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Codes for "The Hodge structure of Berry-phase transport: topology, geometry, and noise"

Zhi-Wei Wang, Samuel L. Braunstein

Published Sep 27, 2026• 0 citations
Publisher

Abstract

These are the codes for our paper "The Hodge structure of Berry-phase transport: topology, geometry, and noise."
DOI
10.5281/zenodo.22997716

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A Short Introduction to: Connections on $$ \mathbb{U} $$ (1) Bundles and Berry’s Phase.

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Local Dirac-node topology is usually described by a winding number or Berry phase, whereas curvature-flux formulations are commonly associated with global band topology. Here we show that the non-Hermitian Zeeman quantum geometric tensor (QGT) provides a curvature-flux representation of local π1π_1π1​ topology. Unlike the conventional Hermitian QGT, whose metric and Berry curvature are fixed respectively by its real symmetric and imaginary antisymmetric parts, the Zeeman QGT must be decomposed according to symmetry under index exchange. This decomposition yields normal and anomalous metric-curvature sectors. The normal sector has the same metric-curvature structure as the conventional Hermitian QGT. The anomalous sector, by contrast, contains two components absent in the conventional QGT: an imaginary symmetric metric-like tensor and a real antisymmetric curvature-like tensor. In a two-dimensional Dirac system, the anomalous Zeeman curvature forms a radial flux field Hodge-dual to the tangential winding field of the Dirac node, thereby converting the local winding invariant into a Gauss-type flux invariant. We further show that the four Zeeman-geometric sectors map one-to-one onto frequency- and symmetry-resolved gyrotropic conductivity channels, with reciprocal kinetic magnetoelectric response providing a complementary probe. These results establish non-Hermitian Zeeman quantum geometry as a measurable framework connecting local Dirac-node topology and transport.
We propose a practical computational framework for detecting structural changes in parameter-dependent topological data. In many applications, such as time-series data analysis, anomaly detection, and monitoring of systems under changing control parameters, persistence diagrams describe the birth and death of topological features at each parameter value, but they do not fully capture how these features are reorganized over time. To address this limitation, we represent homological features by zero modes of the ordinary combinatorial Hodge Laplacian and track the corresponding feature spaces in a common ambient chain space. This allows us to compute curvature and holonomy as descriptors of local reorganization and accumulated memory in evolving topological structures. Curvature highlights parameter regions where homological features mix or change rapidly, while holonomy summarizes the net effect of such changes after a closed cycle. We also establish stability estimates showing that these descriptors are robust under perturbations of the Hodge Laplacian on regular regions. Numerical experiments on controlled time-dependent point-cloud data show that the proposed method detects tracking instability, distinguishes systems with nearly identical persistence diagrams, and captures cycle-level memory invisible to pointwise feature matching. These results suggest that zero-mode transport geometry can serve as a useful computational tool for analyzing dynamic topological data.
A Short Introduction to: Connections on U \mathbb{U} U (1) Bundles and Berry’s Phase.