Quantum Geometry Advances Rotating Shallow Water Equations with Three Band Descriptions

The dynamics of rotating fluids underpin many natural phenomena, from weather patterns to ocean currents, and are typically modelled using the rotating shallow water equations. Sriram Ganeshan of City College, City University of New York and CUNY Graduate Center, alongside Alan T. Dorsey from the University of Georgia, present a novel geometric framework for understanding these equations. Their research develops a full geometric tensor approach to describe wave behaviour, revealing connections between wave polarization, geometric phases and fundamental invariants. This work is significant because it provides a unified description of wave bands and suggests a pathway to experimentally verify these geometric properties using rotating tank experiments, potentially advancing fluid dynamics modelling and our understanding of related physical systems.

Their research develops a full geometric tensor approach to describe wave behaviour, revealing connections between wave polarization, geometric phases and fundamental invariants. This work provides a unified description of wave bands and suggests a pathway to experimentally verify these geometric properties using rotating tank experiments, potentially advancing fluid dynamics modelling and our understanding of related physical systems.

Electron gases share similarities with active-matter fluids, prompting investigation into the topological characteristics of Rossby-Steward Wave Equation (RSWE) wave bands. This research develops a complementary quantum-geometric description by calculating the full quantum geometric tensor (QGT) for the linearized RSWE on an f-plane. The QGT consolidates two aspects of band geometry: its real component defines a metric quantifying the rate of wave polarization change with varying parameters, and its imaginary component represents the Berry curvature controlling geometric phases and topological invariants. Compact, symmetry-guided expressions are derived for all three bands, revealing the transverse structure of the metric and a monopole-like Berry curvature which yields Chern numbers.

Rotating Shallow Water Equations and Linear Wave Connections

The rotating shallow water equations (RSWE) are fundamental to modelling atmospheric and oceanic fluids, offering a simplified yet comprehensive description of large-scale flows and wave behaviour within a thin, rotating fluid layer. These equations capture the interaction between stratification and the Coriolis force, organising movement into fast Poincaré (inertia-gravity) waves and a slower geostrophic mode. Similar linear wave problems also appear in areas such as two-dimensional electron fluids within magnetic fields and active fluids exhibiting parity and time-reversal-breaking stresses, demonstrating broad applicability beyond geophysics. Recent research highlights a connection between linear wave equations in continuous media and the geometric and topological phenomena observed in Bloch bands within condensed matter physics.

Linearisation of the RSWE and formulation as a Hermitian eigenvalue problem reveals a three-band structure supporting Berry curvature and quantized Chern numbers specifically within the Poincaré bands. This perspective clarifies the stability of wave guiding along equatorial and coastal regions, framing it as a bulk, boundary correspondence within a continuum model and linking geophysical wave dynamics to the wider concept of topological phases. Beyond topology, the geometry of these bands is crucial, and is best understood through the quantum geometric tensor (QGT). The QGT, applied to a normalized eigenmode, is a gauge-invariant Hermitian tensor measuring the overlap of projected tangent vectors in Hilbert space, dependent on adiabatic control parameters.

Its symmetric component, the Fubini, Study (FS) metric, quantifies the distinguishability of adjacent eigenstates, establishing a Riemannian structure within the parameter space. The FS metric is increasingly recognised for its importance, influencing response functions, establishing links between geometry and topology, and playing a dynamic role in bands that are flat or nearly flat, including contributions to superfluid weight and related concepts. Researchers are now exploring methods to probe this geometry experimentally, utilising weak, time-periodic parametric driving within rotating tank experiments.

Geometric Description of Rotating Shallow Water Equations

Scientists achieved a complete geometric description of linearized rotating shallow water equations (RSWE) using the full geometric tensor (QGT) on an f-plane. The research establishes a framework unifying band geometry through a metric quantifying wave polarization changes and a Berry curvature controlling geometric phases. Experiments reveal that the QGT provides compact, symmetry-guided expressions for all three bands, highlighting the transverse structure of the metric and a monopole-like Berry curvature that yields specific Chern numbers for the Poincaré bands. This detailed geometric characterization offers new insights into wave dynamics relevant to atmospheric and oceanic modeling.

The team measured the components of the QGT, beginning with calculations based on spectral representation to facilitate generalization to higher spin-S models. Results demonstrate that components of the QGT involving the z-axis are zero, simplifying the analysis and revealing key symmetries within the system. Subsequent calculations, utilizing raising and lowering operators for spin S, yielded specific values for the matrix elements necessary to define the QGT components, specifically Q(m)xx = Q(m)yy = Cm d2, where Cm = S(S + 1) −m2 2, and Q(m)xy = im 2d2. Measurements confirm that the final expression for the QGT, Q(m)ij = Cm ∂id · ∂j d + im 2 d · ∂id × ∂j d, encapsulates the full geometric properties of the system.

The derived FS metric, g(m)ij, exhibits a null direction along the parameter vector d, indicating that radial derivatives vanish and the geometry is confined to a sphere in parameter space. Furthermore, the Berry curvature tensor, F(m)ij, also shares this null direction, demonstrating rotational covariance of the Hamiltonian. The team successfully computed all components of the QGT, providing a comprehensive framework for understanding wave behavior in rotating shallow water systems. The breakthrough delivers a feasible route to probing this geometry in rotating-tank experiments through weak, time-periodic parametric driving, detailing how control parameters (kx, ky, f) influence the QGT, enabling experimental verification of the theoretical predictions. Tests prove the utility of spherical coordinates for simplifying the FS metric, resulting in a diagonal form g(m)(θ, φ, d) = Cm 0 0 0 Cm sin2 θ 0 0 0 0. This detailed geometric description has implications for understanding wave dynamics in diverse physical systems, ranging from atmospheric and oceanic phenomena to two-dimensional electron gases and active-matter fluids.

Quantum Geometry of Rotating Shallow Water Equations

This work establishes a quantum-geometric description of the rotating shallow water equations (RSWE) by calculating the full quantum geometric tensor (QGT) for the three linear wave bands. Researchers moved beyond a purely topological characterization of the linearized RSWE, successfully extracting both geometric data, the Fubini, Study metric, and topological data, the Berry curvature, encoded within the QGT. The resulting framework reveals a consistent structure, where the metric defines distances between states and the Berry curvature governs geometric phases within the parameter space of the RSWE. The significance of this achievement lies in its ability to separate robustly quantized properties, such as integrated curvature, from aspects related to overall metric scale, allowing for systematic comparisons between bands and a clearer understanding of the physical effects driven by topology versus geometry.

Furthermore, the study demonstrates that the RSWE share a “Riemannian + symplectic” structure with multiband quantum systems, suggesting a deeper connection between these seemingly disparate fields. The authors propose a method for experimentally probing this geometry through weak, time-periodic parametric driving in rotating tank experiments, offering a pathway to measure both metric and curvature. Acknowledging limitations, the authors highlight the need to integrate the QGT into a wave-packet theory applicable to slowly varying backgrounds. They anticipate that the Berry curvature will govern post-eikonal corrections to ray trajectories, while the quantum metric will contribute to the diffractive spreading of wave envelopes. Future research will focus on developing this theoretical framework and exploring its implications.