Topology plays a crucial role in understanding the robust properties of condensed matter systems, particularly their immunity to disorder. The quantum Hall effect, characterized by the Chern number, exemplifies this, where the bulk-boundary correspondence ensures the existence of gapless boundary modes in systems with non-zero topological invariants. Recent research has hinted at extending the concept of topology to nonlinear systems; however, a comprehensive understanding of nonlinear topological invariants has been lacking. This paper addresses this gap by proposing a nonlinear extension of the Chern number and investigating its associated bulk-boundary correspondence, particularly focusing on the nonlinearity-induced topological phase transitions that occur beyond the weakly nonlinear regime. The study explores the amplitude dependence of these transitions and provides analytical and numerical confirmations of the nonlinear bulk-boundary correspondence. This research promises to broaden our understanding of topological phenomena in a wide range of systems where nonlinear dynamics are prevalent, including classical and interacting bosonic systems.

While band topology is well-established in linear systems, the ubiquitous presence of nonlinear dynamics in both classical and interacting bosonic systems necessitates a framework that accounts for these effects. Previous research has explored nonlinear effects on topological edge modes, revealing unique phenomena intertwined with solitons and synchronization. Studies on one-dimensional systems have shown nonlinearity-induced topological phase transitions, demonstrating that the existence of topological edge modes can be amplitude-dependent. However, extending topological invariants to nonlinear systems, which lack conventional band structures, has remained a significant challenge. Prior research in two-dimensional nonlinear systems has been limited, leaving a gap in understanding nonlinear topology in these systems. This paper aims to address this gap by establishing a robust framework for understanding two-dimensional nonlinear topological systems.

The authors introduce a nonlinear extension of the eigenvalue problem to define the nonlinear Chern number for two-dimensional systems. They consider the general nonlinear dynamics described by the equation i∂_tΨ(r) = f(Ψ;r), where Ψ(r) is the state variable and f(⋅;r) is a nonlinear function. The U(1) and translational symmetries are imposed on the nonlinear equation. The nonlinear eigenvector and eigenvalue are defined as the state vector components Ψ_j(r) and the constant E that satisfy f_j(Ψ;r) = Eψ_j(r). This nonlinear eigenequation forms the basis for extending the Chern number to nonlinear systems. The Bloch ansatz, Ψ(r) = e^{i**k**⋅**r**}u(**k**), is used to analyze the bulk eigenvectors in lattice systems, allowing for the definition of the nonlinear Chern number, C_N(ω), which depends on the amplitude ω = Σ_i|φ_i(**k**)|. The paper then presents a detailed derivation and analysis of a nonlinear extension of the Qi-Wu-Zhang (QWZ) model, a prototypical topological insulator model. The authors use the Bloch ansatz to obtain the wavenumber-space description of this model and derive exact bulk solutions. From these exact solutions, the nonlinear Chern number is calculated, revealing the phase diagram and the amplitude dependence of the topological phase transitions. Numerical simulations of the nonlinear QWZ model’s dynamics, under both open and periodic boundary conditions, are used to confirm the bulk-boundary correspondence, particularly the relationship between the nonlinear Chern number and the existence of localized edge states. A continuum system analysis using the nonlinear Dirac Hamiltonian, derived as an effective theory from the nonlinear QWZ model, further strengthens the theoretical foundation. The paper also presents an observation protocol of the edge modes via quench dynamics, simulating the time evolution of the system and illustrating the topological phase transition based on the long-term amplitude at edge sites.

The paper's key findings include the successful definition and characterization of a nonlinear Chern number, a crucial step in extending topological concepts to nonlinear systems. The authors rigorously demonstrate the existence of a bulk-boundary correspondence for this nonlinear Chern number, linking the non-zero Chern number to the existence of localized edge modes, even under strong nonlinearity. A significant contribution is the discovery and detailed analysis of nonlinearity-induced topological phase transitions. These transitions depend crucially on the amplitude of the oscillatory modes, a feature absent in linear systems. The authors provide analytical calculations of the nonlinear Chern number and phase diagrams for a proposed nonlinear QWZ model, supplemented by numerical simulations that confirm the theoretical predictions. The bulk-boundary correspondence is validated through both numerical simulations of the nonlinear QWZ model's dynamics and analytical solutions for the nonlinear Dirac Hamiltonian (a low-energy effective theory for the QWZ model). The results show that the nonlinear Chern number accurately predicts the presence or absence of edge-localized modes, even in strong nonlinearity regimes. The authors also demonstrate an experimental observation protocol employing quench dynamics, providing a practical pathway for detecting the topological phases and transitions in experiments. They propose a feasible experimental setup utilizing topological photonics and the Kerr nonlinearity, leveraging the common nature of Kerr nonlinearity in photonic systems.

The findings of this paper significantly advance the understanding of topological phenomena in nonlinear systems. The introduction of the nonlinear Chern number provides a powerful tool for classifying and predicting topological phases in systems where nonlinear effects are dominant. The demonstration of the bulk-boundary correspondence, even in strongly nonlinear regimes, solidifies the connection between bulk topological properties and boundary phenomena. The discovery of nonlinearity-induced topological phase transitions highlights the unique and rich behavior that can arise from nonlinear interactions, going beyond the limitations of linear topological theories. The proposed experimental protocol based on quench dynamics offers a readily implementable approach to verify the theoretical predictions and explore nonlinear topological materials experimentally. The applicability of the findings is broad, potentially impacting various fields where nonlinear dynamics play a central role, including photonics, ultracold atoms, and electrical circuits.

This paper successfully extends the concept of topological invariants to nonlinear systems by introducing the nonlinear Chern number. The authors demonstrate the existence of a bulk-boundary correspondence and nonlinearity-induced topological phase transitions in two-dimensional systems, validated by analytical calculations and numerical simulations. The proposed experimental protocol using quench dynamics offers a practical approach to observe these phenomena. This work opens exciting new avenues for researching and classifying nonlinear topological materials and offers potential for novel applications in various technological domains. Future research could explore stronger nonlinear regimes, investigate the stability of Bloch-ansatz states, and delve into the connection between nonlinear topology and many-body quantum physics.

While the paper presents a comprehensive framework, some limitations exist. The analysis focuses primarily on weakly and moderately nonlinear systems where the Bloch ansatz remains a valid approximation. The behavior in strongly nonlinear regimes, where bulk-localized modes and topological edge solitons might emerge, requires further investigation. The theoretical framework and numerical simulations primarily focus on specific model systems. Further research is needed to determine the extent to which the findings generalize to broader classes of nonlinear systems and different types of nonlinear interactions. The experimental protocol proposed is a theoretical suggestion; further experimental work is needed to verify the predictions.