orders. Our work establishes modular commutators as a powerful numerical probe of chiral topological order in strongly correlated quantum magnets.

Figures (6)

• Figure 1: Partition of a disk-shaped region $ABC$ in the bulk $\Lambda$ into three adjacent subsystems $A$, $B$, and $C$. This setup is used to implement the formulas for the chiral central charge and the TEE. Each subsystem is assumed to be sufficiently large compared to the correlation length so that universal contributions can be reliably extracted.
• Figure 2: (a) Honeycomb lattice for the Kitaev model, with nearest-neighbor bonds labeled by $x$, $y$, and $z$, and a single three-spin term $\sigma^x_i \sigma^y_j \sigma^z_k$ of the three-spin chirality term; (b) Schematic of the kagome lattice. The sites $(i,j,k)$ on a triangular plaquette are ordered counterclockwise to define the scalar spin-chirality interaction.
• Figure 3: Finite-size scaling of (a) the chiral central charge $c_-$ and (b) the TEE $\gamma$ in the ITO phase of the Kitaev model. The numerical results (dots) are obtained from exact diagonalization of finite clusters, and the solid curves show the best fits using exponential (for $c_-$) and power-law (for $\gamma$) ansatz functions. The horizontal dashed lines indicate the theoretical values, $c_- = \frac{1}{2}$ and $\gamma = \ln 2$, respectively.
• Figure 4: Finite-size results for (a) the chiral central charge $c_-(N)$ and (a) the TEE $\gamma(N)$ of the kagome CSL, obtained from exact diagonalization on clusters with $N=12,18,24,30$ sites and periodic boundary conditions. Blue (orange) symbols denote raw data at $\theta=0.2\pi$ ($\theta=0.15\pi$). Black diamonds show the pairwise averages of the two datasets at each $N$, with error bars corresponding to half the inter-$\theta$ difference and thus providing a conservative estimate of systematic uncertainty within the CSL phase. Dashed horizontal lines indicate the universal values expected for the $\nu = 1/2$ bosonic Laughlin state, $c_- = 1$ and $\gamma = \tfrac{1}{2}\ln 2$. The near-convergence of the averaged data for $N=24$ and $N=30$ demonstrates consistency with the universal topological invariants.
• Figure 5: Finite clusters used for exact diagonalization. (a) Honeycomb clusters for the Kitaev model with $N=12,16,18,24,30$ sites. (b) Kagome clusters for the Heisenberg model with scalar spin-chirality, with $N=12,18,24,30$ sites. All clusters are taken with periodic boundary conditions.