Open Quantum Systems as Regular Holonomic $\mathcal{D}$-Modules: The Mixed Hodge Structure of Spectral Singularities

TL;DR

This paper addresses the breakdown of geometric descriptions for open quantum systems at spectral singularities by introducing a Dissipative Mixed Hodge Module (DMHM) framework. By identifying the Liouvillian family with a regular holonomic $\mathcal{D}_X$-module and endowing it with Hodge and weight filtrations, the authors regularize the Quantum Geometric Tensor (QGT) and separate coherent and decay channels. A Brieskorn lattice-based singular extension yields a residue formula for the singular QGT component, linked to the inverse Saito pairing and monodromy, and the complete QGT decomposes into a regular part plus a topological flux supported on the discriminant. They prove symmetry and positivity from the Mixed Hodge Module axioms and demonstrate spectral-sequence degeneration, providing a robust, topological understanding of dissipative topology and practical spectroscopic protocols (HFS and WFS) for tomographic reconstruction. Canonical models and extensive examples illustrate how EPs encode nontrivial cohomological data and offer experimentally accessible fingerprints via the Hodge-Weight Diagram. Overall, the work furnishes a mathematically rigorous, physically interpretable framework for dissipative quantum geometry with concrete implications for sensing and topological quantum devices.

Abstract

The geometric description of open quantum systems via the Quantum Geometric Tensor (QGT) traditionally relies on the assumption that the physical states form a differentiable vector bundle over the parameter manifold. This framework becomes ill-posed at spectral singularities, such as Exceptional Points, where the eigen-bundle admits no local trivialization due to dimension reduction. In this work, we resolve this obstruction by demonstrating that the family of Liouvillian superoperators $\mathcal{L}(k)$ over a complex parameter manifold $X$ canonically defines a \textbf{regular holonomic $\mathcal{D}_X$-module} $\mathcal{M}$. By identifying the physical coherence order with the Hodge filtration and the decay rate hierarchy with the \textbf{Kashiwara filtration}, we show that the open quantum system underlies a \textbf{Mixed Hodge Module (MHM)} structure in the sense of Saito. This identification allows us to apply the \textbf{Grothendieck six-functor formalism} rigorously to dissipative dynamics. We prove that the divergence corresponds to a non-trivial cohomology class in $\text{Ext}^1_{\mathcal{D}_X}$, thereby regularizing the Quantum Geometric Tensor without ad-hoc cutoffs. Specifically, the ``singular component'' of the Complete QGT arises as the residue of the connection on the \textbf{Brieskorn lattice} associated with the vanishing cycles functor.

Figures (5)

• Figure 1: The Canonical Hodge-Weight Stratification. Schematic classification of the local Liouvillian cohomology for a Dissipative Dimer at an Exceptional Point ($H_{\text{eff}} \sim \text{Jordan}_2$). The horizontal axis ($p$) denotes the Hodge number, distinguishing diagonal Populations ($p=0$) from off-diagonal Coherences ($p=1$). The vertical axis ($k$) denotes the Monodromy Weight. While generic phases reside in the pure weight $W_0$ (exponential decay), the topological singularity lifts the coherence to weight $W_2$ (defective polynomial decay). This diagrammatic logic generalizes to the global $\mathcal{D}$-module structure of the open quantum system.
• Figure 2: Regularization of the Quantum Geometric Tensor (QGT) at a Spectral Singularity.(a) The Divergence Problem: The standard Fubini-Study metric $g_{\lambda\lambda}$ (blue trace) exhibits a characteristic algebraic divergence ($\sim |\lambda|^{-2}$) as the system approaches the Exceptional Point ($\lambda \to 0$), rendering the geometry ill-defined. (b) The Complete QGT: Application of the DMHM regularization. The metric is decomposed into a smooth Regular Component ($G_{reg}$, blue dashed) and a distributional Singular Flux (orange). The singular component is not an error but a conserved topological current; its integrated weight is proportional to the Milnor Number$\mu$, counting the number of coalescing eigenstates. This transformation reinterprets the breakdown of the adiabatic limit as the detection of a non-trivial cohomology class in $\mathrm{Ext}^1$.
• Figure 3: Tomographic Resolution of Dissipative Topology via Monodromy Weight Filtration.(a) Hermitian Limit ($\gamma \ll \Delta$): Standard 2D spectroscopic map of the Kitaev chain in the weak dissipation regime. The topological phase is identifiable via characteristic peak splitting, reproducing standard results Gerken2022. (b) The Exceptional Point Limit ($\gamma \approx \Delta$): The system driven to a spectral singularity (EP2). The spectral gap closes ($\text{Re}[\Delta E] \to 0$), causing the signal to merge into a structureless feature ("spectral congestion") where standard invariants are obscured. (c) DMHM Time-Domain Reconstruction: The application of the Monodromy Weight Filtration to the singular response in (b). The signal is decomposed into graded components: the trivial bulk ($W_0$, blue) decays exponentially, while the topological defect is isolated in the Weight-2 subspace ($W_2$, orange). The detection of the anomalous polynomial kinetics ($t \cdot e^{-\gamma t}$) constitutes a rigorous signature of the Rank-2 Jordan block, recovering the topological invariant $N \neq 0$ from the noise floor. (d) Hodge-Weight Phase Diagram: Numerical classification of the $N=40$ chain. Modes are stratified by Hodge number $p$ (Particle-Hole expectation) and Weight $k$ (Jordan rank). The transition to the topological phase is mapped as a discrete jump to Weight 2, validating the global stability of the filtration defined in Theorem 3.6.
• Figure 4: The Structure of the Nearby Cycles $\psi_f\mathcal{M}$. Visualizing the Monodromy Weight Filtration established by the Nilpotent Orbit Theorem. (Left) The Filtration Layers: The vanishing cohomology is stratified by the weight $k$, corresponding physically to the decay rate hierarchy. (Right) The Hard Lefschetz Action: The nilpotent monodromy operator $N$ (vertical arrows) acts as a "lowering operator" for the weight. The curved arrow indicates the Hard Lefschetz Isomorphism, guaranteeing that the decay map $N^k: \text{Gr}_k \to \text{Gr}_{-k}$ is invertible. This symmetry proves that the filtration is canonical and unique, independent of the basis choice.
• Figure 5: Figure 5: Homological Resolution of the Topological Phase (The Diagram Chase). A step-by-step visualization of the proof provided in Appendix \ref{['app:spectral_sequence']}, demonstrating why the dissipative topology is robust. (a) The Filtration Sequence: The Short Exact Sequence of $\mathcal{D}$-modules, $0 \to W_{k-1} \to W_k \to \text{Gr}_k \to 0$, representing the separation of decay channels. (b) The Derived Triangle: The induced Long Exact Sequence in Hypercohomology. The connecting homomorphism $\partial$ (red arrow) represents the "mixing" or extension class between different decay rates. (c) The Monodromy Gap: Visual representation of the Strictness Theorem. Because the nilpotent monodromy $N$ maps $W_k \to W_{k-2}$ (a "gap" of 2), it cannot generate non-trivial extensions between adjacent weights $k$ and $k-1$. (d) Spectral Collapse: Consequently, the differential vanishes ($\partial = 0$). The diagram commutes without mixing, proving that the Weight Filtration is a stable topological invariant even at the singularity, allowing for the distinct labeling of modes in the Hodge-Weight Diagram.

Theorems & Definitions (48)

• Remark 2.1: Abelian Stability vs. Non-Hermitian Pathologies
• Definition 2.2: The System Module
• Remark 2.3
• Definition 2.4: Discriminant Locus
• Proposition 2.5: Non-Semisimple Monodromy
• proof
• Remark 2.6
• Proposition 2.7: Existence of Operations
• proof
• Remark 2.8: The Physical Dictionary