From Divergent Series to Geometry: Resurgence of the Quantum Metric

TL;DR

This work extends resurgence theory to the perturbative expansions of the quantum metric tensor (QMT) in anharmonic oscillators, focusing on quartic, sextic, and $d$-dimensional quartic models. By analyzing high-order perturbation series, the authors identify factorial growth and universal non-perturbative scales, then apply generalized Borel resummation (including $\alpha$-Borel and Borel–Leroy transforms) to extract accurate QMT and energy results, validating against exact diagonalization. They reveal a dominant Borel singularity at $u=-1/3$ (or its scaled counterparts) and demonstrate how non-perturbative corrections scale as $\exp(1/(3\lambda))$ in appropriate variables, with generalized transforms required for higher-order growth. Across quartic, sextic, and multidimensional oscillators, Borel–Padé resummation yields high-precision results for the ground state and meaningful insights for excited states, highlighting a deep connection between quantum geometry and non-perturbative physics. The findings broaden the applicability of resurgence to quantum geometry and suggest extensions to more general quantum systems and time-dependent quantum geometric structures.

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--Padé resummation to the QMT. Comparisons with exact diagonalization confirm that Borel--Padé 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.

Figures (7)

• Figure 2: (a) Ground-state energy computed using the Padé approximant, $P_m[E^{(0)}]$, and the Borel--Padé resummation $\mathcal{TP}_m[E^{(0)}]$ (with $m=100$), compared with the exact numerical energy obtained from Hamiltonian diagonalization. (b) Ground-state energy obtained from the Borel--Padé resummation $\mathcal{TP}_m[E^{0}]$ for $m=50,100,200$, compared with the exact diagonalization result.
• Figure 3: Poles $\lambda^{(ij)}_{\textrm{pol}}$ of the Padé approximant $P_m [g_{ij}^{ (0)}]$ for $m=1,\dots,100$. We have fixed $k=1$.
• Figure 4: (a) Singularities in the Borel plane of $\mathcal{P}_m[E^{(0)}]$ and $\mathcal{P}_m[g_{ij}^{ (0)}]$. An accumulation point is observed around $-1/3$. (b) Zoom of the figure (a) at the accumulation point.
• Figure 5: Metric components obtained by Borel resummation $\mathcal{TP}_m\left[g_{ij}^{ (0)}(\lambda)\right]$ for different truncation orders $m$. As the figures show, the approximation improves as $m$ increases. (a) $g_{11}^{ (0)}$ component, (b) $g_{12}^{ (0)}$ component, and (c) $g_{22}^{ (0)}$ component.
• Figure 6: Metric components obtained by Borel resummation $\mathcal{TP}_{m}[g_{ij}^{ (N)}]$ (solid line) and by exact diagonalization (dashed line), for the exited states $N=1,2,3,4$. (a) $g_{11}^{ (N)}$ component, (b) $g_{12}^{ (N)}$ component, and (c) $g_{22}^{ (N)}$ component.