Instantaneous response and quantum geometry of insulators

TL;DR

This work introduces the time-dependent quantum geometric tensor (tQGT) as a gauge-invariant, unifying framework for the geometric content of insulating linear response. By recasting the Kubo conductivity in the time domain, it shows that a single object, the tQGT, generates generalized optical sum rules—SWM, the f-sum rule, and dielectric responses—through derivatives at $t=0$, tying quantum metric and Berry curvature to observable quantities like optical mass, orbital moment, and dielectric permittivity. The theory is illustrated with Landau levels and lattice models, demonstrating how zero-point motion and spectral weight transfer emerge from geometry and how low-energy truncations must consistently preserve these geometric contributions. This framework clarifies how geometry pervades response across different sum rules, with implications for topological materials, moiré systems, and flat-band physics, where wavefunction details and geometric embedding crucially influence observables. Overall, the tQGT provides a principled, unified approach to quantify and relate geometric properties to measurable electronic responses in gapped quantum materials.

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.

Figures (3)

• Figure 1: a. Time-dependent quantum geometric tensor (tQGT) in Landau levels. The real longitudinal and imaginary Hall parts of the tQGT are identical and oscillate with the same frequency $\omega_c = \sqrt{e B/m}$. b. tQGT in a honeycomb lattice model with nearest neighbor hopping and a $C_{2z}$ breaking mass term $m_z>0$ (details in Appendix I1). Different parts of tQGT now have different time profiles. As a result, geometric quantities arising from different derivatives are all distinct.
• Figure 2: Longitudinal and Hall sum rules $(\mathcal{S}^\eta = \mathcal{S}_{L}^\eta+i \mathcal{S}_{H}^\eta)$ for the two-dimensional Haldane Chern Insulator at half-filling across the topological phase transition that occurs at $|m_z/t_2| = 3\sqrt{3}$. The model includes a nearest neighbor hopping $t$, a next nearest neighbor hopping $t_2$ with flux $\phi = \pi/2$, and an inversion breaking mass $m_z$ (see Appendix I2 for details). The sum rules are normalized by their value at $m_z=0$. The SWM sum rule (black) in the left panel corresponds to ground state quantum metric $g$ and the Chern number $\mathcal{C}$. The metric diverges across the topological phase transition. The discontinuities become more smooth or divergent depending on the energy power $\eta$. Note that the Hall sum rules with $\eta \leq -1$ show an anti-symmetric divergence at the phase transition.
• Figure 3: Transfer of Spectral Weight to higher frequencies in the Lieb lattice model that interpolates to the square lattice with parameter $t_p/t$. At $t_p=t$, the square lattice has no geometric contribution and follows a Drude-like behavior. The inter-band transitions increase ss $t_p$ is reduced. Finally, at $t_p=0$, one of the orbital factors out and the resulting Lieb lattice has a vanishing Drude weight. The optical conductivity, on the other hand, has a non-zero interband contribution coming from the minimal inter-band conductivity of a Dirac cone Sun2018.