Quantum Geometric Tensor in the Wild: Resolving Stokes Phenomena via Floquet-Monodromy Spectroscopy

TL;DR

Open quantum systems with essential singularities cause the local quantum metric to diverge and invalidate standard topological invariants. The authors define the Complete Quantum Geometric Tensor (cQGT) to regularize the geometry by adding a singular Stokes term encoded by a Brieskorn lattice within the Dissipative Mixed Hodge Module framework, and they develop Floquet-Monodromy Spectroscopy (FMS) to extract the Stokes multipliers from controlled time evolution. This enables exact reconstruction of nonperturbative physics via Resurgence Theory and unveils the Stokes Invariant as a robust quantum number that classifies open system topology beyond Chern numbers. The approach is demonstrated in a superconducting qudit, reveals universal Sabbah-type scaling, and extracts Milnor and Tjurina numbers, while exposing modal singularities where topology alone fails and the full singular trace is required. The work resolves the Wild Riemann-Hilbert correspondence in this setting and points to new directions such as geometric k-space combs for non-Hermitian quantum technologies.

Abstract

Standard topological invariants, such as the Chern number and Berry phase, form the bedrock of modern quantum matter classification. However, we demonstrate that this framework undergoes a \textbf{catastrophic failure} in the presence of essential singularities -- ubiquitous in open, driven, and non-Hermitian systems ("Wild" regime). In these settings, the local geometric tensor diverges, rendering standard invariants ill-defined and causing perturbative predictions to deviate from reality by order unity ($\sim 100\%$). We resolve this crisis by introducing the \textbf{Floquet-Monodromy Spectroscopy (FMS)} protocol, a pulse-level control sequence, which experimentally extracts the hidden \textit{Stokes Phenomenon} -- the "missing" geometric data that completes the topological description. By mapping the singularity's Stokes multipliers to time-domain observables, FMS provides a rigorous experimental bridge to \textbf{Resurgence Theory}, allowing for the exact reconstruction of non-perturbative physics from divergent asymptotic series. We validate this framework on a superconducting qudit model, demonstrating that the "Stokes Invariant" serves as the next-generation quantum number for classifying phases of matter beyond the reach of conventional topology.

Figures (11)

• Figure 1: Overview of the Framework. (a) The Crisis: Standard quantum geometry ["Tame", e.g., Chern insulators TKNN1982] successfully classifies regular phases but suffers a Catastrophic Failure at essential singularities ["Wild", e.g., Non-Hermitian Skin Effect Yao2018]. The plot shows the $\mathcal{O}(1)$ error in the standard adiabatic prediction ($\text{Tr}\,M=0$) versus the actual FMS measurement ($S \approx 1$). (b) The Engine: We introduce the Complete QGT ($\mathcal{Q}_{ij}$), which augments the regular local metric $g^{reg}_{ij}$ (Smooth Geometry) with the singular Stokes Invariants$f_{mix}$ (Topological Defect). This structure, rooted in the Brieskorn Lattice (schematic), resolves the singularity. (c) The Resolution: The FMS Protocol extracts these invariants experimentally by driving the system $k(t)$ along a loop $\gamma$ and measuring the non-Abelian monodromy $U(T)$. (d) The Reward: This geometric data resolves the Resurgence Ambiguity, allowing for the exact reconstruction of the non-perturbative wavefunction $\hat{\Psi}_{resum}$ from divergent perturbative series.
• Figure 2: Geometry of the Complete QGT. (a) The Stokes Sectors $\mathcal{V}_j$ divide the parameter space. (b) As the connection moves across a Stokes Ray (dashed), the basis jumps by $S_j$. (c) FMS detects this jump via the Monodromy.
• Figure 3: The Periodic Table of Singularities. Classification of Open Quantum Systems by their Singularity Rank. Comparison of "Tame" (Regular, Rank 0) vs "Wild" (Irregular, Rank 1, 2) classes and their invariants.
• Figure 4: Experimental Protocol and the Breakdown of Adiabaticity. (a) FMS Pulse Sequence: Real-time control pulses for the Rabi drive $J(t)$ (blue) and Detuning $\Delta(t)$ (red) required to encircle the Exceptional Point in a Superconducting Transmon Qudit. (b) The Crisis (Adiabatic Breakdown): While the standard Adiabatic Theorem predicts a trace-less monodromy (Black Dashed, $\text{Tr} M = 0$), the actual dynamics exhibit a Stokes Jump (Red Solid) due to the divergence of the quantum metric at the EP (consistent with observations in polaritons Gianfrate2020). FMS captures this non-perturbative correction ($\text{Im} \text{Tr} M \approx 1$), resolving the geometric origin of the divergence.
• Figure 5: Observation of the Stokes Phase Transition. The topological invariant $S$ jumps from 0 to 1 (left axis, red) while the spectral gap $\Delta E$ (right axis, dashed) remains finite and open. This confirms the "Phantom" nature of the transition.

Theorems & Definitions (3)

• Remark 1: Mathematical Connection
• Theorem 1
• proof : Sketch