Abstract: Fractional Chern insulators (FCIs) in moiré materials present a unique platform for exploring strongly correlated topological phases beyond the paradigm of ideal quantum geometry. While analytical approaches to FCIs and fractional quantum Hall states (FQHS) often rely on idealized Bloch wavefunctions, realistic moiré models lack direct tunability of quantum metric and Berry curvature, limiting theoretical and numerical exploration. Here, we introduce an unsupervised machine learning framework to model interacting Hamiltonians directly through the distribution of single-particle form factors. Using a variational autoencoder (VAE), we show that unsupervised learning can not only distinguish FCI and non-FCI states, but also generate new form factors with distinct topological character, not present in the training set. This latent space enables the generation and interpolation of form factors for topological flatbands with Chern number ∣ C ∣ = 1, enabling the discovery of unobserved many-body states such as charge density waves. Principal component analysis (PCA) further reveals that the dominant patterns in the form factors—reflecting correlations across the Brillouin zone—can be decomposed into components with approximately quantized Chern numbers, providing new insights into the global and topological structure of quantum geometry. Our results highlight the ability of machine learning to generalize and model topological quantum systems, paving the way for the inverse design of form factors with tailored quantum geometry and many-body phases in flatband materials.
Abstract: Fractional Chern insulators (FCIs) in moiré materials present a unique platform for exploring strongly correlated topological phases beyond the paradigm of ideal quantum geometry. While analytical approaches to FCIs and fractional quantum Hall states (FQHS) often rely on idealized Bloch wavefunctions, realistic moiré models lack direct tunability of quantum metric and Berry curvature, limiting theoretical and numerical exploration. Here, we introduce an unsupervised machine learning framework to model interacting Hamiltonians directly through the distribution of single-particle form factors. Using a variational autoencoder (VAE), we show that unsupervised learning can not only distinguish FCI and non-FCI states, but also generate new form factors with distinct topological character, not present in the training set. This latent space enables the generation and interpolation of form factors for topological flatbands with Chern number ∣ C ∣ = 1, enabling the discovery of unobserved many-body states such as charge density waves. Principal component analysis (PCA) further reveals that the dominant patterns in the form factors—reflecting correlations across the Brillouin zone—can be decomposed into components with approximately quantized Chern numbers, providing new insights into the global and topological structure of quantum geometry. Our results highlight the ability of machine learning to generalize and model topological quantum systems, paving the way for the inverse design of form factors with tailored quantum geometry and many-body phases in flatband materials.
The discovery and understanding of new quantum phases has time and again transformed both fundamental physics and technology, yet progress often relies on slow, intuition-based theoretical considerations or experimental serendipity. Here, we introduce a general gradient-based framework for targeted phase discovery. We define a differentiable function, dubbed "target-phase loss function", which encodes fingerprints of a quantum state, thereby recasting phase search as a tractable optimization problem in Hamiltonian space. The method is broadly applicable to a wide range of symmetry-broken and topological orders and can be interfaced with most many-body numerical solvers. As a demonstration, we apply it to spinless fermions on the kagome lattice using exact diagonalization and discover two distinctive fractional Chern insulators (FCIs): (i) at filling ν=1/3, a "non-ideal" Abelian FCI whose band geometry lies far beyond the Landau-level mimicry paradigm and all recent generalizations; and (ii) at ν=1/2, a non-Abelian FCI stabilized purely by finite-range two-body interactions. These results provide the first explicit realization of such types of FCIs and establish a versatile paradigm for systematic quantum-phase discovery.
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 σB. 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.
Quantum geometry - including both the quantum metric and Berry curvature - arises from the non-zero overlap of well-defined eigenstates and their adiabatic evolution across the Brillouin zone. It has revolutionized condensed matter physics and material science by explaining quantum Hall effects, establishing the modern theory of polarization, and enabling a systematic search for topological materials on a large scale. Yet, despite these advances, we lack a framework for leveraging quantum geometry in the strongly interacting regime. Bridging this gap is critical: if we can harness geometric responses in correlated metals, we stand to engineer desirable transport and optical properties using existing material platforms, thus bypassing the complex task of designing materials with tailored quantum geometry from scratch. In this thesis, we take steps toward such a framework. First, we analyze the resilience of topological boundary modes in the presence of electronic correlations, identifying when interactions preserve, diminish, or destroy boundary modes. Second, we reveal geometric fingerprints of fluctuations in magnetically ordered systems, tying the electric quantum metric to the formation of instabilities and chiral quasi-particle excitations. Third, we generalize quantum geometry to describe families of many-body wave functions, providing a new algorithm to compute state-manifold curvatures suited to interacting phases. Our approach combines analytical theory with numerical methods, including density functional theory and tensor-network simulations, and is supported by open-source software developed during the PhD. Together, these results advance the topological classification of interacting phases and extend quantum geometry from single-particle bands to correlation functions, providing tools to design materials and devices with targeted geometric responses.
Fractional Chern insulators have attracted broad interest as lattice analogs of fractional quantum Hall states without Landau levels. However, low-filling fractional Chern insulators are fragile because charge-ordered phases can compete strongly with the fractional topological liquid. Here, we propose a center-decorated kagome model, motivated by geometry-tunable artificial lattices, in which the center-site hopping t2 provides a direct knob for the quantum geometry of an isolated C=1 flat band. Here quantum geometry refers to the Berry curvature and Fubini--Study metric, which determine the form factors of interactions projected into the Chern band. Exact diagonalization shows that tuning t2 away from the flatness-optimized kagome limit reduces the trace-condition deviation, suppresses competing charge order, and enhances the many-body stability at both ν=1/3 and the more fragile ν=1/5 filling. At ν=1/5, this stability-enhanced window persists under nearby interaction profiles, including variations of the dominant third-neighbor repulsion and weak nearest-neighbor admixtures. Low-energy spectra, spectral flow, quasihole and entanglement counting, static structure factors, and the quantized total many-body Chern number Ctot=1 consistently support Laughlin-like fractional Chern insulators. These results identify quantum-geometry engineering as a route to stabilizing dilute fractional Chern insulators beyond band-flatness optimization alone.
Quantum geometry - including both the quantum metric and Berry curvature - arises from the non-zero overlap of well-defined eigenstates and their adiabatic evolution across the Brillouin zone. It has revolutionized condensed matter physics and material science by explaining quantum Hall effects, establishing the modern theory of polarization, and enabling a systematic search for topological materials on a large scale. Yet, despite these advances, we lack a framework for leveraging quantum geometry in the strongly interacting regime. Bridging this gap is critical: if we can harness geometric responses in correlated metals, we stand to engineer desirable transport and optical properties using existing material platforms, thus bypassing the complex task of designing materials with tailored quantum geometry from scratch. In this thesis, we take steps toward such a framework. First, we analyze the resilience of topological boundary modes in the presence of electronic correlations, identifying when interactions preserve, diminish, or destroy boundary modes. Second, we reveal geometric fingerprints of fluctuations in magnetically ordered systems, tying the electric quantum metric to the formation of instabilities and chiral quasi-particle excitations. Third, we generalize quantum geometry to describe families of many-body wave functions, providing a new algorithm to compute state-manifold curvatures suited to interacting phases. Our approach combines analytical theory with numerical methods, including density functional theory and tensor-network simulations, and is supported by open-source software developed during the PhD. Together, these results advance the topological classification of interacting phases and extend quantum geometry from single-particle bands to correlation functions, providing tools to design materials and devices with targeted geometric responses.
A flat band with Chern number C = 0 , well isolated from the rest of Hilbert space by a gap much larger than interaction strength, is an unexpected context for fractional quantum Hall physics. In this work, we demonstrate the emergence of the fractional Chern insulator (FCI) states in such a trivial flat band using large-scale exact diagonalization and infinite density matrix renormalization group simulations. Hosted by an anisotropic lattice, this band has a sharp peak for the quantum geometry. At 1/3 hole filling, we find the unexpected FCI with 3-fold ground-state degeneracy and σ H = − 1 / 3 ( e 2 / h ) . The momentum-space carrier distribution shows that the quantum geometry peak tends to push the interacting holes away, toward other regions of the Brillouin zone with the nearly "ideal" quantum geometry, underlying the formation of the FCI in the C = 0 flat band. In comparison, we find only trivial ground states when the quantum geometry peak becomes less avoidable. Our work not only shows that FCl states could exist without the single-particle topology, relying on the more intrinsic quantum geometry, but also suggests a possibly generic scenario to explore FCIs in otherwise trivial flat bands.