Refining Heuristic Predictors of Fractional Chern Insulators using Machine Learning

TL;DR

This work tackles predicting the stability of fractional Chern insulators (FCIs) from single-particle band geometry. By combining large-scale exact diagonalization with an interpretable machine-learning framework based on Kolmogorov--Arnol'd networks, the authors extract compact symbolic formulas that relate the FCI quality metric $\mathcal{L}$ to the band-geometry descriptors $T$ and $\sigma_B$ for checkerboard and kagome lattices at $\nu = 1/3$. The resulting expressions achieve over 80% classification accuracy and competitive regression performance, revealing model-dependent trends (e.g., $\sigma_B$ can stabilize FCIs in checkerboard but destabilize them in kagome), and remain effective even with data-scarce datasets. This approach provides a general methodology for deriving simple, phenomenological laws that connect many-body phase stability to physically meaningful descriptors, enabling rapid hypothesis testing and targeted design of quantum phases.

Abstract

We develop an interpretable, data-driven framework to quantify how single-particle band geometry governs the stability of fractional Chern insulators (FCIs). Using large-scale exact diagonalization, we evaluate an FCI metric that yields a continuous spectral measure of FCI stability across parameter space. We then train Kolmogorov-Arnold networks (KANs) -- a recently developed interpretable neural architecture -- to regress this metric from two band-geometric descriptors: the trace violation $T$ and the Berry curvature fluctuations $σ_B$. Applied to spinless fermions at filling $ν=1/3$ in models on the checkerboard and kagome lattices, our approach yields compact analytical formulas that predict FCI stability with over $>80 \%$ accuracy in both regression and classification tasks, and remain reliable even in data-scarce regimes. The learned relations reveal model-dependent trends, clarifying the limits of Landau-level-mimicking heuristics. Our framework provides a general method for extracting simple, phenomenological "laws" that connect many-body phase stability to chosen physical descriptors, enabling rapid hypothesis formation and targeted design of quantum phases.

Figures (4)

• Figure 1: Log-log plots of the dataset used for ML analysis, representing the distribution of trace violation $T$ and Berry curvature deviation $\sigma_B$, colored by the corresponding FCI quality metric $\mathcal{L}$. Data is generated from ED calculations using the Hamiltonian on (a) the checkerboard lattice (\ref{['eq:checkerboard_model']} and (b) kagome lattice (\ref{['eq:kagome_model']}). Insets show a schematic representation of the lattice models, with arrows indicating hoppings with positive phase.
• Figure 2: Example of a KAN [3,[1,1],1] architecture. In KAN notation, an architecture with $n$ layers is described by a list of $n$ entries, with the $i$-th entry specifying the number of addition nodes in the $i$-th layer. Layers with two numbers (i.e. [1,1]) indicate the number of additive and multiplicative nodes at that layer, respectively.
• Figure 3: (a) ML pipeline for training KANs and extracting accurate symbolic formulas. (b) Representative KAN networks for the checkerboard and kagome models. The momentum-space functions $T(\mathbf{k})$ and $\mathcal{F}(\mathbf{k})$ (plotted respectively in blue and red, in each Brillouin zone) are used to compute $T$ and $\sigma_B$, which are then fed into the networks as inputs, with output the FCI metric $\mathcal{L}$. (c) Regression and classification performance of networks and symbolic formulas, measured in terms of the RMSE and accuracy, respectively.
• Figure 4: Average RMSEs and accuracies as a function of training dataset size, with error bars in grey.