Imagine a vast, flat city made of tiny, interconnected houses (these are the optical waveguides). In the world of physics, scientists have been studying how "energy" or "light" moves through these cities. For a long time, they believed there were two distinct rules for how this energy could get trapped in the corners or along the edges of the city:

The Dipole Rule: Think of this like a city with a strong wind blowing from left to right. The energy gets pushed to the top or bottom edges, like leaves piling up against a wall.
The Quadrupole Rule: Think of this like a city with four distinct corners where energy loves to hide, regardless of the wind. It's a more complex pattern where the energy gets stuck specifically in the four corners of the grid.

Until now, physicists thought you could only have one of these rules at a time in a single city. If your city had a "wind" (dipole), it couldn't have the special "corner traps" (quadrupole) in the same way, and vice versa. They were considered mutually exclusive.

The Big Discovery
The authors of this paper, Konstantin Rodionenko, Maxim Mazanov, and Maxim Gorlach, have built a theoretical "city" that breaks this rule. They designed a system where both the Dipole Rule and the Quadrupole Rule exist at the same time.

How did they do it? (The Analogy)
Imagine each house in their city isn't just a simple room. Instead, every house has two separate rooms inside it:

Room A (The "S" room): A round, symmetrical room where light can spin freely.
Room B (The "P" room): A dumbbell-shaped room where light has a specific direction (like a dipole).

By carefully arranging these double-room houses in a specific grid pattern and connecting them with different strengths of "hallways" (couplings), the authors created a situation where:

The "P" rooms create a dipole effect (pushing energy to the top and bottom edges).
The "S" rooms, interacting with the "P" rooms in a specific way, create the quadrupole effect (trapping energy in the corners).

It's as if the city has a wind blowing north-south while simultaneously having four magical corners that catch the wind.

The "Wannier" Lens
To prove this wasn't just a mathematical trick, the scientists used a special "lens" called Wannier functions. You can think of this as a way to look at the city through different glasses:

Through one pair of glasses, the city looks like a simple dipole system (energy on the edges).
Through another pair of glasses, the city looks like a quadrupole system (energy in the corners).

The paper shows that you can mathematically separate the city's behavior into these two distinct "layers" or "sub-sectors." In one layer, the rules for the dipole apply; in the other, the rules for the quadrupole apply. They coexist peacefully in the same physical space.

The Proof
The team didn't just do the math on paper. They simulated a real-world version of this using lasers and glass.

They imagined writing these "houses" into a piece of glass using a super-fast laser (a technique called femtosecond laser writing).
They ran computer simulations of light traveling through this glass structure.
The Result: The light behaved exactly as predicted. It appeared on the top and bottom edges (the dipole signature) and got trapped in the four corners (the quadrupole signature) at the same time.

Why This Matters (According to the Paper)
The paper concludes that this "coexistence" is real and robust. It means that nature allows for more complex combinations of topological states than we previously thought. Just as a collection of electric charges can have a net charge, a dipole, and a quadrupole all at once, a quantum system can now be shown to host both dipolar and quadrupolar topological protections simultaneously.

The authors also noted that this structure is resistant to "disorder" (like a few broken hallways or slightly misaligned houses), meaning the special corner and edge states stay protected even if the city isn't perfect.

In Summary
The paper demonstrates that the "Dipole" and "Quadrupole" topological phases are not enemies that cancel each other out. Instead, they can be partners living in the same structure, creating a system that is protected by both types of rules at once. This was proven through a specific model of light-carrying waveguides and confirmed by detailed computer simulations.

Technical Summary: Coexistence of Dipolar and Quadrupolar Higher-Order Topology

Problem Statement

Two-dimensional higher-order topological insulators (HOTIs) are traditionally classified into two distinct, non-overlapping categories: dipolar and quadrupolar topological insulators. This classification stems from the modern theory of polarization, where the bulk quadrupole moment is generally considered ill-defined in the presence of a nonzero bulk dipole polarization due to the origin-shift law. While classical charge distributions can simultaneously possess various multipole moments, quantum periodic systems have been thought to preclude the simultaneous existence of well-defined dipolar and quadrupolar higher-order topologies within a single phase. The central problem addressed is whether this theoretical limitation can be overcome to demonstrate a system where both dipolar and quadrupolar topological features coexist and are independently definable.

Methodology

The authors propose and analyze a specific two-dimensional lattice model to resolve this tension. The methodology involves three primary stages:

Two-Orbital Model Construction: The authors construct a theoretical model based on a square lattice where each site hosts a pair of degenerate modes (orbitals): an $s$ -mode (full rotational symmetry) and a $p$ -mode (inversion-odd, dipole-like). This system is formed by combining two copies of a canonical orbital quadrupole insulator rotated by $\pi/2$ relative to each other. The model includes intra-orbital couplings ( $s-s$ and $p-p$ ) and inter-orbital couplings ( $s-p$ ), governed by parameters $\kappa$ , $\gamma$ , and $\Delta$ , respectively, alongside a geometric dimerization parameter $\alpha$ .
Topological Invariant Calculation: To characterize the topology, the authors employ the nested Wilson loop technique. They calculate Wannier bands ( $\nu_x$ $ν_{x}$ and $\nu_y$ $ν_{y}$ ) and subsequently define projectors onto specific Wannier subsectors (dipolar and quadrupolar). By examining the commutation relations of the projected position operators within these subsectors, they determine the nature of the topology:
- Dipolar Subsector: Projected position operators commute, allowing for a well-defined 2D Wannier center and bulk polarization.
- Quadrupolar Subsector: Projected position operators do not commute, indicating a quantized bulk quadrupole moment.
Experimental Implementation Proposal: To validate the theoretical predictions, the authors propose a photonic implementation using arrays of evanescently coupled optical waveguides. They first analyze a "two-orbital" meta-atom design and then refine this into a "single-orbital" model using rhombic four-site meta-atoms. This single-orbital design is shown to be fabricable using femtosecond laser writing techniques in borosilicate glass. Full-wave numerical simulations (COMSOL) are performed to confirm the existence of the predicted topological phases and boundary states.

Key Contributions and Results

1. Demonstration of Coexisting Topologies

The primary result is the identification of a "mixed dipolar-quadrupolar" phase where the system simultaneously exhibits nonzero bulk dipole polarization and a quantized bulk quadrupole moment.

Wannier Subsector Decomposition: The authors demonstrate that while the total system has a nonzero dipole moment (making the global quadrupole moment origin-dependent), the system can be decomposed into distinct Wannier subsectors. In the dipolar subsector, the position operators commute, yielding a quantized polarization $P = (0, 0.5)$ . In the quadrupolar subsector, the operators do not commute, yielding a quantized quadrupole moment $q_{xy} = 0.5$ .
Adiabatic Connection: The topology of the mixed phase is shown to be adiabatically connected to a pair of decoupled 2D quadrupolar and dipolar HOTI layers, despite the absence of a spatially homogeneous unitary transformation that directly splits the system.

2. Distinct Boundary Signatures

The coexistence of these topologies leads to a unique spectrum of boundary states in a finite lattice:

Corner States: Driven by the bulk quadrupole moment, four zero-energy corner states appear. The authors note that due to the lack of $M_x$ mirror symmetry in their specific lattice, the left and right corner modes are inequivalent; two are perfectly localized at single sites, while the other two exhibit a structure analogous to edge states.
Edge States: Driven by the bulk dipole polarization, $2(N-1)$ zero-energy edge states appear on the upper and lower edges of the sample (where the polarization vector is non-zero). The left and right edges remain devoid of such states.
Robustness: The energies of both corner and edge states are shown to be robust against disorder in couplings and on-site symmetric orbital detuning.

3. Photonic Realization

The paper provides a concrete pathway for experimental realization:

Single-Orbital Model: By utilizing a rhombic four-site meta-atom with specific coupling strengths ( $J, J', \alpha$ ), the system mimics the mixed topology near energies $\pm J$ .
Fabrication: The proposed structure can be fabricated using femtosecond laser writing. Numerical simulations of a realistic $4 \times 4$ unit-cell lattice confirm the presence of topological bandgaps and the localization of corner and edge states, validating the theoretical predictions.

Significance and Claims

The paper claims to resolve the theoretical tension between classical multipole intuition and quantum topological classification. It asserts that the limitation preventing the coexistence of dipolar and quadrupolar HOTIs is not fundamental but rather a consequence of how invariants are defined in the presence of nonzero polarization.

By introducing the concept of Wannier subsectors, the authors show that the bulk quadrupole moment remains well-defined and physically meaningful even in the presence of a bulk dipole, provided the system is analyzed within the appropriate subspace. This finding enriches the variety of known topological phases, allowing for the combination of different multipole topologies within a single structure.

The authors further suggest that this coexistence may lead to novel physical phenomena, such as non-Abelian Bloch oscillations, which are typically associated with quadrupole topology but absent in standard dipolar HOTIs. They propose that in a mixed dipole-quadrupole insulator, such non-Abelian effects could manifest for a specific class of initial conditions, opening a new avenue for studying higher-order topological dynamics.

The work is presented as a proof-of-concept supported by full-wave numerical simulations, establishing a foundation for future experimental verification in photonic systems.

Coexistence of dipolar and quadrupolar higher-order topology