From Divergent Series to Geometry: Resurgence of the Quantum Metric

TL;DR

This paper investigates the divergent perturbative series of the quantum metric tensor in anharmonic oscillators, demonstrating how resurgence techniques can accurately resum these series and reveal non-perturbative geometric information.

Contribution

It extends resurgence methods from energy spectra to quantum geometric tensors, providing new insights into non-perturbative quantum geometry.

Findings

Divergent QMT series exhibit factorial growth.

Borel--Padé resummation accurately reproduces ground state QMT.

Resurgence analysis reveals universal non-perturbative scales.

Abstract

In this work, we analyze perturbative expansions of the quantum metric tensor (QMT) in anharmonic oscillators, focusing on quartic, sextic, and $d$-dimensional models. Using high-order perturbation theory, we show that the divergent QMT series exhibit factorial growth. Our analysis identifies universal non-perturbative scales, with coefficients displaying large-order behavior consistent with resurgence theory. Then, we apply resurgence and Borel--Pad\'e resummation to the QMT. Comparisons with exact diagonalization confirm that Borel--Pad\'e resummations yield accurate results, especially for the ground state. For completeness, we also present the analysis of the energy eigenvalues in the examples. Our findings extend resurgent techniques from energies to the QMT, highlighting the interplay between quantum geometry and non-perturbative physics.