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Quantum geometry beyond projective single bands
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abstract
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 Pl\"ucker 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.
Forward citations
Cited by 7 Pith papers
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Multi-State Geometry of Density Matrices and Rectification Sum Rules
Rectification current in insulators obeys a zero-temperature sum rule expressed as a ground-state cumulant minus a many-body multi-state geometric tensor, generalizing known single-particle results.
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Odd-integer (meronic) Euler phases in C2T-symmetric three-band systems occur only when the Brillouin zone boundary conditions are non-trivial and anisotropic, and quench dynamics reveals these obstructions through lin...
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Orbital magnetization reveals multiband topology
The quantum-geometric part of orbital magnetic susceptibility fingerprints multiband Euler topology, providing a proposed doping-dependent experimental probe.
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Identifying geometric third-order nonlinear transport in disordered materials
A catalog of 20 third-order nonlinear-transport mechanisms plus a scaling-law fingerprint table for identifying geometric vs. disorder-dominated responses in experiments.
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Enhancement of exciton radius near a band-gap closing through quantum geometry
Near a band-gap closing, quantum-geometry suppression of the electron–hole Coulomb overlap enlarges the exciton radius and its diamagnetic response, as shown in a Lieb-lattice model.
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Nonlinear Odd Viscoelastic Effect
Strains in two orthogonal directions produce dissipationless momentum flow in the third direction, with magnitude set by multiband quantum geometry and integer topological invariants.
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Probing Tensor Monopoles and Gerbe Invariants in Three-Dimensional Topological Matter
The Hopf invariants of 3D topological insulators are identified with Dixmier-Douady invariants of tensor Berry connections, which quantize magnetoelectric and shift-current responses.
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