🔬 condensed matter

Topological invariants and topological charges in photonic systems

This paper proposes a general, symmetry-based real-space Hamiltonian framework that bridges the gap between global topological invariants and local far-field defects in photonic systems, enabling the intuitive design of structures with arbitrary topological properties across both localized and delocalized mode limits.

Original authors: Kristian Arjas, Grazia Salerno, Päivi Törmä

Published 2026-01-15

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CC BY 4.0

Original authors: Kristian Arjas, Grazia Salerno, Päivi Törmä

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to understand the shape of a complex piece of music. Usually, you might listen to the whole song to get the "vibe" (the global feeling), or you might zoom in on a specific, jarring note or a sudden silence (a local defect). In the world of light and photonic crystals, scientists have been doing the same thing: studying the overall "topology" (the global shape of light waves) and the "topological charges" (local whirlpools or vortices in the light's polarization).

The problem is that these two ways of looking at light have been like two different languages. One language describes the light as a smooth, flowing river (global topology), while the other describes it as a collection of tiny, spinning tops (local defects). Until now, translating between these two languages required heavy, slow computer simulations that didn't offer much physical intuition.

The New "Universal Translator"
The authors of this paper have built a new "translator" or a universal framework. Think of it as a master blueprint (a mathematical tool called a Hamiltonian) that can describe light in two ways at once:

The "Atomic" View: Like a tight-knit group of friends passing a ball to their immediate neighbors (short-range connections).
The "Photonic" View: Like a crowd where everyone can shout across the room to anyone else (long-range connections).

This blueprint is special because it is built on symmetry. Imagine a kaleidoscope; no matter how you rotate it, certain patterns repeat. The authors use these repeating patterns (symmetries) to design the blueprint. This allows them to control the energy of different "modes" (ways the light can vibrate) independently, almost like tuning individual instruments in an orchestra to create a specific harmony.

Connecting the Near and the Far
One of the most exciting parts of this work is how it connects what happens inside the material (the "near field") with what happens outside in the air (the "far field").

The Near Field: Imagine the light trapped inside a maze. The authors can now predict how the light spins and twists inside this maze.
The Far Field: Imagine the light escaping the maze and flying out into the world. Usually, predicting exactly how it looks when it escapes is hard. But this new framework shows that the "twists" and "vortices" inside the maze leave a fingerprint on the light that escapes.

They discovered that you can look at the escaping light and tell if the light inside had a "global" topological property (like a specific winding number, similar to how many times a ribbon is twisted). It's like looking at the smoke coming out of a chimney and being able to tell exactly how the fire is burning inside, even without seeing the fire itself.

The "SSH" Analogy
To test their idea, they used a model called the "2D SSH model." Think of this as a grid of springs and masses.

In a trivial (boring) version, the springs are all the same, and the system is stable but uninteresting.
In a topological (interesting) version, they tweak the springs so that the edges of the grid start to vibrate in a special way that the middle doesn't.

The authors showed that even if you change the springs to be very long-range (connecting far-away masses), the "edge vibrations" remain robust. They also showed that if you break the symmetry (like adding a magnetic twist), the light escaping from the edge becomes "circularly polarized" (spinning in a specific direction), which is a clear sign of a topological state.

Why This Matters (According to the Paper)
The paper claims this framework is a powerful design tool. Because the blueprint is based on symmetry, scientists can now:

Design Light: Create structures with almost any topological property they want by simply adjusting the symmetry parameters.
Bridge the Gap: Connect the simple, intuitive models used by physicists (tight-binding) with the complex, realistic models used by engineers (long-range diffraction).
Apply Everywhere: While they tested this with light (photons), the math works for any system where you can engineer connections between points, such as mechanical metamaterials (structures that move in specific ways) or electrical circuits.

In short, the authors have created a "symmetry-based Lego set" for light. It allows them to snap together different pieces to build structures where the global behavior (the whole structure) and local behavior (the individual pieces) are perfectly understood and connected, making it much easier to design robust, next-generation photonic devices.

Technical Summary: Topological invariants and topological charges in photonic systems

Problem Statement
Topological photonics currently operates with two distinct conceptual frameworks: global topology, defined by invariants (e.g., Chern numbers, Zak phases) obtained by integrating over the entire Brillouin zone (BZ), and local topology, defined by defects such as polarization vortices in far-field emission. While global invariants are typically calculated using numerical solutions to Maxwell's equations (e.g., finite element methods) or tight-binding (TB) models, and local defects are often analyzed via empty-lattice approximations (ELA) or diffractive orders, these approaches lack a unified physical intuition. Existing methods to connect short-range TB models with long-range ELA models are computationally expensive and do not easily allow for the independent control of mode energies at high-symmetry points (HSPs). Furthermore, there is a need for a framework that bridges the gap between near-field Hamiltonian descriptions and far-field radiation properties, particularly for vector fields where polarization plays a critical role.

Methodology
The authors propose a general framework based on a real-space effective Hamiltonian, $\hat{H} = \hat{H}_k + \hat{H}_c$ , capable of describing electric fields as vectors in both near- and far-fields.

Hamiltonian Construction:
- $\hat{H}_k(k)$ (Momentum-dependent part): This term approximates the empty-lattice approximation (ELA) by modeling the dispersion of freely propagating modes. It is constructed via an inverse Fourier transform of desired band structures, allowing for long-range couplings ( $M$ neighbors) that evolve the dispersion from a standard cosine TB form to a linear light-cone dispersion.
- $\hat{H}_c$ (Coupling part): This momentum-independent matrix shifts the energies of bands associated with different symmetry modes at HSPs. It is constructed using group-theory-based projection matrices ( $P_{IR}$ ) derived from the irreducible representations (IRs) of the lattice's symmetry group (e.g., $C_4$ or $C_{4v}$ ). This allows for the independent tuning of mode energies for specific IRs, effectively opening gaps and ordering bands.
Scalar and Vector Extensions:
- The model is first developed for scalar fields (e.g., out-of-plane components) and then extended to vector fields (in-plane electric fields).
- For vector fields, the basis is expanded to include circular polarization states ( $|L\rangle, |R\rangle$ ). The representation matrices for symmetry operations are modified to act on both the lattice sites and the polarization basis, enabling the treatment of polarization vortices.
Far-Field Connection:
- The far-field emission is calculated as a coherent sum of the near-field eigenstates on the unit cell sites.
- The authors define two types of curvatures: the conventional Berry curvature ( $B_{NF}$ ) derived from near-field eigenvectors and the "radiation curvature" ( $B_{FF}$ ) derived from far-field Stokes parameters.
- Topological charges (winding numbers) of polarization vortices are calculated at HSPs to analyze local topology.
Application to 2D-SSH Models:
- The framework is applied to construct 2D Su-Schrieffer-Heeger (SSH) models with tunable couplings, including long-range interactions and time-reversal symmetry (TRS) breaking terms to simulate magnetic flux lattices.

Key Results

Unified Framework: The proposed Hamiltonian successfully bridges the gap between TB and ELA models. By increasing the range of couplings ( $M$ ), the model smoothly transitions from cosine dispersion to linear light-cone dispersion without generating new band crossings, preserving topological invariants.
Control of Topological Phases: By independently tuning the energies of IRs at HSPs via $\hat{H}_c$ $\hat{H}_{c}$ , the authors demonstrate the ability to design structures with specific topological properties:
- Trivial Bands: Configurations where local topological charges exist but sum to zero, resulting in zero Zak phase and Chern number.
- Zak Phase (Z): Non-trivial Zak phases ( $Z=1$ ) are achieved by creating a mismatch in inversion symmetry eigenvalues between $\Gamma$ and $X$ points. The authors establish a direct relationship between the Zak phase and local topological charges: $Z/\pi = (q_\Gamma + q_X) \mod 2$ .
- Chern Number (C): By breaking TRS (introducing complex intracell couplings), degeneracies at HSPs are split into circularly polarized states, generating non-zero Chern numbers.
Local vs. Global Topology: The study reveals that while the distribution of Berry curvature differs between near-field and far-field (due to sensitivity to polarization variations in the far-field), the integrated global topological invariants (Chern number) remain robust and identical in both regimes.
Edge States: Simulations of systems with open boundary conditions confirm the existence of topological edge states. These states are shown to be robust against structural deformations and defects. The far-field emission from these edge states is fully spin-polarized and traces the shape of the sample edge in momentum space, distinct from the speckled emission of bulk modes.

Significance and Claims
The paper claims to provide a "general framework" that enables the design of photonic structures with "almost arbitrary topological properties" by leveraging symmetry principles. The primary significance lies in:

Physical Intuition: Offering an analytical, Hamiltonian-based approach that connects real-space couplings to momentum-space topology, avoiding the "black box" nature of purely numerical Maxwell solvers.
Bridging Scales: Establishing a natural link between local topological defects (polarization vortices in the far-field) and global topological invariants (Chern numbers, Zak phases).
Versatility: The approach is not limited to photonic crystals but is applicable to any synthetic engineered material with controllable real-space couplings, such as circuit lattices and mechanical metamaterials.
Design Tool: The method serves as a symmetry-guided design tool where the deviation of IR energies from the empty-lattice limit can be used as fitting parameters to achieve desired topological phases, facilitating the experimental realization of protected topological photonic systems.

Technical Summary: Topological invariants and topological charges in photonic systems

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