Quantum geometry quantifies how the single-particle Bloch wavefunction changes in phase and amplitude across the Brillouin Zone. In multi-orbital systems where bands have strongly mixed orbital composition, quantum geometry plays a vital role in determining the ground state and low-energy properties of interacting electronic systems. In this work, we show that Mott metal-insulator transitions, as well as transitions between different magnetic orders within the Mott insulating phase, can be driven by the quantum geometry of the underlying Bloch band, thereby providing a mechanism complementary to conventional bandwidth-tuned Mott transitions. By studying the Kane-Mele-Hubbard model using exact diagonalization, we demonstrate that in in half-filled and topologically-trivial bands, quantum geometric properties of the Bloch states alone can act as a tuning knob for Mott metal-to-insulator and affect the competition between ferromagnetism and antiferromagnetism. We show that both transitions may be heuristically understood via non-local Coulomb scattering in a basis of exponentially localized Wannier functions. These results highlight the role of quantum geometry beyond topological settings as a governing principle for conventional Mott and magnetic physics in multi-orbital and moiré materials.
Quantum geometry quantifies how the single-particle Bloch wavefunction changes in phase and amplitude across the Brillouin Zone. In multi-orbital systems where bands have strongly mixed orbital composition, quantum geometry plays a vital role in determining the ground state and low-energy properties of interacting electronic systems. In this work, we show that Mott metal-insulator transitions, as well as transitions between different magnetic orders within the Mott insulating phase, can be driven by the quantum geometry of the underlying Bloch band, thereby providing a mechanism complementary to conventional bandwidth-tuned Mott transitions. By studying the Kane-Mele-Hubbard model using exact diagonalization, we demonstrate that in in half-filled and topologically-trivial bands, quantum geometric properties of the Bloch states alone can act as a tuning knob for Mott metal-to-insulator and affect the competition between ferromagnetism and antiferromagnetism. We show that both transitions may be heuristically understood via non-local Coulomb scattering in a basis of exponentially localized Wannier functions. These results highlight the role of quantum geometry beyond topological settings as a governing principle for conventional Mott and magnetic physics in multi-orbital and moiré materials.
We construct a Hubbard model with a nearly flat band whose quantum geometry can be tuned independent of the energy dispersion and the Coulomb interaction. We show that, when the nearly flat band is half filled, the exact ground state of the model exhibits ferromagnetism and that this ferromagnetism is stabilized by the quantum metric through the spin stiffness. Furthermore, we demonstrate that tuning the quantum geometry alone drives a magnetic phase transition. Our nonperturbative results without resorting to mean-field approximations reveal the quantum-geometric origin of ferromagnetism and the underlying many-body physics in dispersive-band systems.
We discuss the appearance of quantum orders in the Hubbard model for interacting electrons, at half-filling. Such phases do not have local order parameters and need to be characterized by the quantum mechanical properties of their ground state. On one hand, we study the Mott transition from a metal to a spin liquid insulator in two dimensions, of potential relevance to some layered organic compounds. The correlation-driven transition occurs at fixed filling and involves fractionalization of the electron: upon entering the insulator, a Fermi surface of neutral spinons coupled to an internal gauge field emerges. We focus on the transport properties near the quantum critical point and find that the emergent gauge fluctuations play a key role in determining the universal scaling. Second, motivated by a class of three-dimensional transition metal oxides, the pyrochlore iridates, we study the interplay of non-trivial band topology and correlations. Building on the strong spin orbit coupling in these compounds, we construct a general microscopic Hubbard model and determine its mean-field phase diagram, which contains topological insulators, Weyl semimetals, axion insulators and various antiferromagnets. We also discuss the effects many-body correlations on theses phases. We close by examining a fractionalized topological insulator that combines the two main themes of the thesis: fractionalization and non-trivial band topology. Specifically, we study how the two-dimensional protected surface states of a topological Mott insulator interact with a three-dimensional emergent gauge field. Various correlation effects on observables are identified.
Higher vortexability is often viewed as a route to topological flat bands with higher-Landau-level-like quantum geometry. Here we emphasize a complementary perspective: it provides a tunable multiband structure in which wave function geometry can be varied continuously while the band dispersion, degeneracy, and topology remain fixed. We perform systematic exact-diagonalization studies of many-body phases in fractionally filled higher vortexable moiré systems, retaining the full flat-band Hilbert space rather than projecting onto a single band. The multiband treatment reveals a cascade of Abelian and non-Abelian phases at zero magnetic field, including integer and fractional exciton insulators, Abelian fractional Chern insulators, Moore-Read and Read-Rezayi states. At fixed filling, different phases are connected through transitions or crossovers driven solely by changes in wave function geometry, highlighting quantum geometry itself as a direct tuning parameter between competing topological states. At fillings associated with Moore-Read and Read-Rezayi states, our calculations show that interband mixing shifts the optimal quantum geometry regime without suppressing non-Abelian topological order under screened Coulomb interaction. Our results establish higher vortexable moiré bands as a tunable platform for exploring geometry-driven multiband topological phases at zero magnetic field.
Recent studies have revealed that the quantum geometry of electronic bands determines the electromagnetic properties of noninteracting insulators and semimetals. However, the role of quantum geometry in the optical responses of interacting electron systems remains largely unexplored. Here, we examine the interplay between Coulomb interactions and Bloch-band quantum geometry in clean metals. We demonstrate that the low-frequency optical conductivity of a correlated metal encodes the structure of Bloch wave functions at the Fermi surface. This response originates from integrating out highly off-resonant interband scattering processes enabled by Coulomb interactions. The resulting quantum-geometric contribution appears generically in multiband systems but becomes the dominant effect in the optical conductivity for a parabolic band. We consider a dilute correlated metal near a topological band inversion and show that the doping dependence of optical absorption can measure how the orbital character of Bloch wave functions changes at the Fermi surface. Our results illustrate how the confluence of quantum geometry and Coulomb interactions can enable optical processes and enrich the physics of Fermi liquids.
This thesis reports original research on strongly interacting fermions described by Hubbard-like models which can be realized with ultracold atom experiments. In particular, two main directions have been pursued: (i) the control of magnetic and topological properties in a honeycomb lattice through a suitable tuning of the trapping potential, and (ii) the possibility to observe orbital-selective Mott physics in a multicomponent Fermi system where the SU(N) symmetry is broken in a controlled way. The first line of research is based on the idea that increasing the strength of a trapping potential, one can trap a strongly interacting fermions in a limited portion of space, thereby changing their average density and the related magnetic properties. In particular it was demonstrated that this confinement protocol can be used to designeffective “nanoflakes” which are reminiscent of a solid-state proposal [1]. The second step was to extend the same philosophy to the so-called Kane-Mele model, a spinful version of the paradigmatic model proposed by Haldane which presents quantum Hall states. This model can be realized with cold-atom systems using artificial gauge fields. In this thesis we consider a Kane-Mele-Hubbard model where the electronic structure of the Kane-Mele model is supplemented by the onsite Hubbard interaction, allowing us to study the interplay between the correlations and magnetism induced by the repulsion, with the non-trivial topological properties. Also for this model, we follow the strategy discussed above, namely the possibility to engineer an effective “nanoflake” by tuning the lattice potential. The final outcome in this case is even reacher, because it leads, among other results, to a situation where the effective flake hosts an inhomogeneous state which presents at the same time topologically non-trivial properties and magnetism, realizing a spin-Chern insulator [2]. Besides the remarkable application for topological matter, artificial gauge fields, which rely on the coupling of spin states, offer also great insides into mutli-orbital physics. In the second main research line reported in the thesis we study in collaboration with the experimental group of Leonardo Fallani, the controlled symmetry breaking on a 173Yb many-body system, where coupling between spin states is exploited to simulate the mechanism relevant for the physics of multi-orbital materials, such as iron-based superconductors. In order to reveal this mechanism in cold atoms we focus on interacting SU(3) lattice fermions where the SU(3) spin symmetry is explicitly broken by a Raman coupling between the spin states. Combining experiments and theory, we demonstrate that this setup reveals orbital-selective physics and it shows that the SU(3)-symmetry-breaking Raman coupling favours Mott localization and selective properties [3]. Our results show a clear evidence of orbital-selective correlations, which reflect in contrasting transport properties in different orbitals, the extreme case being an orbital-selective Mott state where only some of the orbitals are Mott localized while others remain metallic. Moreover, the coupling between spin states favors Mott localization with respect to the symmetric case, hence it reduces the critical interaction needed for a full Mott localization.
We study by dynamical mean-field theory the ground state of a quarter-filled Hubbard model of two bands with different bandwidths. At half-filling, this model is known to display an orbital selective Mott transition, with the narrower band undergoing Mott localization while the wider one being still itinerant. At quarter-filling, the physical behavior is different and to some extent reversed. The interaction generates an effective crystal field splitting, absent in the Hamiltonian, that tends to empty the narrower band in favor of the wider one, which also become more correlated than the former at odds with the orbital selective paradigm. Upon increasing the interaction, the depletion of the narrower band can continue till it empties completely and the system undergoes a topological Lifshitz transition into a half-filled single-band metal that eventually turns insulating. Alternatively, when the two bandwidths are not too different, a first order Mott transition intervenes before the Lifshitz's one. The properties of the Mott insulator are significantly affected by the interplay between spin and orbital degrees of freedom.