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.
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.
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.
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.
Quantum geometry characterizes the variation of wavefunctions in momentum space through their overlaps and relative phases, providing a general framework for understanding many transport and optical properties. It is generally formulated in terms of interband matrix elements, which, entering the response functions, allow obtaining experimental access to the quantum geometric tensor. Recently, it has been emphasized that quantum geometry can also be interpreted in terms of quantum dipole fluctuations in the ground state driven by interband mixing. Here, we extend this picture to include contributions from many-body collective fluctuations, in which propagators and response vertices are dressed dynamically by the interaction with collective modes. Focusing on the Berry curvature, we show that contributions from collective fluctuations can be experimentally distinguished from bare band-geometric contributions, via specific antisymmetric channels in inelastic scattering spectra. We further identify the non-commutative properties of transverse quantum fluctuations as well as non-local-time interactions as the generators of this dynamical curvature in the susceptibility response.
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.
Quantum geometry has emerged as a powerful framework for understanding topological, optical and transport phenomena, revealing connections between the quantum geometric tensor (QGT) and linear response functions. Building on these developments, recent works formulate quantum geometry so that the parameter space is extended from the Brillouin zone to the space of external perturbing fields that modify the ground state. Here we develop a self-consistent diagrammatric route to determine the QGT for generic parameter spaces, including collective fluctuations or structural distortions. With this in view, we generalize previous approaches to build the geometric tensor from interacting vertex correlations that provide the spectral metric and curvature. The formalism presented here extends the interaction-dressed vertex approach (previously applied to electromagnetic responses in momentum space) to collective electronic and structural deformation fields, complementing ongoing efforts to quantify many-body contributions to quantum geometry.
In this work we propose that some strongly correlated quantum materials can spontaneously generate an effective geometrical structure described by a metric tensor g_{μν}^{eff}. Unlike engineered metamaterials or external spacetime analogues, this emergent geometry arises intrinsically from the many‑body quantum state. We develop a theoretical framework that connects the low‑energy effective action of fermionic quasiparticles to a curved‑space Dirac equation, with curvature determined self‑consistently by the electronic density and interaction parameters. The resulting geometry influences transport, optical, and topological properties, suggesting a novel platform for exploring quantum gravity concepts in table‑top experiments.