Abstract Quantum geometry is a differential geometry based on quantum mechanics. It
is related to various transport and optical properties in condensed matter
physics. The Zeeman quantum geometry is a generalization of quantum geometry
including the spin degrees of freedom. It is related to electromagnetic
cross responses. Quantum geometry is generalized to non-Hermitian systems
and density matrices. Especially, the latter is quantum information
geometry, where the quantum Fisher information naturally arises as quantum
metric. We apply these results to the $X$-wave magnets, which include $d$%
-wave, $g$-wave and $i$-wave altermagnets as well as $p$-wave and $f$-wave
magnets. They have universal physics for anomalous Hall conductivity,
tunneling magneto-resistance and planar Hall effect. We also study
magneto-optical conductivity, magnetic circular dichroism and Friedel
oscillations in the $X$-wave magnets. Various analytic formulas are derived
in the case of two-band Hamiltonians. This paper presents a review of recent
progress together with some
original results

Magnetic phase transitions driven by quantum geometry

Exploring many-body quantum geometry beyond the quantum metric with correlation functions: A time-dependent perspective

We develop a symmetry-based framework to compute band-geometric quantities in multiband systems whose low-energy Hamiltonians realize arbitrary SU ( 2 ) representations. Exploiting the existence of a quantization axis, we use the Wigner-Eckart theorem to identify the allowed matrix elements and obtain compact analytic expressions for the quantum geometric tensor, the orbital magnetic moment, and the associated orbital transport coefficients. The formalism applies to multifold fermions as well as to gapped SU ( 2 ) models. Its versatility is illustrated through explicit calculations in representative SU ( 3 ) and SU ( 4 ) settings, where orbital Edelstein and orbital Hall responses emerge naturally from the antisymmetric components of the band geometry. These results establish a direct connection between the algebraic structure of the Hamiltonian and intrinsic orbitronic phenomena.

Nematic enhancement of superconductivity in multilayer graphene via quantum geometry

Sliding-tuned quantum geometry in moiré systems: Nonlinear Hall effect and quantum metric control

Quantum geometry in low-energy linear and nonlinear optical responses of the magnetic Rashba semiconductor (Ge,Mn)Te

Magnetoelectric coupling enables the manipulation of magnetization by electric fields and polarization by magnetic fields. While typically found in heavy element materials with large spin-orbit coupling, recent experiments on rhombohedral-stacked pentalayer graphene have demonstrated a longitudinal magnetoelectric coupling (LMC) without spin-orbit coupling. Here, we develop a microscopic theory of LMC in layered quantum materials and identify how it is controlled by a "layer-space" quantum geometry. Focusing on rhombohedral multilayer graphene systems, we find that the interplay between LMC and valley-polarized order produces a butterfly shaped magnetic hysteresis controlled by out-of-plane electric field: a signature of LMC and a multiferroic valley order. Furthermore, we identify a nonlinear LMC in rhombohedral multilayer graphene under time-reversal symmetry, while the absence of centrosymmetry enables the generation of a second-order nonlinear electric dipole moment in response to an out-of-plane magnetic field. Our theoretical framework provides a quantitative understanding of LMC, as well as the emergent magnetoelectric properties of rhombohedral multilayer graphene.

Graphene-based multilayer systems serve as versatile platforms for exploring the interplay between electron correlation and topology, thanks to distinctive low-energy bands marked by significant quantum metric and Berry curvature from graphene's Dirac bands. Here, we investigate Mott physics and local spin moments in Dirac bands hybridized with a flat band of localized orbitals in functionalized graphene. Via hybridization control, a topological transition is realized between two symmetry-distinct site-selective Mott states featuring local moments in different Wyckoff positions, with a geometrically enforced metallic state emerging in between. We find that this geometrically controlled real-space switching of local moments and associated metal-insulator physics may be realized through proximity coupling of epitaxial graphene on SiC(0001) with group IV intercalants, where the Mott state faces geometrical obstruction in the large-hybridization limit. Our Letter shows that chemically functionalized graphene provides a correlated electron platform, very similar to the topological heavy fermions in graphene moiré systems but at significantly enhanced characteristic energy scales.

In these lecture notes for the Les Houches School on Quantum Geometry I give an introductory overview of non-perturbative aspects of topological string theory. After a short summary of the perturbative aspects, I first consider the non-perturbative sectors of the theory as unveiled by the theory of resurgence. I give a self-contained derivation of recent results on non-perturbative amplitudes, and I explain the conjecture relating the resurgent structure of the topological string to BPS invariants. In the second part of the lectures I introduce the topological string/spectral theory (TS/ST) correspondence, which provides a non-perturbative definition of topological string theory on toric Calabi–Yau manifolds in terms of the spectral theory of quantum mirror curves.

Accurately and efficiently predicting the equilibrium geometries of large molecules remains a central challenge in quantum computational chemistry, even with hybrid quantum-classical algorithms. Two major obstacles hinder progress: the large number of qubits required and the prohibitive cost of conventional nested optimization. In this work, we introduce a co-optimization framework that combines Density Matrix Embedding Theory (DMET) with Variational Quantum Eigensolver (VQE) to address these limitations. This approach substantially reduces the required quantum resources, enabling the treatment of molecular systems significantly larger than previously feasible. We first validate our framework on benchmark systems, such as H4 and H2O2, before demonstrating its efficacy in determining the equilibrium geometry of glycolic acid (C2H4O3)─a molecule of a size previously considered intractable for quantum geometry optimization. Our results show the method achieves high accuracy while drastically lowering computational cost. This work thus represents a significant step toward practical, scalable quantum simulations, moving beyond the small, proof-of-concept molecules that have historically dominated the field. More broadly, our framework establishes a tangible path toward leveraging quantum advantage for the in silico design of complex catalysts and pharmaceuticals.

Quantum geometry from the Moyal product: Quantum kinetic equation and nonlinear response

The optical selection rules dictate symmetry-allowed and forbidden transitions, playing a decisive role in engineering exciton quantum states and designing optoelectronic devices. While both the real (quantum metric) and imaginary (Berry curvature) parts of quantum geometry contribute to optical transitions, the conventional theory of optical selection rules in solids incorporates only Berry curvature. Here, we propose quantum-metric-based optical selection rules. We unveil a universal quantum metric and oscillator strength correspondence for linear polarization of light and establish valley-contrasted optical selection rules that lock orthogonal linear polarizations to distinct valleys. Tight-binding and first-principles calculations confirm our theory in two models (altermagnet and Kane-Mele) and monolayer d-wave altermagnet V_{2}SeSO. This work provides a quantum-metric paradigm for valley-based spintronic and optoelectronic applications.

The motion of a quantum particle constrained to a two-dimensional non-compact Riemannian manifold with nontrivial metric can be described by a flat-space Schrödinger-type equation at the cost of introducing local mass and metric and geometry-induced effective potential with no classical counterpart. For a metric tensor periodically modulated along one dimension, the formation of bands is demonstrated and transport-related quantities are derived. Using S-matrix approach, the quantum conductance along the manifold is calculated and contrasted with conventional quantum transport methods in flat spaces. The topology, e.g., whether the manifold is simply connected, compact or non-compact shows up in global, non-local properties such as the Aharonov–Bohm phase. The results vividly demonstrate emergent phenomena due to the interplay of reduced-dimensionality, particles quantum nature, geometry, and topology.

The shift current is a non-linear photocurrent generally associated with the underlying quantum geometry. However, a topological origin for the shift photocurrent in non-centrosymmetric systems has recently been proposed. The corresponding topological classification goes beyond the ten-fold paradigm and is associated with the presence of a reverting Thouless pump (RTP). In this work we perform a first-principles computational analysis of antiferromagnetic monolayer within the family of MXenes, Ti[Formula: see text]C[Formula: see text]. This material is centrosymmetric, however, magnetic ordering violates inversion symmetry. We demonstrate evidence of an RTP in each spin-sector which has been perturbed, destroying quantization of the invariant. Nevertheless, a giant spin-resolved shift current persists. We further investigate the mid-gap edge states and classification of the system as a fragile topological insulator to which trivial bands have been coupled.

Second harmonic Hall response in insulators: Interband quantum geometry and breakdown of Kleinman's conjecture

Ideal quantum geometry of the surface states of rhombohedral graphite and its effects on the surface superconductivity

Interfaces in heterostructures possess inherent inversion asymmetry and display diverse physical effects, however, the pristine in-plane mirror symmetries of the constituent layers are usually preserved at the interface. On-demand manipulation of these symmetries remains challenging. Here, we demonstrate a strategy to control the in-plane mirror symmetries of interfaces by engineering the crystallographic orientation of heterostructures. We design a workhorse system with a new orientation, i.e., the LaAlO3/SrTiO3 heterostructure with metallic interfaces in the (112)-plane. Such a high index orientation leads to the breaking of all the pristine mirror symmetries except the mirror plane perpendicular to the [1bar{1}0] direction ({M}_{[1bar{1}0]}), resulting in the Cs point symmetry with a metallic conduction. Consequently, this interface exhibits a giant nonlinear Hall effect characterized by a large Berry curvature dipole, a circular photogalvanic effect, and current-induced out-of-plane magnetization, all functional at room temperature. The magnitude of the nonlinear Hall effect rivals the Weyl and Dirac systems. Our work establishes a new strategy in exploring emerging electronic properties with nontrivial quantum geometry by designing the interface symmetry.

Td-MoTe2, a semimetallic material with broken inversion symmetry, allows a rich variety of nonlinear transport phenomena which may manifest intriguing quantum geometry of the band structure. Here, we report the observation of a nonlinear planar Hall effect in Td-MoTe2 over a broad temperature range (2 to 150 K). The nonlinear Hall voltage scales quadratically (linearly) with the applied electric (magnetic) field and shows distinct angular dependence on the in-plane magnetic field: a cosine (sine) relationship when the electric field aligns with the a axis (b axis). Combined first-principles calculations and scaling analysis of experimental data reveal two scaling terms, including an intrinsic contribution from Berry connection polarizability of band structure and an extrinsic nonlinear Drude-like contribution. This study not only expands the understanding of nonlinear Hall transport mechanisms in low-symmetry materials but also suggests potential applications in precision magnetic field sensing and electronic devices exploiting directional charge transport.

Gaplessness from disorder and quantum geometry in gapped superconductors