Nonlinear Edge States in Topological Systems

• Nonlinear edge states are localized modes at lattice boundaries whose properties vary with amplitude and nonlinear interactions, leading to frequency shifts and self-induced transitions.
• They arise from mechanisms like Kerr nonlinearity and cubic spring effects, enabling adjustable energy spectra, multistability, and robustness under symmetry constraints.
• These states facilitate practical applications in optical switching and signal routing, with experimental realizations in photonic, mechanical, and quantum platforms.

Nonlinear edge states are spatially localized modes at the boundaries of discrete or continuum lattices whose properties—existence, spectrum, spatial profile, stability—crucially depend on the interplay between topological band structure and intrinsic nonlinear interactions. Unlike their linear counterparts, which are fully determined by bulk invariants and lattice symmetries, nonlinear edge states exhibit amplitude-dependent energetics, localization, multistability, and even self-induced transitions. These phenomena have been realized in photonic, phononic, polaritonic, atomic, mechanical, and electronic platforms, and their description unifies nonlinear lattice theory, topological phase analysis, and nonlinear dynamics.

1. Theoretical Foundations: Nonlinear Extensions of Topological Models

The occurrence of edge states in topological lattices is classically exemplified by the paradigmatic Su-Schrieffer-Heeger (SSH) chain, Aubry-André-Harper (AAH) models, and their 2D generalizations (e.g., honeycomb, Kagome, Floquet-driven architectures). Nonlinearity enters these frameworks in several forms:

• On-site Kerr (cubic) interaction in photonic, mechanical, or exciton-polariton systems: e.g.,

$$i\,\frac{d a_n}{dt} = \omega_0 a_n + t_{n,n-1} a_{n-1} + t_{n,n+1} a_{n+1} + \gamma |a_n|^2 a_n + F_n(t),$$

with $\gamma>0$ corresponding to self-focusing and $\gamma<0$ to self-defocusing.

• Distributed cubic spring nonlinearities in elastic and mechanical lattices:

$$m\ddot u_n + k_{n}(u_{n} - u_{n-1}) + k_{n+1}(u_{n} - u_{n+1}) + \alpha [ (u_n - u_{n-1})^3 + (u_n - u_{n+1})^3 ] = 0.$$

• Nonlinear spatially modulated couplings for engineered self-induced transitions, e.g.,

$$K_{n,n+1} = K_0 + \alpha (|a_{1,n+1}|^2 + |a_{2,n}|^2)$$

in staggered SSH chains.

In these systems, nonlinearity produces amplitude-dependent shifts in the effective parameters dictating the topological phase and edge state characteristics.

2. Nonlinear Edge-State Continuations: Amplitude Tuning and Self-Induced Topological Transitions

In weakly nonlinear regimes, continuation from linear edge modes governs the behavior of nonlinear edge states:

• Frequency/energy shift: Nonlinearity alters the mode's frequency. For Kerr or cubic nonlinearity, this is typically quadratic in amplitude: $\Delta \omega \sim \gamma |A|^2$.
• Self-induced topological transition: In models such as the nonlinear SSH chain, above a threshold intensity $I_{\text{th}} = (v - K_0)/\alpha$, the system can undergo a transition to a topological phase—closing and reopening its spectral gap, with concomitant appearance of nonlinear edge states (Hadad et al., 2015).
• Nonlinear plateau: Nonlinear edge states may decay into a finite plateau within the bulk rather than strictly to zero, reflecting local gap closure by nonlinearity (e.g., Hadad et al. (Hadad et al., 2015)).

The critical amplitude where an edge state collides with the bulk band, resulting in delocalization, is determined by nonlinear eigenvalue continuation: $$\omega^{\text{edge}}(A_c) = \omega^{\text{bulk}}(\mu^*, A_c), \quad \partial_\mu \omega^{\text{bulk}}(\mu^*, A_c) = 0,$$ where $A_c$ is the critical amplitude and $\mu^*$ the relevant bulk wavenumber (Rosa et al., 2022, Pal et al., 2017).

3. Stability, Bifurcation, and the Role of Symmetry

Stability of nonlinear edge states encompasses spectral (Floquet) and dynamical criteria:

• Weakly nonlinear limit: Stability can be inferred from the eigenvalues of the linearization (Bogoliubov–de Gennes analysis). For on-site Kerr nonlinearity, edge states become unstable as their energy approaches the bulk band—oscillatory (complex quartet) or modulational instabilities appear (Ma et al., 2020, Manda et al., 2021).
• Floquet (time-periodic) systems: Nonlinear edge states in periodically driven lattices display unique bifurcation phenomena. For example, a threshold nonlinearity $g_c$ produces a transition from a stable attractor edge state to an unstable repeller, associated with the collision of Floquet multipliers on the unit circle—the so-called "Floquet bifurcation" (Mochizuki et al., 2019, Mochizuki et al., 2021).
• Chiral symmetry and topological protection: Nonlinear edge states retain protection when the nonlinearity respects generalized chiral (sublattice) symmetry; this prevents their drift in energy and ensures the persistence of zero-energy midgap localization (Jezequel et al., 2021). In contrast, generic on-site Kerr interactions can destabilize edge modes above critical amplitude.
• Disorder robustness: Chiral disorder preserves the midgap frequency and stability of nonlinear edge states, whereas generic disorder degrades protection but may leave high-amplitude, stiffening-induced stable regimes intact (Chaunsali et al., 2020).

4. Experimental Realizations and Prototypical Systems

Nonlinear edge states have been investigated and observed in a variety of experimental platforms:

• Electromagnetic/photonic lattices (varactors, SRRs, waveguides): Intensities above threshold self-tune the edge resonance via nonlinearity, with observed spectral shifts up to tens of MHz and gradual edge-to-bulk delocalization at high pump (Dobrykh et al., 2018).
• Photonic graphene with electromagnetically-induced transparency (EIT): Discrete nonlinear Schrödinger dynamics with local Kerr and gain allow realization of edge solitons at the zigzag edge, which are robust to engineered defects (Zhang et al., 2019).
• Floquet topological insulator waveguide arrays: Power-tuned Kerr nonlinearity produces self-localized, unidirectional edge wavepackets (soliton-like states) with distinct propagation properties across a threshold in input power (Mukherjee et al., 2020).
• Polaritonic condensates in microcavity double-wave (DW) chains: Driven-dissipative Gross–Pitaevskii equations predict multistability between different edge and bulk-localized condensate modes under resonant pumping—facilitating local control of topological states beyond linear band theory (Schneider et al., 2023).
• Mechanical/phononic lattices: Alternating mass–spring chains and hexagonal lattices with cubic spring nonlinearities demonstrate amplitude-dependent tuning of gap modes; softening nonlinearity can induce delocalization, while hardening shifts the mode deeper into the gap (Pal et al., 2017, Rosa et al., 2022).
• Quantum walk and driven atomic lattice realizations: Nonlinear quantum walks on synthetic lattices, Floquet-engineered cold atom platforms, and photonic lattices simulate topological edge-state bifurcations, lifetimes, and crossover from stabilization to chaos (Mochizuki et al., 2019, Mochizuki et al., 2021).

5. Nonlinear Edge States Beyond Single-Particle Physics: Doublons and Self-Trapping

Spatial modulation of the nonlinearity or interaction strength can support bound edge modes of pairs of particles (doublons), even in the absence of single-particle topology:

• Bound photon pair edge states: In a dimerized Bose–Hubbard chain with alternating $U_j$, one obtains doublon bands and edge-localized doublon states. The mechanism is nonlinear self-localization, with the critical condition $U/J>1$ set by perturbative detuning (Lyubarov et al., 2019). These edge-bound doublons, in contrast to single-particle edge states, do not derive from bulk Zak phase but from graph connectivity in quasi-particle configuration space.
• Breakdown of conventional topological invariants: Standard single-particle Zak phase fails to predict the presence of doublon edge states in nonlinear systems; instead, effective defect criteria based on many-body virtual processes must be employed (Gorlach et al., 2016).
• Coexistence with breathers: In nonlinear modulated chains, edge states and discrete breathers (bulk-localized, amplitude-induced states) can coexist, with transitions between the two observed as amplitude grows (Rosa et al., 2022).

6. Multidimensional Generalizations and Higher-Order Topology

The interplay of nonlinearity and higher-dimensional lattice structure yields complex localized states:

• 2D lattices supporting edge and corner states: Stacking modulated 1D chains (e.g., DW chains) in a second spatial direction results in multi-wave 2D arrays exhibiting both edge and higher-order (corner) modes. Kerr nonlinearity leads to multistability: for given pump strength/resonant condition, different edge or corner states may be stable and switchable (Schneider et al., 2023).
• Topological valley and chiral edge solitons: In photonic lattices with type-II Dirac cones, nonlinear valley-Hall edge solitons emerge, supported by positive group-velocity dispersion in the edge band and gap opening by inversion symmetry breaking (Zhong et al., 2020). Such states combine symmetry-protected boundary transport and robust nonlinear localization.

7. Practical Consequences, Limitations, and Outlook

The unique qualities of nonlinear edge states interface topological robustness with strong intensity-dependence, enabling functionalities not possible in linear systems:

• Amplitude-controlled routing and switching: Nonlinear edge states can act as threshold-based switches for energy or information, with their existence or properties turning on/off with the local field amplitude (Dobrykh et al., 2018, Hadad et al., 2015, Pal et al., 2017).
• Multistability and self-induced selection: Nonlinearity can support multiple coexisting edge/corner states, or induce transitions between localization at different sites or regions for the same parameter set (pump, detuning) (Schneider et al., 2023).
• Interaction-enabled quantum states: Strongly-correlated bound edge states (e.g., photon pair doublons) and their experimental signatures are directly attributable to engineered spatial modulation of nonlinear interactions (Gorlach et al., 2016, Lyubarov et al., 2019).
• Robustness and fragility: Topologically protected edge modes can lose robustness under strong nonlinearity or disorder unless symmetry conditions are respected. In Floquet and dissipative systems, edge state lifetimes become finite and tunable (Mochizuki et al., 2021).
• Design principles: Achieving robust nonlinear edge localization requires careful tuning of bulk and nonlinear parameters: gap size, coupling strength, nonlinearity sign/amplitude, and symmetry-engineering to prevent destabilizing bifurcations or hybridization with bulk modes.

These insights collectively delineate the emerging class of nonlinear topological phases, their critical thresholds, multistability, and routes to localization, delocalization, and dynamical transitions. Systematic analytical frameworks now exist to estimate edge state thresholds, frequency shifts, and finite-size scaling in 1D, 2D, and multi-chain arrangements (Song et al., 11 Nov 2025), forming a quantitative foundation for designing topologically robust nonlinear waveguides, logic circuits, and dynamical devices across photonic, mechanical, and quantum engineered platforms.