Nonlinearity-Induced Topological Edge States

• Nonlinearity-induced topological edge states are spatially localized modes whose existence, stability, and spectral position are dynamically controlled by nonlinear interactions.
• They extend classical topological models by incorporating amplitude-dependent phase transitions, bifurcations, and novel localized states not present in the linear regime.
• Experimental implementations in photonic lattices, exciton-polariton condensates, and electrical circuits validate these phenomena, demonstrating robust tunability and multistability.

Nonlinearity-induced topological edge states are spatially localized modes at the boundaries of lattice or continuum systems whose existence, spectral position, stability, and dynamical properties are fundamentally governed or dramatically altered by the presence of nonlinear interactions. Unlike in linear topological insulators, where edge states are fixed by linear band topology and protected by symmetries such as chiral, time-reversal, or parity, nonlinearity can (i) tune, (ii) stabilize or destabilize, (iii) induce, or (iv) destroy such edge-localized excitations. These phenomena appear across diverse platforms: photonics, exciton-polariton condensates, electrical circuits, mechanical systems, atomic lattices, and more. The role of nonlinearity is notably pronounced in systems with Kerr nonlinearity, saturable nonlinearities, or nonlinear coupling, and can lead to phenomena such as multistability, nonlinearity-induced phase transitions, and the emergence of new edge or corner states not present in the linear regime.

1. Model Systems and Theoretical Foundations

Nonlinearity-induced edge states arise in a variety of systems modeled by nonlinear extensions of paradigmatic linear topological Hamiltonians. Common frameworks include:

• Nonlinear SSH-like Models: The Su-Schrieffer-Heeger (SSH) model with onsite cubic (Kerr) nonlinearity is frequently studied, both in photonic and mechanical contexts (Ma et al., 2020, Chaunsali et al., 2020, Manda et al., 2021). The governing equations typically read

$$i\dot{a}_j = t\,b_j + t'\,b_{j-1} + |a_j|^2 a_j, \quad i\dot{b}_j = t\,a_j + t'\,a_{j+1} + |b_j|^2 b_j$$

under appropriate open boundary conditions.

• Generalized Aubry-André-Harper (AAH) and Multimode Models: Systems with modulated intersite couplings mapped to AAH models, or higher-dimensional lattices constructed from stacking such chains, support edge and corner states whose properties can be controlled by nonlinearity (Schneider et al., 2023).
• Continuum and Spinor Models: Nonlinear Dirac equations and Gross–Pitaevskii-type equations are essential for systems with spinor degrees of freedom and nonlinear local interactions, especially in the context of photonics and exciton-polariton condensates (Smirnova et al., 2019, Schneider et al., 2023).
• Nonlinear Chern Insulators and Topolectric Circuits: Nonlinear eigenvalue problems, characterized by amplitude-dependent parameters, enable the definition of amplitude-tunable topological invariants (e.g., nonlinear Chern numbers), leading to nonlinearity-induced bulk-boundary correspondence (Sone et al., 2023, Zangeneh-Nejad et al., 2019, Li et al., 13 Dec 2024).

The mathematical analysis leverages:

• Nonlinear dynamical equations (often of discrete Gross–Pitaevskii or nonlinear Schrödinger type),
• Fixed-point and bifurcation analysis for localizing modes,
• Linear stability analysis (spectral, Floquet, Krein signature) for assessing robustness,
• Adiabatic perturbation theory and extension of topological invariants (nonlinear Zak phase, nonlinear Chern number).

2. Regimes of Nonlinearity and Edge State Classification

Nonlinearity can act in several distinct regimes, leading to qualitatively different edge phenomena:

• Weak Nonlinearity (Perturbative Edge Solitons): Edge states bifurcate continuously from linear topological modes. For example, in the SSH model, antisymmetric or symmetric edge solitons retain phase and spatial profiles of their linear ancestors, and their existence persists as long as the nonlinearity does not close the bulk gap or push the eigenvalue into the continuum (Ma et al., 2020, Li et al., 13 Dec 2024). Spectral tuning and multistability appear, with nonlinear-induced eigenfrequency shifts isolated at the system edge (Dobrykh et al., 2018, Schneider et al., 2023).
• Strong Nonlinearity (Nontopological or Novel Localized States): At higher intensities, new families of edge states can emerge that are not adiabatically connected to the linear regime. These “nontopological” edge solitons can have antisymmetric, symmetric, or strongly asymmetric profiles, highly confined at the system boundary, and their existence is entirely dependent on nonlinear self-localization rather than inherited topological protection (Li et al., 13 Dec 2024). In some cases, nonlinearity also enables the formation of in-gap solitons or edge states in otherwise trivial lattices by creating effective internal interfaces (Tuloup et al., 2020).
• Nonlinearity-Induced Topological Transitions: A remarkable effect is the ability of nonlinearity to induce phase transitions between trivial and topological phases. For instance, in 2D nonlinear Chern insulators, the topological invariant (nonlinear Chern number) depends explicitly on the amplitude, and edge states appear or disappear as the excitation strength crosses critical thresholds (Sone et al., 2023). In SOTIs, nonlinear coupling can drive phase transitions yielding quantized changes in polarization and enabling dynamically reconfigurable edge and corner states (Zangeneh-Nejad et al., 2019).
• Nonlinearity-Induced Edge-State Destruction: Strong nonlinearity can also lead to loss of edge-state localization via hybridization with bulk modes, delocalization, or crossover to dynamical instability. For example, Floquet edge states in driven nonlinear systems can undergo bifurcations from long-lived attractors to rapidly decaying repellers, correlated with eigenvalue collisions and pseudo-Hermiticity breaking in the Floquet operator (Mochizuki et al., 2019, Mochizuki et al., 2021).

3. Experimental Realizations and Observations

A range of platforms supports the experimental observation and manipulation of nonlinearity-induced edge states:

• Photonic Systems: Arrays of coupled nonlinear resonators (with varactor diodes to introduce Kerr nonlinearity), laser-written photonic lattices (inducing nonlinear index changes via photorefractive effects), and periodically modulated waveguide arrays for Floquet topological phases (Dobrykh et al., 2018, Mukherjee et al., 2020, Xia et al., 2020, Maczewsky et al., 2020).
• Exciton-Polariton Condensates: Driven-dissipative systems, such as microcavities with DW or SSH-type modulated potentials, explore the synergy of nonlinearity and topology, yielding multistable edge and corner states via polariton–polariton interactions (Schneider et al., 2023).
• Electrical Circuits: Nonlinear trimer circuit lattices with voltage-dependent capacitive diodes realize SSH-like or trimer models, supporting both topological and nontopological edge solitons, and allow direct site-resolved voltage measurements (Li et al., 13 Dec 2024, Zangeneh-Nejad et al., 2019).
• Mechanical Metamaterials and Quasicrystals: Variants incorporating nonlinear elements (e.g., cubic nonlinearity, variable-capacitance diodes in Toda quasicrystals) have been constructed, demonstrating robust edge states and nonlinear phase transitions (Manda et al., 2021, Ezawa, 2022).

Key experimental approaches include quench dynamics (preparing and evolving initial localized states), pump–probe spectroscopy, and spatially resolved measurements of electromagnetic fields, voltages, or other observables. The corresponding observations confirm theoretical predictions regarding edge state localization, multistability, tunability, bifurcations, and amplitude-controlled phase transitions.

4. Stability, Bifurcations, and Robustness

The linear stability of nonlinear edge states governs their observability and functional utility:

• Stability Domains: In weakly nonlinear regimes and for suitable parameter ranges (such as coupling ratios below a critical threshold in SSH-type models), edge solitons are linearly stable and robust to perturbations (Ma et al., 2020). For stiffening nonlinearity in mechanical analogues, regimes exist where high-amplitude edge states remain strictly localized (Chaunsali et al., 2020).
• Onset of Instability: As nonlinearity increases, edge states may lose their spatial segregation due to oscillatory (complex quartet) instabilities, edge–bulk eigenvalue collisions, or delocalization into the bulk, especially near gap edges or when edge-state energy approaches the continuum (Ma et al., 2020, Chaunsali et al., 2020, Mochizuki et al., 2019). Floquet systems are particularly susceptible to additional bifurcations, unique to their time-periodic bandstructure, and pseudo-Hermiticity breaking (Mochizuki et al., 2019, Mochizuki et al., 2021).
• Role of Disorder: Chiral-symmetry-preserving disorder tends to maintain edge state stability and spectral position, while generic disorder can degrade localization and promote instability by shifting eigenvalues or facilitating mixing between edge and bulk components (Chaunsali et al., 2020, Mochizuki et al., 2021).
• Transition to Chaos and Thermalization: In mechanical and similar lattices, even linearly stable edge states may eventually undergo dynamic delocalization and thermalization under sufficiently strong perturbations, with energy equipartition and signatures of underlying symmetry retained in the renormalized dispersion (Manda et al., 2021).

5. Extension of Topological Invariants and Bulk-Boundary Correspondence

Nonlinear physics necessitates revision of several key topological concepts:

• Nonlinear Topological Invariants: Nonlinear Zak phase and nonlinear Chern number generalize their linear counterparts by incorporating amplitude-dependence and state-dependence. The sum of nonlinear Zak phases can recover quantization lost through chiral symmetry breaking (Tuloup et al., 2020). The nonlinear Chern number becomes functionally dependent on mode intensity, meaning the “topological phase diagram” itself depends on excitation amplitude (Sone et al., 2023).
• Amplitude-Dependent Bulk-Boundary Correspondence: The appearance (or vanishing) of robust edge modes is tied not just to the static structure of the bulk bands but also to the excitation amplitude and resultant nonlinear correction terms (Sone et al., 2023, Zangeneh-Nejad et al., 2019).
• Order Parameters in Nonlinear Systems: Quantized local topological indices can be constructed from generalized chiral symmetry or block-off-diagonal structures, allowing explicit identification and protection of edge modes even in nonlinear settings (Jezequel et al., 2021).
• Nonlinear Transitions and Hysteresis: Amplitude-controlled phase transitions induced by nonlinearity (e.g., closure and reopening of gaps with consequent polarization jumps or corner state creation) have no linear analogue and permit reversible, on-demand activation of edge or corner states (Zangeneh-Nejad et al., 2019, Sone et al., 2023).

6. Dynamical and Interaction-Induced Effects

Nonlinearity also directly affects edge state dynamics and their interaction with other modes:

• Interaction with Solitons and Bulk Modes: Gap solitons and edge states coexist and interact in nonlinear Dirac models, allowing for energy transfer, excitation, and nonlinear mixing phenomena (Smirnova et al., 2019). In photonic SSH lattices, nonlinearity unlocks coupling into otherwise protected edge or defect channels, a process unavailable in the linear regime (Xia et al., 2020).
• Nonlinear Interference and Topological Protection: Nonlinearity can challenge the foundational notion of topological protection. For instance, when two wave packets (an edge mode and a bulk mode) overlap with constructive interference in a nonlinear Floquet insulator, the resultant intensity can induce local band inversion and irreversible scattering of the edge mode into the bulk, depending on the relative phase (Michen et al., 2023). This violates the linear superposition principle and introduces a new, genuinely nonlinear mechanism for loss (or recovery) of edge state robustness.
• Floquet Topological Phenomena Under Nonlinearity: In periodically driven systems, the interplay between nonlinear amplitude-induced band renormalization and time-periodic modulation leads to unique bifurcation diagrams, stability boundaries, and transitions between long-lived and short-lived edge modes, often controlled by quantum or classical Krein signatures (Mochizuki et al., 2021, Mochizuki et al., 2019).

7. Applications, Implications, and Future Directions

The ability to engineer, control, and exploit nonlinearity-induced topological edge states opens several research and application avenues:

• Reconfigurable Photonic and Electronic Devices: Dynamic tunability of edge and corner states with intensity, multistability, and on-demand phase transitions are promising for high-fidelity switching, logic, and routing (Zangeneh-Nejad et al., 2019, Maczewsky et al., 2020).
• Higher-Order Topological Phases and Corner Modes: Stacking lower-dimensional topological systems in the nonlinear regime promotes emergent higher-order topological phenomena, including robust, localized corner states in 2D and beyond (Schneider et al., 2023, Zangeneh-Nejad et al., 2019).
• Extension to Exotic Platforms: The principles demonstrated in electrical circuits are readily transferrable to photonic, mechanical, acoustic, and cold-atom systems, broadening the experimental capability and potential impact (Li et al., 13 Dec 2024, Ezawa, 2022).
• Theoretical Classification and Nonlinear Topological Matter: The field is moving toward a generalized theoretical framework where topological classification naturally incorporates nonlinear interactions, amplitude dependence of invariants, and new dynamical bulk-boundary correspondences. This encompasses both “inherited” (from linear topology) and “emergent” (genuinely nonlinear) edge phenomena (Sone et al., 2023, Jezequel et al., 2021).
• Robust Quantum and Classical Information Processing: Nonlinearity-induced edge modes exhibiting strong localization, switching, and stability properties may be harnessed for robust information transfer, low dissipation devices, and robust signal processing immune to certain kinds of disorder and defects.

In summary, nonlinearity profoundly enriches the physics of topological edge states—enabling spectral tuning, phase transitions, new forms of localization, and novel dynamical behaviors. The experimental maturity in realizing and probing these phenomena across multiple platforms signals a broad scope for applications and a need for systematic theoretical extension of topological concepts into the nonlinear, amplitude-dependent regime.