The past few years have seen a revived interest in quantum geometrical characterizations of band structures due to the rapid development of topological insulators and semi-metals. Although the metric tensor has been connected to many geometrical concepts for single bands, the exploration of these concepts to a multi-band paradigm still promises a new field of interest. Formally, multi-band systems, featuring in particular degeneracies, have been related to projective spaces, explaining also the success of relating quantum geometrical aspects of flat band systems, albeit usually in the single band picture. Here, we propose a different route involving Plücker embeddings to represent arbitrary classifying spaces, being the essential objects that encode $all$ the relevant topology.This paradigm allows for the quantification of geometrical quantities directly in readily manageable vector spaces that a priori do not involve projectors or the need of flat band conditions. As a result, our findings are shown to pave the way for identifying new geometrical objects and defining metrics in arbitrary multi-band systems, especially beyond the single flatband limit, promising a versatile tool that can be applied in contexts that range from response theories to finding quantum volumes and bounds on superfluid densities as well as possible quantum computations.
The past few years have seen a revived interest in quantum geometrical characterizations of band structures due to the rapid development of topological insulators and semi-metals. Although the metric tensor has been connected to many geometrical concepts for single bands, the exploration of these concepts to a multi-band paradigm still promises a new field of interest. Formally, multi-band systems, featuring in particular degeneracies, have been related to projective spaces, explaining also the success of relating quantum geometrical aspects of flat band systems, albeit usually in the single band picture. Here, we propose a different route involving Plücker embeddings to represent arbitrary classifying spaces, being the essential objects that encode all the relevant topology.This paradigm allows for the quantification of geometrical quantities directly in readily manageable vector spaces that a priori do not involve projectors or the need of flat band conditions. As a result, our findings are shown to pave the way for identifying new geometrical objects and defining metrics in arbitrary multi-band systems, especially beyond the single flatband limit, promising a versatile tool that can be applied in contexts that range from response theories to finding quantum volumes and bounds on superfluid densities as well as possible quantum computations.
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.
W HAT THEORETICAL APPROACHES ARE APPROPRIATE for describing topological quantum matter? The answer depends on the type of topological phase we want to study, e.g., non-interacting or interacting, and on the type of information we are interested in, e.g., system-specific properties in the case of a theory designed to provide support for a particular experiment or type of experiment. Non-interacting topological phases can be described within topological band theory using the basic tools of the band theory of solids. However, the complexity of the model, i.e., how much detail has to be included, depends on whether we are only interested in the topological properties of the system or, in addition, we need to consider certain nontopological features. For example, predicting whether or not a specific material is a topological insulator requires detailed knowledge of its band structure, which can be acquired using density functional theory (DFT) methods. Similar treatments may be required to account for certain experimental features involving both topological and non-topological contributions. On the other hand, to understand the classification of topological insulators, it is sufficient to consider “bare bone” effective models, such as continuum Dirac models or simple tight-binding models. Topologically, these descriptions with extremely different levels of complexity are equivalent whenever one can smoothly connect the corresponding Hamiltonians (up to the addition/subtraction of trivial bands) without closing the insulating gap. However, the effective models provide only a crude description of experimentally relevant properties such as energy gaps, band dispersion, and real space properties of boundary states. In some sense, the situation is analogous to the characterization of a given mathematical object within topology, which only deals with properties that are robust against deformations, and geometry, which accounts for properties described by distances, angles, and shapes.
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.
We uncover the quantum geometric structure of graviton Wigner functions and stress-energy tensors via quantum field theory in curved spacetime, revealing the phase-space geometry of spacetime and relativistic quantum-state manifolds. We show that the polarization mode expansion of the underlying graviton field operator is fully captured by quantum geometry, as encoded in the spinor-helicity formalism, thereby establishing the geometric framework from the outset. Applying this to a weak gravitational background, we demonstrate the roles of the quantum metric and connection in describing graviton transport beyond the chiral vortical effect. We also clarify the role of the quantum metric in normalizing the perfect-fluid graviton stress tensor within this approach. This work paves the path for explorations of multistate Hilbert-space geometry in high-energy and gravitational physics. In addition, this framework naturally encompasses lower-spin excitations, allowing for a unified quantum geometric treatment of bosonic and fermionic many-body systems in condensed matter and particle physics.
Quantum geometry, a nascent field, seeks to formalize the geometric properties of quantum systems through the lens of topology. This paper explores the development of a novel mathematical framework for defining and analyzing these geometric properties, explicitly incorporating topological concepts to enhance understanding and provide a more robust and insightful approach compared to traditional geometric methods. We propose a new methodology predicated on the observation that certain geometric features exhibit topological behavior, offering a fundamentally different perspective on the structure of quantum systems. The paper outlines the core mechanism, detailing how topological aspects will be integrated into the definition of geometric properties and how this will lead to a deeper comprehension of quantum phenomena. The investigation will focus on establishing a rigorous mathematical foundation for this approach, demonstrating its potential for improving the analysis of complex quantum systems.