Quantum Metric in Step Response

TL;DR

This paper proposes a novel method to directly measure the quantum metric in quantum materials through relaxation from constrained equilibrium, overcoming previous experimental challenges and enabling new insights into quantum geometry.

Contribution

It introduces a new approach using relaxation from constrained equilibrium to directly measure the quantum metric, which was previously difficult to access experimentally.

Findings

Proposes relaxation from constrained equilibrium as a measurement method.

Links the quantum metric to the time-dependent quantum geometric tensor.

Discusses potential to reveal other geometric properties in insulators.

Abstract

Quantum geometry of Bloch wavefunctions has gained considerable interest with the discovery of moir\'e materials that exhibit bands flattened by quantum interference. The quantum metric, the symmetric part of the quantum geometric tensor, influences several observables, such as the dielectric constant, superfluid stiffness and optical spectral weight. However, a direct measurement of the metric itself has remained elusive so far. In linear response functions such as the conductivity, the matrix elements of the metric typically appear convoluted with energy prefactors, preventing finding an observable that is directly proportional to the total quantum metric. The only observable that may extract it is the integrated optical spectral weight weighted by the inverse frequency, a generalized sum rule known as the Souza-Wilkens-Martin (SWM) sum rule. However, the sum rule comes with experimental challenges, such as requiring a large spectrum of frequency resolution. In this work, we propose relaxation from constrained equilibrium as a method to directly measure the symmetric part of the time-dependent quantum geometric tensor (tQGT), which at $t=0$ is the quantum metric. Additionally, we comment on other geometric properties of insulators that are absent in the frequency expansions of conductivity in insulators but can, in principle, be revealed in step response.