🔬 mesoscale physics

Hybrid-order nonlinear topological phases

This paper establishes a multiband bulk-boundary correspondence for nonlinear systems by introducing an auxiliary-system formalism that enables the classification of hybrid-order topological phases, demonstrating how uniform and dimerized stacking of 2D nonlinear eigenvalue systems can engineer distinct 3D first-, second-, and third-order topological states.

Original authors: Yu-Peng Ma, Ming-Jian Gao, Jun-Hong An

Published 2026-07-07

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CC BY 4.0

Original authors: Yu-Peng Ma, Ming-Jian Gao, Jun-Hong An

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 building a complex structure out of Lego bricks. In the world of physics, these "bricks" are often waves of energy (like light or sound) moving through a material. For a long time, scientists have known how to arrange these bricks to create "topological phases." Think of these as special, super-sturdy patterns where the edges of your structure behave differently than the middle. Usually, if you have a "gap" in the energy levels (like a missing rung on a ladder), you expect to find a special path for energy to travel right along the edge of that gap. This is called the "bulk-boundary correspondence."

However, things get messy when you introduce nonlinearity. In simple terms, nonlinearity means the material changes its behavior depending on how hard you push it. It's like a spring that gets stiffer the more you stretch it. When this happens, the old rules break down. The "map" scientists used to predict where the energy paths would go becomes blurry, especially when there are multiple gaps in the energy ladder.

This paper by Yu-Peng Ma, Ming-Jian Gao, and Jun-Hong An solves this puzzle by inventing a new "translator" tool.

The Magic Translator (The Auxiliary System)

The authors realized they couldn't read the nonlinear system directly. So, they built a fictional, linear "twin" system (called an auxiliary system) that acts like a translator.

The Problem: The real system is a tangled knot of equations where the answer changes the question.
The Solution: They created a linear twin that is easy to read. If they find a special path in the twin, they know that same path exists in the real, messy system. This allows them to map out the "topology" (the shape and structure) of the system even when it's nonlinear.

The 2D Discovery: Two Types of Roads at Once

Using this translator, they looked at a flat, two-dimensional (2D) grid. They discovered something surprising: Two different types of energy highways can exist at the same time in the same system, but in different "gaps."

The Highway (First-Order): In one energy gap, they found a "gapless" highway running along the entire edge of the square. It's like a road that goes all the way around the perimeter.
The Corner Store (Second-Order): In a different energy gap, they found energy trapped only in the four corners of the square. It's like a store that only exists at the very tips of the building, nowhere else.

Usually, you'd expect a system to be either a "highway system" or a "corner system," but not both simultaneously. This paper shows they can coexist, creating a "hybrid-order" phase.

Stacking the Layers: Building a 3D Tower

The real magic happens when they stack these 2D squares on top of each other to make a 3D tower. How they stack them changes the shape of the energy paths:

Scenario A: The Uniform Stack (The Smooth Tower)
If they stack the layers perfectly evenly (like a neat stack of pancakes), the "highways" from the 2D layers turn into side-surface roads on the 3D tower. The "corner stores" turn into hinge roads (running along the vertical edges where two walls meet).
- Result: A 3D tower with both side roads and hinge roads.
Scenario B: The Dimerized Stack (The Alternating Tower)
If they stack them in an alternating pattern (strong bond, weak bond, strong bond, like a zipper), the energy gets squeezed even tighter.
- The "corner stores" from the 2D layers get squeezed down to the very top and bottom tips of the 3D tower. This creates a Third-Order phase (energy trapped at the 0-dimensional corners of a 3D object).
- The "highways" get squeezed into the hinges (the vertical edges). This creates a Second-Order phase.
- Result: A 3D tower where energy lives only at the very corners and the vertical edges, but not on the flat faces.

Why This Matters

The paper doesn't claim to build a new phone or cure a disease. Instead, it provides a new theoretical framework. It proves that in nonlinear systems (which are common in real-world things like lasers, mechanical metamaterials, and electrical circuits), you can have multiple "orders" of topological protection happening at once.

By using their "translator" method and by carefully engineering how layers are stacked (stacking engineering), scientists can now design materials that host these complex, hybrid states. It's like discovering that by arranging your Lego bricks in a specific, alternating pattern, you can make a tower that has secret doors on the corners and secret tunnels on the edges, all at the same time.

Technical Summary: Hybrid-order nonlinear topological phases

Problem Statement
The bulk-boundary correspondence (BBC), which links bulk topological invariants to boundary modes, is well-established for linear systems but remains an open question in nonlinear regimes. Specifically, in nonlinear-eigenvalue systems—where nonlinearity arises via the eigenvalue itself rather than the eigenstate amplitude—the conventional topological framework breaks down. While recent studies have established a BBC for single-gap boundary modes in such systems using auxiliary linear mappings, a complete description of multigap topologies is absent. It remains unclear whether different orders of topological boundary modes (e.g., gapless edge states and gapped corner states) can coexist within the distinct band gaps of a single nonlinear-eigenvalue system, and how these phases manifest when extended to three dimensions.

Methodology
The authors address this by developing an auxiliary-system formalism to enable topological classification of nonlinear-eigenvalue systems.

Auxiliary Mapping: They consider systems governed by the equation $H|\psi\rangle = \omega S(\omega)|\psi\rangle$ , where $H$ is a Hermitian Hamiltonian and $S(\omega)$ is an overlap matrix dependent on the eigenvalue $\omega$ . By defining a pencil matrix $P(\omega) = H - \omega S(\omega)$ , they map the nonlinear problem to a formally linear eigenvalue problem: $P(\omega)|\phi\rangle = \lambda|\phi\rangle$ .
Topological Characterization: The zero line of the auxiliary spectrum ( $\lambda = 0$ ) corresponds to the physical eigenvalues ( $\omega$ ) of the original system. The authors define topological invariants (Chern numbers and mirror-graded winding numbers) within the auxiliary system at specific fixed eigenvalues to characterize the boundary modes of the original nonlinear system.
Model Construction:
- 2D Model: A spinless square-lattice model with two degrees of freedom per unit cell is analyzed. The Hamiltonian includes linear and nonlinear hopping terms, as well as nonlinear mass terms.
- 3D Extension: The 2D system is stacked along the third dimension ( $z$ -axis) using a Su-Schrieffer-Heeger (SSH) chain configuration. Two stacking configurations are investigated: uniform stacking ( $t_1 = t_2$ ) and dimerized stacking ( $t_1 < t_2$ ).

Key Contributions and Results

2D Hybrid-Order Phase:
In the 2D nonlinear system, the authors identify a hybrid-order topological phase where two distinct topological features coexist within different band gaps of the same spectrum:
- At the eigenvalue $\omega = 0$ , the system exhibits first-order gapless boundary states (edge modes), characterized by a non-zero Chern number ( $C_1$ ) in the auxiliary system.
- At the eigenvalue $\omega = -M^{-1}$ , the system exhibits second-order gapped corner states, characterized by a non-zero mirror-graded winding number ( $W_2$ ) in the auxiliary system.
- Numerical simulations confirm the simultaneous presence of these states, with corner states localized at the four corners and edge states forming gapless branches along the boundaries.
3D Hybrid-Order Phases via Stacking Engineering:
Stacking the 2D system along the $z$ -axis yields distinct 3D hybrid-order realizations depending on the stacking configuration:
- Uniform Stacking ( $t_1 = t_2$ ): The 2D edge states evolve into side-surface states, and the 2D corner states transform into hinge states. This results in a coexisting 3D first-order and 3D second-order topological phase.
- Dimerized Stacking ( $t_1 < t_2$ ): The SSH chain induces localization along the $z$ $z$ -axis.
  - The 2D gapless boundary states localize at the hinges of the 3D system, forming a 3D second-order topological phase.
  - The 2D gapped corner states become further confined to the two ends of the $z$ -axis, manifesting as 3D third-order corner states.
- This configuration demonstrates the coexistence of 3D second-order and 3D third-order topological phases.

Significance and Claims
The paper claims to establish a multigap bulk-boundary correspondence for nonlinear-eigenvalue systems. By utilizing the auxiliary-system formalism, the authors provide a complete topological characterization that resolves the ambiguity of topological descriptions in nonlinear regimes with multiple gaps.

The primary significance lies in identifying stacking engineering as a versatile platform for exploring hybrid-order topological phases. The work demonstrates that nonlinear systems can support the simultaneous existence of boundary modes of different orders (e.g., edge, hinge, corner, and surface states) within a single physical system. This expands the known landscape of topological phases beyond the single-gap, linear paradigms, offering a theoretical framework for designing devices that combine the functionalities of different topological orders. The results are conceptually applicable to various physical platforms where nonlinear-eigenvalue problems naturally arise, including dispersive photonic, phononic, circuit, and mechanical systems.

The Magic Translator (The Auxiliary System)

The 2D Discovery: Two Types of Roads at Once

Stacking the Layers: Building a 3D Tower

Why This Matters

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