Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2512.20253

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

Prasoon Saurabh

quant-phcond-mat.mes-hallmath-phSubmitted Dec 23, 2025• Announced Dec 24, 2025• 0 citations
arXivHTMLPDF

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.

Comments
19 pages, 11 figures

Related papers

Hybrid Quantum-Classical Neural Networks for Discovering Hidden Floquet Topological Invariants

Lee Cheng, Chang Lei

The discovery of topological phases in non-equilibrium quantum systems, particularly those driven by periodic fields (Floquet systems), holds immense potential for next-generation quantum technologies. However, identifying and classifying these phases is notoriously challenging, as the associated topological invariants are often non-local, abstract, and lack straightforward experimental observables. Traditional theoretical methods typically rely on idealized models and exact symmetries, which break down in the presence of disorder, interactions, or finite-size effects inherent to real materials. Here, we propose a novel paradigm for the discovery of hidden Floquet topological invariants using hybrid quantum-classical neural networks. Our framework synergistically combines a parameterized quantum circuit, acting as a variational ansatz for the Floquet quasienergy states, with a classical neural network that learns to map the system's time-resolved dynamics onto a robust, data-driven topological classifier. By training the hybrid model on simulated measurement data—such as time-of-flight images or spin-resolved photoelectron spectra—the classical network distills the essential, often obscured, features of the system's unitary evolution without prior knowledge of the underlying Hamiltonian. Crucially, we demonstrate that this approach can uncover novel invariants for anomalous Floquet topological phases that are invisible to conventional frequency-domain analyses. Furthermore, the trained network provides a practical, noise-resilient protocol for real-time classification, enabling the experimental detection of topological transitions solely from short-time dynamics. This work establishes a powerful, generalizable framework for accelerating the discovery and characterization of emergent topological order in complex driven quantum matter, bridging the gap between abstract topological theory and experimental reality.

•Jul 14, 2026•0 citations

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

Prasoon Saurabh

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.

math-phCross-listedarXiv:2512.19487•Dec 23, 2025•0 citations

A Gauge-Covariant Geometric Framework for Non-Hermitian Quantum Systems

Gargi Das, Sudhanshu Shekhar, Bhabani Prasad Mandal

We develop a comprehensive, gauge-covariant geometric framework for non-Hermitian quantum systems in the quasi-Hermitian regime, that is, the region of parameter space where the non-Hermitian Hamiltonian admits a real spectrum and a positive-definite metric operator. We build this framework by elevating the Dyson map to a central geometric object. This map is the transformation that converts a non-Hermitian Hamiltonian into an equivalent Hermitian one. From it we construct the Dyson connection and decompose it into Hermitian and anti-Hermitian parts, identified respectively as {\it stretching } and {\it rotation } components. This decomposition cleanly separates the genuine physical metric deformations from the unitary gauge redundancies. Working with manifestly gauge-covariant states, we then derive the complex non-Hermitian Berry phase and the quantum geometric tensor (QGT), and show that the non-Hermitian geometric curvature originates from the non-commutativity of the stretching components at the operator level. We further analyse the geometric singularities near an exceptional point (EP) and uncover a distinct hierarchy of divergences. For a general two-level non-Hermitian model, the quantum metric tensor (QMT) exhibits a leading-order divergence $\sim |ε_μ|^{-2}$, while the Berry curvature shows a weaker, subleading divergence $\sim |ε_μ|^{-3/2}$, with $ε_μ$ denoting the parameter displacement from the EP along an individual parameter axis $μ$. Finally, we examine physical realizations of this model, including the non-Hermitian Su--Schrieffer--Heeger (SSH) and Hatano--Nelson (HN) models, where exact analytical results confirm the predicted critical scaling laws and illustrate the metric-deformation-driven non-Hermitian geometries.

quant-pharXiv:2606.15922•Jun 16, 2026•0 citations

Realization of anomalous Floquet topological phases with ultracold atoms

Karen Wintersperger, Christoph Braun, F. Nur Ünal +5 more

Coherent control via periodic modulation, also known as Floquet engineering, has emerged as a powerful experimental method for the realization of novel quantum systems with exotic properties. In particular, it has been employed to study topological phenomena in a variety of different platforms. In driven systems, the topological properties of the quasienergy bands can often be determined by standard topological invariants, such as Chern numbers, which are commonly used in static systems. However, due to the periodic nature of the quasienergy spectrum, this topological description is incomplete and new invariants are required to fully capture the topological properties. Most prominently, there exist two-dimensional anomalous Floquet systems that exhibit robust chiral edge modes, despite all Chern numbers are equal to zero. Here, we realize such a system with bosonic atoms in a periodically-driven hexagonal lattice and infer the complete set of topological invariants from energy gap measurements and local Hall deflections.

•Feb 23, 2020•4 citations

Floquet quantum geometry in periodically driven topological insulators

Peng He, Jian-Te Wang, Jiangbin Gong +1 more

Quantum geometry plays a fundamental role across many branches of modern physics, yet its full characterization in nonequilibrium systems remains a challenge. Here, we propose a framework for quantum geometry in Floquet topological insulators by introducing a time-resolved quantum metric tensor, defined via the trace distance between micromotion operators in momentum-time space. For class A in two spatial dimensions, we find a general inequality linking the Floquet quantum metric tensor and the Floquet topology: the associated quantum volume is bounded below by the Floquet topological invariant. This relation is found to also hold in class AIII in one dimension, where the Floquet geometric tensor may be notably reduced due to time-reflection symmetry. This work will be useful in digesting the general aspects of quantum geometry in periodically driven systems in connection with their topological characterization.

arXiv:2602.00569•Jan 31, 2026•0 citations

Quantum Geometric Bounds in Non-Hermitian Systems

Milosz Matraszek, Wojciech J. Jankowski, Jan Behrends

We identify quantum geometric bounds for observables in non-Hermitian systems. We find unique bounds on non-Hermitian quantum geometric tensors, generalized two-point response correlators, conductivity tensors, and optical weights. We showcase these findings in topological systems with non-Hermitian Chern numbers. We demonstrate that the non-Hermitian geometric constraints on response functions naturally arise in open quantum systems governed by out-of-equilibrium Lindbladian dynamics. Our findings are relevant to experimental observables and responses under the realistic setups that fall beyond the idealized closed-system descriptions.

quant-pharXiv:2512.23708•Dec 30, 2025
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 (∼100%\sim 100\%∼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.
The discovery of topological phases in non-equilibrium quantum systems, particularly those driven by periodic fields (Floquet systems), holds immense potential for next-generation quantum technologies. However, identifying and classifying these phases is notoriously challenging, as the associated topological invariants are often non-local, abstract, and lack straightforward experimental observables. Traditional theoretical methods typically rely on idealized models and exact symmetries, which break down in the presence of disorder, interactions, or finite-size effects inherent to real materials. Here, we propose a novel paradigm for the discovery of hidden Floquet topological invariants using hybrid quantum-classical neural networks. Our framework synergistically combines a parameterized quantum circuit, acting as a variational ansatz for the Floquet quasienergy states, with a classical neural network that learns to map the system's time-resolved dynamics onto a robust, data-driven topological classifier. By training the hybrid model on simulated measurement data—such as time-of-flight images or spin-resolved photoelectron spectra—the classical network distills the essential, often obscured, features of the system's unitary evolution without prior knowledge of the underlying Hamiltonian. Crucially, we demonstrate that this approach can uncover novel invariants for anomalous Floquet topological phases that are invisible to conventional frequency-domain analyses. Furthermore, the trained network provides a practical, noise-resilient protocol for real-time classification, enabling the experimental detection of topological transitions solely from short-time dynamics. This work establishes a powerful, generalizable framework for accelerating the discovery and characterization of emergent topological order in complex driven quantum matter, bridging the gap between abstract topological theory and experimental reality.
Open Quantum Systems as Regular Holonomic D\mathcal{D}D-Modules: The Mixed Hodge Structure of Spectral Singularities
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 L(k)\mathcal{L}(k)L(k) over a complex parameter manifold XXX canonically defines a \textbf{regular holonomic DX\mathcal{D}_XDX​-module} M\mathcal{M}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 ExtDX1\text{Ext}^1_{\mathcal{D}_X}ExtDX​1​, 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.
We develop a comprehensive, gauge-covariant geometric framework for non-Hermitian quantum systems in the quasi-Hermitian regime, that is, the region of parameter space where the non-Hermitian Hamiltonian admits a real spectrum and a positive-definite metric operator. We build this framework by elevating the Dyson map to a central geometric object. This map is the transformation that converts a non-Hermitian Hamiltonian into an equivalent Hermitian one. From it we construct the Dyson connection and decompose it into Hermitian and anti-Hermitian parts, identified respectively as {\it stretching } and {\it rotation } components. This decomposition cleanly separates the genuine physical metric deformations from the unitary gauge redundancies. Working with manifestly gauge-covariant states, we then derive the complex non-Hermitian Berry phase and the quantum geometric tensor (QGT), and show that the non-Hermitian geometric curvature originates from the non-commutativity of the stretching components at the operator level. We further analyse the geometric singularities near an exceptional point (EP) and uncover a distinct hierarchy of divergences. For a general two-level non-Hermitian model, the quantum metric tensor (QMT) exhibits a leading-order divergence ∼∣εμ∣−2\sim |ε_μ|^{-2}∼∣εμ​∣−2, while the Berry curvature shows a weaker, subleading divergence ∼∣εμ∣−3/2\sim |ε_μ|^{-3/2}∼∣εμ​∣−3/2, with εμε_μεμ​ denoting the parameter displacement from the EP along an individual parameter axis μμμ. Finally, we examine physical realizations of this model, including the non-Hermitian Su--Schrieffer--Heeger (SSH) and Hatano--Nelson (HN) models, where exact analytical results confirm the predicted critical scaling laws and illustrate the metric-deformation-driven non-Hermitian geometries.
Coherent control via periodic modulation, also known as Floquet engineering, has emerged as a powerful experimental method for the realization of novel quantum systems with exotic properties. In particular, it has been employed to study topological phenomena in a variety of different platforms. In driven systems, the topological properties of the quasienergy bands can often be determined by standard topological invariants, such as Chern numbers, which are commonly used in static systems. However, due to the periodic nature of the quasienergy spectrum, this topological description is incomplete and new invariants are required to fully capture the topological properties. Most prominently, there exist two-dimensional anomalous Floquet systems that exhibit robust chiral edge modes, despite all Chern numbers are equal to zero. Here, we realize such a system with bosonic atoms in a periodically-driven hexagonal lattice and infer the complete set of topological invariants from energy gap measurements and local Hall deflections.
Quantum geometry plays a fundamental role across many branches of modern physics, yet its full characterization in nonequilibrium systems remains a challenge. Here, we propose a framework for quantum geometry in Floquet topological insulators by introducing a time-resolved quantum metric tensor, defined via the trace distance between micromotion operators in momentum-time space. For class A in two spatial dimensions, we find a general inequality linking the Floquet quantum metric tensor and the Floquet topology: the associated quantum volume is bounded below by the Floquet topological invariant. This relation is found to also hold in class AIII in one dimension, where the Floquet geometric tensor may be notably reduced due to time-reflection symmetry. This work will be useful in digesting the general aspects of quantum geometry in periodically driven systems in connection with their topological characterization.
We identify quantum geometric bounds for observables in non-Hermitian systems. We find unique bounds on non-Hermitian quantum geometric tensors, generalized two-point response correlators, conductivity tensors, and optical weights. We showcase these findings in topological systems with non-Hermitian Chern numbers. We demonstrate that the non-Hermitian geometric constraints on response functions naturally arise in open quantum systems governed by out-of-equilibrium Lindbladian dynamics. Our findings are relevant to experimental observables and responses under the realistic setups that fall beyond the idealized closed-system descriptions.