Polarization Topological Index Spaces

• Polarization topological index spaces are a conceptual framework linking quantized polarization features with topological invariants and index theories across various physical systems.
• They employ geometric representations like the Poincaré sphere and its higher-order variants to characterize structured beams and vortices using quantized winding numbers and indices.
• Their applications in optical sensing, photonic communications, and electronic materials leverage robust, quantized markers to achieve precise control and novel device functionalities.

Polarization topological index spaces constitute a conceptual and formal framework linking the quantized or protected features of polarization—whether in electronic, photonic, or hybrid systems—to topological invariants and index theories. In both real and momentum space, the structure, dynamics, and transformation properties of polarization fields encode distinct topological information: this includes winding numbers of polarization vortices, quantized indices for structured beams, and robust markers of band topology in crystalline or engineered photonic materials. Recent research has demonstrated the emergence, classification, and practical manipulation of polarization indices in advanced materials and structured light, with implications spanning sensing, communication, and condensed matter physics.

1. Geometric Representations and Topological Constructs

Polarization states, in their uniform form, are most naturally visualized as points on the standard Poincaré sphere, with fixed azimuth (γ) and ellipticity (χ) parameters. For spatially structured fields (e.g., vector beams, beams with vortex singularities), the use of higher-order Poincaré spheres (HOPS) is standard; a point on an HOPS corresponds to a distinct structured polarization distribution, potentially featuring phase singularities or nontrivial spatial topology (Umar et al., 4 Jul 2025). Optical elements such as $q$-plates, characterized by topological charge $q$, can effect transformations between uniform and structured states, mapping between spheres and their higher-order analogues.

In advanced optical experiments, interferometric arrangements and phase-shifters allow the controlled generation and observation of polarization state trajectories with nontrivial topology in Stokes space. For example, the evolution under a generic SU(2) rotation operator,

$$\hat{\mathcal{D}}(\mathbf{\hat{n}},\delta\phi) = \cos\left(\frac{\delta\phi}{2}\right)\mathbf{1} - i\,\sin\left(\frac{\delta\phi}{2}\right)(\hat{\bm\sigma} \cdot \mathbf{\hat{n}}),$$

traces circles (𝕊¹), tori (𝕊¹ × 𝕊¹), or more complex manifolds according to the degrees of phase control and axis specification, as confirmed in Mach–Zehnder–based optical setups (Saito, 2023).

2. Topological Indices: Definition and Physical Meaning

Structured polarization fields and elements are characterized by discrete, quantized topological indices. Key examples include:

• Winding number (vortex charge $q$) in polarization vortices: Defined by

$$q = \frac{1}{2\pi}\oint_C dk\, \nabla_k[\arg(E(k))],$$

where $E(k)$ is the complex far-field polarization vector; $q$ counts net rotation of the polarization vector around a closed loop in momentum space and is robust to moderate perturbations (Nazarov et al., 21 Jul 2025).

• Poincaré–Hopf (PH) index ($\eta$): For structured beams, this quantifies the winding of the polarization azimuth $\gamma$ around singularities in the transverse plane,

$\eta = \frac{1}{2\pi} \oint \nabla\gamma \cdot dl,$

serving as an identifier of singularities and spatially structured polarization (Umar et al., 4 Jul 2025).

• Semilocal hybrid polarization (SHP): In systems with non-trivial band topology, the SHP $P^h_\beta(r_j,k_\perp)$ at position $r_j$ and momentum component $k_\perp$ is computed as an integrated change in the hybrid Wannier charge center,

$$P^h_\beta(r_j,k_\perp) = -\frac{ef}{\Omega_0} \sum_n^{\mathrm{occ}} \int_0^{x(r_j)} \left[ \partial_{\bar{x}} w^h_n,_\beta(\dots) \right] dx',$$

with total local polarization in a subcell obtained upon integration over $k_\perp$ (Jankowski et al., 25 Apr 2024).

• Index of projection operators in topological bands: For symmetry class A Hamiltonians, the analytical index is given as

$$\operatorname{ind}(\hat{\mathbb{P}}(k)) := \dim \ker(\hat{\mathbb{P}}(k)) - \dim \ker(\hat{\mathbb{P}}^\perp(k)) = n-m,$$

relating the structure of valence and conduction bands to topological invariants (e.g., Chern number) in polaritonic systems (Rips, 2023).

These indices serve as markers of both the local geometry of polarization fields (e.g., at singularities, domain walls, or defects) and global topological phases (e.g., in the Brillouin zone or under certain symmetry constraints).

3. Holonomic and Nonholonomic Polarization Transformations

Holonomic transformations in polarization optics are defined as those in which the polarization trajectory remains entirely on a specific topological manifold, such as a Poincaré sphere or its higher-order variant (HOPS). The holonomy condition enforces matching among the structured beam (charge $l$), the order of the topological sphere ($\eta$), and the optical element's (e.g., $q$-plate) charge ($q$): $l = \eta = q$ If this constraint is met, the polarization transformation corresponds to an SO(3) rotation on the HOPS; the system output preserves the structured topological signature (Umar et al., 4 Jul 2025). In contrast, nonholonomic transformations involve transitions where the output state no longer resides on the original topological sphere, leading to potential loss or inversion of topological charge (as, e.g., when a half-waveplate with $q=0$ acts on a beam with $\eta \neq 0$).

4. Topological Features in Sensing, Photonic, and Electronic Systems

In photonic structures, the topological charge associated with polarization vortices is harnessed for robust sensing. Polarization vortices attached to bound states in the continuum (BICs) possess winding numbers (±1 in practical cases) and exhibit a migration in the momentum plane (k-space) according to a square-root law,

$\Delta\varphi = \sqrt{A + B \Delta n_m},$

where $\Delta n_m$ is the change in refractive index, $A$ and $B$ are system-dependent parameters, and $\Delta\varphi$ marks the k-space shift of the vortex. This non-analytic migration offers high angular sensitivity, often rivaling or surpassing spectral sensitivity, and forms the basis of advanced dielectric refractive index sensors (Nazarov et al., 21 Jul 2025).

Similarly, in electronic topological materials, polarization indices capture the interplay between band topology (e.g., non-zero Chern number) and real-space polarization textures. Semilocal hybrid polarizations reveal that, even in "Wannier obstructed" bands, local polarization can be defined and its winding controlled, providing direct access to the quantized Chern number and enabling the coexistence of band topology with nontrivial polar textures, including merons or skyrmions (Jankowski et al., 25 Apr 2024).

5. Classification and Mathematical Structure of Index Spaces

The mathematical structure underlying polarization topological index spaces is rooted in the classification of maps from physical parameter spaces (momentum, real space, or the space of optical configurations) into topological manifolds such as spheres or Grassmannians. The index spaces are often parameterized by:

• The order of the relevant geometric manifold (e.g., Poincaré sphere or HOPS of order $\eta$)
• The topological charge or winding number (e.g., $q$ for vortices, $\eta$ for PH index)
• The transformation properties under optical elements (e.g., Jones matrices for $q$-plates),
• The symmetry class and band structure of the underlying Hamiltonian (class A, class D, etc.).

For multi-band photonic or polaritonic systems, the mapping from the Brillouin zone into Grassmannian spaces,

$$f: \mathbb{T}^D \rightarrow U(n+m)/(U(n)\times U(m)),$$

enables classification via homotopy and cohomology, connecting topological invariants (such as the sum of Chern numbers and the analytical index of the projection operator) to the presence of protected edge modes and emergent phases (Rips, 2023).

6. Physical Realizations and Applications

Structured topological index spaces have immediate utility in engineering optical and electronic systems:

• In optical communications and quantum optics, holonomically constrained transformations preserve the integrity of orbital angular momentum channels, minimizing cross-talk and enabling robust information transfer (Umar et al., 4 Jul 2025).
• Polarization vortices provide a highly sensitive "angular marker" for index sensing, because their topological charge ensures robustness under perturbations and their position in k-space can be tracked with high precision (Nazarov et al., 21 Jul 2025).
• In condensed matter and moiré materials, local polarization textures can be exploited to design edge transport, reconfigurable topological currents, and complex domain-wall phenomena (Jankowski et al., 25 Apr 2024).

These applications leverage the robust, quantized features of polarization index spaces—whether their invariance under smooth deformations, their switching upon topological phase transitions, or the controlled manipulation via structured optical elements and external fields.

7. Outlook and Broader Implications

The concept of polarization topological index spaces integrates geometric, algebraic, and physical aspects, providing a foundation for both classification and manipulation of structured polarization phenomena. An emerging theme is the coexistence and interplay of band topology (momentum-space invariants) with real-space polar topologies, illuminating new paths for device design in photonic and electronic systems. Formulations in terms of projection operator indices, homotopy classes, and semilocal hybrid polarizations provide a unified language to describe this richness, connecting abstract mathematical frameworks with experimentally observable quantities. As research advances, further integration of these index spaces into sensor design, quantum control, and material engineering is anticipated, revealing new classes of robust, functionally rich physical systems.