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Instantaneous Response and Quantum Geometry of Insulators (2403.07052v3)

Published 11 Mar 2024 in cond-mat.mes-hall, cond-mat.mtrl-sci, and cond-mat.str-el

Abstract: We present the time-dependent Quantum Geometric Tensor (tQGT) as a comprehensive tool for capturing the geometric character of insulators observable within linear response. We show that tQGT describes the zero-point motion of bound electrons and acts as a generating function for generalized sum rules of electronic conductivity. It therefore enables a systematic framework for computing the instantaneous response of insulators, including optical mass, orbital angular momentum, and dielectric constant. This construction guarantees a consistent approximation across these quantities upon restricting the number of occupied and unoccupied states in a low-energy description of an infinite quantum system. We outline how quantum geometry can be generated in periodic systems by lattice interference and examine spectral weight transfer from small frequencies to high frequencies by creating geometrically frustrated flat bands.

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Summary

  • The paper introduces a novel time-dependent Quantum Geometric Tensor (tQGT) that provides a basis-independent definition of optical sum rules.
  • It links quantum geometry with the zero-point motion of bound electrons by deriving non-perturbative bounds between geometric and observable quantities.
  • The framework extends from a trivial atomic insulator to flat-band systems, offering insights relevant to superconductivity and strongly correlated electron behavior.

Unifying Quantum Geometry with Electronic Responses in Insulators

The paper under consideration advances a novel framework at the intersection of quantum geometry and electronic structure theory, particularly concerning insulators. At its core, the authors present a time-dependent Quantum Geometric Tensor (tQGT), a construct aimed at encapsulating the instantaneous response and zero-point motion of bound electrons within insulators. The introduction of tQGT signifies a pivotal step towards a basis-independent definition of the optical sum rule in low-energy manifold states. The research delineates how quantum geometric phenomena are intrinsically linked to the zero-point motion of bound electrons.

The significance of this research is underscored by its ability to extend existing formalism to capture generalized sum rules previously identified in the literature. Through various time-derivatives of tQGT, the authors derive non-perturbative bounds between quantum geometry and observable physical quantities. This formal progression holds potential implications for a better comprehension of quantum geometric contributions across strongly correlated electron systems.

Notably, the paper puts forth an innovative scheme to induce zero-point motion using a trivial atomic insulator as a starting point. Importantly, this results in the formulation of a flat-band system. Such systems are notably interaction-prone not merely through their density of states but also from a real-space perspective, which incorporates the range and complexity of projected interactions. This insight aligns with contemporary discussions on flat-band superconductivity and electronic correlation phenomena.

This research, by bridging quantum geometry with the tangible behavior of electronic systems under insulation, yields significant implications both theoretically and practically. It adds to the foundation for further explorations into the manipulation of electronic structures for diverse applications, particularly in the context of strongly correlated systems and superconductivity. Future developments could build on this framework to establish new paradigms in understanding and potentially manipulating quantum materials for technological benefits.

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