Main

Topology is utilized to realize the robust properties of materials that are immune to disorders1,2. A prototypical example of topological materials is the quantum Hall effect3,4, which was discovered in a two-dimensional semiconductor under a magnetic field. In such a two-dimensional system, the Chern number characterizes the topology of the band structure and the corresponding gapless boundary modes. This bulk–boundary correspondence lies at the heart of the robustness of topological devices utilizing boundary modes. Recent studies have also explored topological phenomena in a variety of platforms, such as photonics5, electrical circuits6, ultracold atoms7, fluids8 and mechanical lattices9.

Although band topology has been well explored in linear systems, nonlinear dynamics is ubiquitous in classical10,11,12,13,14,15 and interacting bosonic systems16,17. For example, nonlinear interactions can naturally emerge in the mean-field analysis of bosonic many-body systems, such as the Gross–Pitaevskii equations. Recent research has also studied the nonlinear effects on topological edge modes11,12,18,19,20,21,22,23,24,25,26,27,28 and revealed unique topological phenomena intertwined with solitons29,30,31,32,33,34,35 and synchronization36,37,38. Nonlinearity can further modify the conventional notion of topological phases; it has been found that one-dimensional systems can exhibit nonlinearity-induced topological phase transitions, where the existence of topological edge modes depends on the amplitude of the oscillatory modes39,40,41,42,43. Although these previous studies have indicated the existence of topological edge modes in nonlinear systems, one cannot straightforwardly extend the topological invariants to nonlinear systems because they have no band structures, at least in the conventional sense. In addition, nonlinear topology in two-dimensional systems18,30,31,35 is much less understood than that in one-dimensional systems.

In this paper, we introduce the notion of the nonlinear Chern number and reveal its relation to the bulk–boundary correspondence. To define the nonlinear Chern number of two-dimensional systems, we consider the nonlinear extension of the eigenvalue problem41,43,44 and make an analogy to band structures. Although it is not obvious that the nonlinear eigenvalue problem elucidates the bulk–boundary correspondence in nonlinear topological insulators, we theoretically prove the bulk–boundary correspondence of the nonlinear Chern number in weakly nonlinear systems. Furthermore, in stronger nonlinear regimes where nonlinear terms are larger compared with the linear bandgap, we find that the nonlinearity-induced topological phase transition can occur in two-dimensional systems (Fig. 1). We analytically show the nonlinear bulk–boundary correspondence, which states that the non-zero nonlinear Chern number predicts the existence of localized edge modes in semi-infinite systems even under strong nonlinearity. Since the nonlinearity-induced topological phases are disconnected from the linear limits under adiabatic deformations, our results show the existence of genuinely nonlinear topological phases.

Fig. 1: Schematic of the nonlinear Chern insulators and the nonlinearity-induced topological phase transition.
figure 1

Although the topology of a non-interacting linear system can be characterized by the Chern number that is computed from its eigenvectors, the topology of a nonlinear system is classified by the nonlinear Chern number, which utilizes the nonlinear extension of the eigenequation. In weakly nonlinear regions (that is, small amplitude), the nonlinear Chern number predicts the existence of edge modes corresponding to those in linear systems. Specifically, when nonlinear systems exhibit edge-localized steady states, both nonlinear and linear Chern numbers are non-zero (top). If we inject higher energy into the system and consider the eigenmodes with large amplitudes, the nonlinear band structure can become gapless. At such a gapless point, a nonlinearity-induced topological phase transition can occur, where topological boundary modes appear with the non-zero nonlinear Chern number (bottom). The nonlinearity-induced topological phases exhibit boundary modes that cannot be predicted from the linear Chern number. Therefore, such topological phases are genuinely unique to nonlinear systems.

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The scope of this paper applies to a broad class of two-dimensional systems with U(1) gauge and spatial translation symmetries, which can be realized in a variety of experimental setups. Similar to how the Thouless–Kohmoto–Nightingale–den Nijs formula4 triggered the research of a variety of topological materials, the nonlinear Chern number is expected to open up the research stream of nonlinear topological materials including their systematic classification. From the experiment point of view, one can realize nonlinear Chern insulators with the U(1)-gauge symmetry in, for example, photonics11,12,18,20,22,24,27,30,31,32,33,34, ultracold atoms16,17,27 and electrical circuits6,36,37, where both linear band topology and nonlinear effects have been investigated. In particular, since the Kerr nonlinearity10 is fairly common in photonic systems, it should be possible to extend the current topological photonic devices to nonlinear ones.

Nonlinear eigenvalue problem and nonlinear Chern number

We consider the nonlinear extension of the eigenvalue equations41,43,44 and define the nonlinear Chern number by using nonlinear eigenvectors. We start from the general nonlinear dynamics:

$${\rm{i}}{\partial }_{t}{{\varPsi}}_{j}({{{\bf{r}}}})={f}_{j}({\bf{\Psi}} ;{{{\bf{r}}}}),$$
(1)

where Ψj(r) is the state variable and fj(⋅; r) is a nonlinear function of the state vector Ψ. In lattice systems, r denotes a representative point in each unit cell of the lattice (Fig. 2a). Then, j represents the internal degrees of freedom that include, for example, sublattices and effective spin degrees of freedom. When we consider continuum systems, r should simply represent the location and j corresponds to the internal degrees of freedom such as spins. For example, the Gross–Pitaevskii equation in the continuous space is given by f(Ψ; r) = −∇2Ψ(r)/(2m) + VΨ(r) + (4πa/m)∣Ψ(r)∣2Ψ(r), where V is a potential and m and a are the mass and scattering length, respectively. The nonlinear function f(⋅; r) depends on Ψ(r) and its derivative, and has no internal degrees of freedom. Since the quantum Hall system has U(1) and translation symmetries, we impose them on the nonlinear equation to study the analogy of such a prototypical topological insulator. Concretely, the U(1) symmetry is represented as fj(eiθΨ; r) = eiθfj(Ψ; r), which is satisfied in, for example, the Kerr-like nonlinearity κ∣Ψj(r)∣2Ψj(r) (ref. 10). The translational symmetry in lattice systems is defined as fj(Ψ; r + a) = fj(Φ; r), where a is a lattice vector and Φ is the translated state variable: Φj(r) = Ψj(r + a). The translational symmetry in continuum systems is also defined in the same equation, whereas fj(Φ; r) still remains dependent on r due to, for example, the periodic potential. We also focus on conservative dynamics analogous to Hermitian Hamiltonians where the sum of squared amplitudes ∑j,r∣Ψj(r)∣2 is preserved.

Fig. 2: Phase diagram and dynamics of the nonlinear QWZ model.
figure 2

a, Schematic of the nonlinear QWZ model. The model has two sublattices (black circles) at each lattice point encircled by the blue ellipse. The green lines represent the linear couplings. We use the notation Ψi(x, y) to represent the state variable at each sublattice, where (x, y) is the location of the representative point of each lattice point denoted by the red cross. b, Analytical demonstration of the phase diagram of the nonlinear QWZ model. The horizontal axis represents the parameter of the mass term and the vertical axis corresponds to the strength of nonlinearity. The colour of each separated region represents the difference in the nonlinear Chern number. c, Numerical demonstration of the absence of edge modes in the topologically trivial parameter region. We simulate the dynamics of the prototypical model of a nonlinear Chern insulator starting from an initial state localized at the left edge. We impose the open boundary condition in the x direction and the periodic boundary condition in the y direction. The figure shows the snapshot at t = 1. The colour shows the absolute value of the components of the state vector at each site. The parameters used are u = 3 and κ = 0.1, and the average amplitude is w = 0.1, which corresponds to the red square in b. d, Numerical demonstration of the existence of the long-lived localized state in the weakly nonlinear topological insulator. The figure shows the snapshot of the simulation at t = 1. The sites at the left edge show large amplitudes, which indicates the existence of the edge-localized state. The parameters used are u = −1 and κ = 0.1, and the average amplitude is w = 0.1, which corresponds to the blue circle in b.

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In accordance with the nonlinear dynamical system in equation (1), the nonlinear eigenvector and eigenvalue are defined as the state vector Ψ with components Ψj(r) and the constant E that satisfy

$${f}_{j}({\bf{\Psi}} ;{{{\bf{r}}}})=E{{\varPsi}}_{j}({{{\bf{r}}}}).$$
(2)

We term the equation as a nonlinear eigenequation and analyse its bulk–boundary correspondence below. We note that we can regard the nonlinear eigenvector as a periodically oscillating steady state Ψj(r; t) = e−iEtΨj(r) of the nonlinear system when the eigenvalue is real.

To extend the Chern number to nonlinear systems, we introduce the eigenvalue problems in the wavenumber space, which is analogous to the linear eigenequation of the Bloch Hamiltonian. In a lattice system with translation symmetries, we assume an ansatz state41 that we name the Bloch ansatz: Ψj(r) = eik·rψj(k). In linear systems, the Bloch theorem guarantees that every eigenvector is given by the form of the Bloch ansatz. On the other hand, in nonlinear systems, there can be nonlinear eigenvectors out of the description of the Bloch ansatz, including bulk-localized ones. Despite the existence of such localized modes, here we only focus on nonlinear bulk eigenvectors described by the Bloch ansatz and show that even such periodic bulk solutions can exhibit topological phenomena inherent to nonlinear systems, namely, the nonlinearity-induced topological phase transition. Under this ansatz and U(1) symmetry, one can rewrite the nonlinear eigenequation as fj(k, ψ(k)) = E(k)ψj(k) parametrized by k (Supplementary Note 1 provides a detailed derivation). We note that in finite periodic systems, the Bloch-ansatz solution is still an eigenvector, whereas it can be unstable under superposition with other eigenstates.

To capture the nonlinearity-induced topological phase transition depending on amplitudes, we focus on a special solution of the nonlinear eigenvector at each k whose sum of the squared amplitudes ∑j∣ψj(k)∣2 = w is fixed independently of wavenumber k. We note that the assumption of such fixed-amplitude Bloch-ansatz solutions is consistent with the perturbative calculation of the nonlinear eigenvectors (Methods). By using fixed-amplitude nonlinear eigenvectors, we define the nonlinear Chern number as

$${C}_{{{{\rm{NL}}}}}(w)=\frac{1}{2\uppi {\rm{i}}}\int\,{\nabla }_{{{{\bf{k}}}}}\times \left(\frac{1}{\sqrt{w}}\left\langle \psi ({{{\bf{k}}}})\right\vert \right){\nabla }_{{{{\bf{k}}}}}\left(\frac{1}{\sqrt{w}}\left\vert \psi ({{{\bf{k}}}})\right\rangle \right){{\rm{d}}}^{2}{{{\bf{k}}}}.$$
(3)

We note that this definition is reduced to the conventional linear Chern number if f defined in equation (2) is a linear function. It is also noteworthy that since special solutions of nonlinear eigenvectors should exist at arbitrary w in ordinary nonlinear systems, we can define the nonlinear Chern number at any positive w except for gap-closing points. One can prove that the nonlinear Chern number is an integer by embedding the nonlinear eigenvectors into an eigenspace of a linear Bloch Hamiltonian (Supplementary Note 2). The main purpose of this paper is to show the bulk–boundary correspondence for this nonlinear Chern number. Since the eigenvector can be changed by the amplitude w, the nonlinear Chern number also depends on w (Fig. 1). Therefore, the nonlinear Chern number can predict the nonlinearity-induced topological phase transition by the change in amplitude of nonlinear systems, which is absent in linear systems22,25,39,40,41,42,43. Although here we define the nonlinear Chern number in lattice systems, we can also define that in continuum systems (Methods).

Since the parameter w is unique to nonlinear systems, it is non-trivial how to relate the amplitude w under the periodic boundary condition to that under the open boundary condition to formulate the bulk–boundary correspondence. There are two possible choices: one can equate w under the periodic boundary condition to either the average amplitude wave ≡ ∑r,j∣Ψj(r)∣2/L or the edge amplitude wedge ≡ ∑j∣Ψj(redge)∣2 in the model under the open boundary condition, the latter of which is used to calculate the corresponding edge modes. As shown later, both definitions can predict the topological phase transition in the continuum limit.

Previous research on one-dimensional nonlinear systems43 has indicated the appearance of higher-order correction terms in the topological invariant due to multifrequency effects in nonlinear eigenvectors. However, under the U(1) symmetry, higher-order terms are excluded from the nonlinear Chern number. We also note that the Bloch ansatz does not describe bulk-localized solutions that can be obtained in strongly nonlinear systems, whereas the ansatz still captures nonlinearity-induced topological phenomena. Therefore, we mainly focus on weakly and more strongly nonlinear systems where the nonlinear terms are smaller than the linear terms (Supplementary Note 3 shows the Bloch-wave-like solutions of the bulk modes in this parameter region).

Nonlinear Chern number calculated from exact solutions

To investigate the bulk–boundary correspondence, that is, the correspondence between the non-zero nonlinear Chern number and the existence of the edge-localized steady state, we propose and analyse the nonlinear extension of the Qi–Wu–Zhang (QWZ) model45 (Methods provides the real-space description of the model). By using the Bloch ansatz, we rigorously obtain its wavenumber-space description:

$$\begin{array}{l}f({{{\bf{k}}}},{\bf{\uppsi}} ({{{\bf{k}}}}))\\=\left(\begin{array}{cc}u+\kappa w+\cos {k}_{x}+\cos {k}_{y}&\sin {k}_{x}+{\rm{i}}\sin {k}_{y}\\ \sin {k}_{x}-{\rm{i}}\sin {k}_{y}&-(u+\kappa w+\cos {k}_{x}+\cos {k}_{y})\end{array}\right)\left(\begin{array}{c}{\psi }_{1}({{{\bf{k}}}})\\ {\psi }_{2}({{{\bf{k}}}})\end{array}\right),\end{array}$$
(4)

where w is the squared amplitude w = ∣ψ1(k)∣2 + ∣ψ2(k)∣2; u and κ are dimensionless parameters of the linear and nonlinear mass terms, respectively. Here we introduce the staggered Kerr-like nonlinearity ±κw to the linear Chern insulator model45.

To calculate the nonlinear Chern number, we focus on special solutions where the squared amplitude w is fixed independent of wavenumber k. Then, we can regard equation (4) as a linear equation and thus can analytically obtain the exact bulk solutions of the nonlinear eigenvalues and eigenvectors as the linear QWZ model. Using the exactly obtained nonlinear eigenvectors, we calculate the nonlinear Chern number and obtain the phase diagram shown in Fig. 2b (Methods shows the detailed calculation and Supplementary Note 6 provides the numerical confirmation). The amplitude dependence of the nonlinear Chern number indicates the existence of the nonlinearity-induced topological phase transition in the nonlinear QWZ model. We note that since we calculate the nonlinear Chern number from the exact nonlinear eigenvectors, our result shows the existence of the nonlinearity-induced topological phase transition without any approximations. Such an analytical demonstration of the nonlinearity-induced topological phase transition is achieved by considering nonlinear equations with the form

$${\rm{i}}{\partial }_{t}{\psi }_{n}\left({{{\bf{k}}}};t\right)=\mathop{\sum}\limits_{m}{f}_{nm}\left(\mathop{\sum}\limits_{j}{\left\vert {\psi }_{j}\left({{{\bf{k}}}};t\right)\right\vert }^{2}\right){\psi }_{m}\left({{{\bf{k}}}};t\right),$$
(5)

where ψj and k are the state variables and wavenumber, respectively. Under the existence of a more general nonlinear term, we may also define and calculate the nonlinear Chern number by appropriately defining w (Supplementary Notes 4 and 5).

Bulk–boundary correspondence in weakly nonlinear systems

We first numerically confirm the bulk–boundary correspondence in weakly nonlinear systems. We simulate the dynamics of the nonlinear QWZ model (equation (4)) with weak nonlinearity, where the nonlinear Chern number is the same as that in the linear limit κw → 0. In the topological phase (CNL = ±1; Fig. 2d), we find a long-lived localized state that corresponds to a topological edge mode in the QWZ model. Meanwhile, in the case of CNL = 0 (Fig. 2c), the edge-localized initial state is spread to the bulk, which indicates the absence of edge modes. We also confirm the bulk–boundary correspondence from the perspective of the nonlinear band structure (Fig. 3; Methods provides the numerical method), which implies the utility of the nonlinear band structure to detect topological edge modes.

Fig. 3: Nonlinear band structure and edge modes of the weakly nonlinear QWZ model.
figure 3

a, We numerically calculate the nonlinear band structure of the topologically trivial system under the open boundary condition in the x direction and the periodic boundary condition in the y direction. In the data shown in a and b, we relate the average amplitude wave to the amplitude of the bulk modes w. One can confirm the absence of gapless modes. The parameters used are u = 3, κ = 0.1 and w = 0.1. b, Nonlinear band structure of the topologically non-trivial system is numerically calculated. There are gapless modes that connect the upper and lower bulk bands. The parameters used are u = −1, κ = 0.1 and w = 0.1. c, Spatial distribution of the gapless mode is presented. The eigenvalue of the localized mode corresponds to the red circle in the band structure in b.

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In fact, the bulk–boundary correspondence between the nonlinear Chern number and the gapless edge modes can be established in general weakly nonlinear systems. We mathematically show the bulk–boundary correspondence under weak nonlinearity compared with the linear bandgap. Methods and Supplementary Note 7 describe the details of the theorem and its proof.

Nonlinearity-induced topological phase transition

We next show that nonlinearity-induced topological phase transitions occur in the stronger nonlinear regime, where the nonlinear Chern number becomes nonzero and topological edge modes appear at a critical amplitude. First, we consider continuum systems and analytically show such nonlinearity-induced topological phase transitions and the bulk–boundary correspondence under stronger nonlinearity. Specifically, we derive the effective theory of the low-energy dispersion of the nonlinear QWZ model as

$${\rm{i}}{\partial }_{t}{\boldsymbol{\Psi}} ({{{\bf{r}}}})=\hat{H}({\boldsymbol{\Psi}} ({{{\bf{r}}}})){\boldsymbol{\Psi}} ({{{\bf{r}}}}),$$
(6)
$$\begin{array}{l}{\hat{H}}({\bf{\Psi}} ({{{\bf{r}}}}))=\\ \left(\begin{array}{cc}m+\kappa\left(| {{\varPsi}}_{1}({{{\bf{r}}}}){| }^{2}+| {{\varPsi}}_{2}({{{\bf{r}}}}){| }^{2}\right)&-{\rm{i}}{\partial }_{x}+{\partial }_{y}\\ -{\rm{i}}{\partial }_{x}-{\partial }_{y}&-m-\kappa\left(| {{\varPsi}}_{1}({{{\bf{r}}}}){| }^{2}+| {{\varPsi}}_{2}({{{\bf{r}}}}){| }^{2}\right)\end{array}\right),\end{array}$$
(7)

where m = u + 2 and Ψ(x, y) = (Ψ1(x, y), Ψ2(x, y))T is the state-vector function at location (x, y) (Methods and Supplementary Note 8 provide the derivation). This state-dependent Hamiltonian has a similar structure to the Dirac Hamiltonian, except for the nonlinear mass term κ(∣Ψ1∣2 + ∣Ψ2∣2), and thus, we term it the nonlinear Dirac Hamiltonian. In general, the nonlinear Dirac Hamiltonian should describe the low-energy dispersions of a broad class of nonlinear topological insulators, and thus, its localized modes unveil the existence of topological edge modes in various continuum systems. By considering the right semi-infinite system which has an open boundary at x = 0 and are periodic in the y direction, and assuming the ansatz \({({{\varPsi}}_{1}(x,y),{{\varPsi}}_{2}(x,y))}^{{\rm{T}}}\) \(={{\rm{e}}}^{{\rm{i}}{k}_{y}y}\phi (x){(1/\sqrt{2},-{\rm{i}}/\sqrt{2})}^{{\rm{T}}}\) (E = ky), we can analytically obtain the spatial distribution of the gapless mode of the nonlinear Dirac Hamiltonian as

$$\begin{array}{l}{({{\varPsi}}_{1}(x,y),{{\varPsi}}_{2}(x,y))}^{{\rm{T}}}\\ ={{\rm{e}}}^{{\rm{i}}({k}_{y}y+\theta )}\sqrt{\frac{1}{-(\kappa /m)+D{{\rm{e}}}^{-2mx}}}{(1/\sqrt{2},-{\rm{i}}/\sqrt{2})}{\,}^{{\rm{T}}},\end{array}$$
(8)

where D is the integral constant and –(κ/m) + De−2mx must be positive.

Figure 4 summarizes the nonlinear Chern numbers (Methods) and the behaviours of gapless modes in the nonlinear Dirac Hamiltonian at different parameters. In the case of m < 0, where the Chern number is C = 1/2 in the linear limit, we obtain the localized states at the left side as in the linear case. These localized states are consistent with the bulk–boundary correspondence in weakly nonlinear systems, as shown earlier. We can also check that no localized modes appear in the case of positive m and κ, where the nonlinear Chern number is CNL = −1/2 at any amplitude.

Fig. 4: Phase diagram and localized modes of the nonlinear Dirac Hamiltonian.
figure 4

a, Phase diagram of the nonlinear Dirac Hamiltonian, which demonstrates the nonlinear bulk–boundary correspondence in the continuum model. The vertical axis represents the amplitude, and the horizontal axes correspond to the parameters of the nonlinear Dirac Hamiltonian. The blue curved surface shows the phase boundary that separates a trivial phase without boundary modes and a topological phase exhibiting localized modes at the left boundary. The red surfaces present the boundaries where the sign of the parameters of the Dirac Hamiltonian changes. The red lines show the phase boundaries at the surfaces of w = 1 and w = 2. In the linear limit (w = 0), the topological phases are separated by the m = 0 axis, whereas the nonlinearity modifies the boundary of the topological phases. b–d, Representative shape of the localized mode in each of the topologically non-trivial parameter regions. b, When m is negative and κ is positive, we obtain a localized mode in the small-amplitude region, which is regarded as a counterpart of a conventional topological edge mode. We set m, κ and D as m = −0.5, κ = 1 and D = 3. c, When both m and κ are negative, a localized mode appears independent of the amplitude as in linear topological insulators. We set m = −0.5, κ = −1 and D = 3. d, When m is positive and κ is negative, the nonlinear Dirac Hamiltonian exhibits the nonlinearity-induced topological phase transition. We obtain an unconventional localized mode if the amplitude is larger than a critical value. In this localized mode, there exist non-vanishing amplitudes even in the limit of x → ∞. We set m = 0.5, κ = −2 and D = −3. e, We can also obtain anti-localized modes in the topologically trivial phase, which are unique to nonlinear systems. We set m = 0.5, κ = −2 and D = −20.

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To discuss the bulk–boundary correspondence under stronger nonlinearity, we must relate the amplitude w under the periodic boundary condition to \({w}_{{{{\rm{ave}}}}}=\int\nolimits_{0}^{L}{\rm{d}}x| \phi (x){| }^{2}/L\) or wedge = ∣ϕ(0)∣2 of the edge-localized mode obtained under the open boundary condition. In fact, either choice can exactly predict the phase boundary as shown in the case of m > 0, κ < 0, where we obtain the localized state with the residual amplitude \(\sqrt{| m/\kappa | }\) in the limit of x → ∞. In this case, we obtain a gapless homogeneous mode \(\phi (x)=\sqrt{| m/\kappa | }\) at the phase boundary. Since both wave and wedge of such a homogeneous mode are \({w}_{{{{\rm{ave/edge}}}}}\) \(=\sqrt{| m/\kappa | }\) and satisfy m + κwave/edge = 0, both definitions of the amplitude predict the nonlinearity-induced topological phase transition associated with the amplitude-dependent Chern number (Supplementary Note 9). We note that the non-vanishing amplitude in the limit of x → ∞ indicates that it is impossible to normalize the edge mode. In finite systems, however, such a non-vanishing localized mode can be normalized and thus can robustly emerge. We also obtain anti-localized modes satisfying ∣ϕ(0)∣ < ∣ϕ(x)∣ for D > 0 (Fig. 4e), which is unique to nonlinear systems.

We can also confirm the bulk–boundary correspondence in lattice systems (Supplementary Notes 10 and 12–17). Specifically, by making the correspondence between the bulk amplitude w and edge amplitude wedge, the non-zero nonlinear Chern number corresponds to the existence of localized zero modes. We analytically show such bulk–boundary correspondence in semi-infinite systems and numerically confirm it in finite systems of the nonlinear QWZ model.

Observation protocol of edge modes via quench dynamics

One can observe the topological properties via quench dynamics, which directly detects the existence of topological edge modes. In quench dynamics, one only has to excite the edge sites at homogeneous amplitudes and observe the dynamics without any other external interactions (Fig. 5a). To confirm the correspondence between the existence of nonlinear edge modes and the localized states in quench dynamics, we numerically simulate the quench dynamics of a nonlinear QWZ model (equation (4)) at various parameters. Figure 5b,c shows the time evolution of the quench dynamics with and without nonlinear edge modes. We also obtain the phase diagrams (Fig. 5d,e), which are classified by the amplitude at the edge of the sample in the long-time limit. Here we consider two initial states equivalent to the nonlinear edge modes at u + κw = ±1 (Methods). We confirm that the localized states remain in the topological cases and vanish in the trivial cases. Identifying the bulk amplitude w as wedge = ∑j,y∣Ψj(0, y)∣2/Ly (Ly is the system size in the y direction) of the initial state, we confirm that the nonlinear Chern number CNL(w) roughly corresponds to the phase boundaries in the quench dynamics, which indicates that the nonlinear Chern number and its nonlinearity-induced topological phase transition can predict the existence of experimentally observable localized states.

Fig. 5: Phase diagram of the quench dynamics of the nonlinear QWZ model.
figure 5

a, Schematic of the experimental protocol of the quench dynamics. First, one excites the edge sites by, for example, applying lasers to the edge resonators in nonlinear topological photonic insulators. Then, one observes the nonlinear dynamics without external fields and confirms the existence or absence of a long-lived localized state. b, Time evolution of the quench dynamics in a trivial phase. We use the parameters u = −2.5 and κ = 0.25, and set the initial edge amplitude as w = 1, which corresponds to the white square in d. We confirm the absence of localized edge modes. c, Time evolution of the quench dynamics of nonlinear edge modes. We use the parameters u = −2.5 and κ = 1.5, and set the initial edge amplitude as w = 1, which corresponds to the white circle in d. We confirm that a localized state remains for a long time, which indicates the existence of edge modes. d,e, Simulation of the quench dynamics and a plot of the amplitude remaining at the edge sites in the long-term limit after a quench in the u–κw plane. The light-blue lines indicate the parameters where we can obtain the exact edge-localized solutions. The white numbers show the nonlinear Chern number, and the grey lines represent their phase boundaries, where we relate the amplitude w to wedge = ∑j∣Ψj(x = 0)∣2 at t = 0. These lines agree with the boundary of the topological phase, and thus, the quench dynamics shows the shift in the phase boundary by nonlinearity. We take the different initial configurations corresponding to the edge modes in the linear limit at u = −1 in d and u = 1 in e. Thus, the phase diagrams of the quench dynamics reproduce the topological phases with different Chern numbers in these panels (C = −1 for d and C = 1 for e).

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Possible experimental setups of nonlinear Chern insulators

Nonlinear Chern insulators are expected to be realized by using topological photonics with the Kerr nonlinearity. In particular, we can replace the nonlinear term in equation (4) by on-site Kerr nonlinear terms, that is, κ∣Ψj(x, y)∣2Ψj(x, y), and such Kerr nonlinearity is feasible in various photonic systems. Supplementary Note 18 further discusses a possible optical setup and the analytical demonstration of the nonlinearity-induced topological phase transition under the on-site Kerr nonlinearity.

Discussions

Our results indicate the existence of unique topological phenomena beyond the weakly nonlinear regime. There remain intriguing issues to establish the topological classification of nonlinear systems including further strongly nonlinear cases where nonlinearity is even stronger than the linear couplings. Since such strong nonlinearity can induce bulk-localized modes18,35,42,46 and topological edge solitons30,31,32,33,34,35 that are out of the description of the Bloch ansatz, it is unclear whether or not the nonlinear Chern number still fully works. We can also discuss the linear stability of the Bloch-ansatz state, which may be related to the topological phase transition28 (Supplementary Note 19). It may also be intriguing to investigate the connection to many-body quantum physics because nonlinear terms can be derived from the mean-field approximation16,17 or the Kohn–Sham equation47,48 of interacting systems in general (Supplementary Note 20). Therefore, the nonlinear topology may also be useful to understand the topology of interacting systems.

Methods

Justification of the Bloch ansatz via perturbation analysis

If the nonlinear term is small compared with the linear bandgap, one can regard the nonlinear effect as a perturbation to the linear band structure. Under such an assumption, one can perturbatively calculate the nonlinear eigenvectors. To show the perturbation calculation protocol of the nonlinear eigenvalue problem, we rewrite the nonlinear eigenequation as

$$H({\boldsymbol{\Psi}} ){\boldsymbol{\Psi}} =({H}^{\,0}+\kappa {H}_{{{{\rm{NL}}}}}({\boldsymbol{\Psi}} )){\boldsymbol{\Psi}} =E\,{\boldsymbol{\Psi}} .$$
(9)

We consider the perturbation expansion by κ as follows:

$${\boldsymbol{\Psi}} ={\boldsymbol{\Psi}}^{(0)}+\kappa {\boldsymbol{\Psi}}^{(1)}+\cdots \,,$$
(10)
$$E={E}^{\,(0)}+\kappa {E}^{\,(1)}+\cdots \,.$$
(11)

One can determine Ψ(0) and E(0) from the eigenvalue and eigenvector of the linear Hamiltonian H0 as

$${H}^{\,0}{\boldsymbol{\Psi}}^{\,(0)}=E{\boldsymbol{\Psi}}^{\,(0)}.$$
(12)

Then, the first-order perturbation is calculated from the eigenequation of (H0 + κHNL(Ψ(0))) as

$$\left({H}^{\,0}+\kappa {H}_{{{{\rm{NL}}}}}({\boldsymbol{\Psi}}^{(0)})\right)\left({\boldsymbol{\Psi}}^{(0)}+\kappa {\boldsymbol{\Psi}}^{(1)}\right)=\left({E}^{(0)}+\kappa {E}^{(1)}\right)\left({\boldsymbol{\Psi}}^{(0)}+\kappa {\boldsymbol{\Psi}}^{(1)}\right).$$
(13)

One can confirm the consistency between equations (9) and (13) by substituting equations (10) and (11) into the former equation.

In translation-invariant systems, the non-perturbed eigenvector Ψ(0) is described by a Bloch wave Ψ(0) = eik·xψ(0), due to the Bloch theorem of linear systems. Since the Bloch wave exhibits no site dependence of the amplitude, one can assume that the nonlinear term κHNL(Ψ) is also uniform, and thus, the whole effective Hamiltonian H0 + κHNL(Ψ) still has translational symmetry. Therefore, in weakly nonlinear systems, one can believe that all of the nonlinear eigenvectors are described by the Bloch ansatz. We note that in strongly nonlinear regimes, there can be localized modes that cannot be described by the Bloch ansatz35,42,46. However, the periodic solutions obtained from the Bloch ansatz are still exact nonlinear eigenvectors under the periodic boundary conditions.

Nonlinear Chern number in continuum systems

In continuum systems with a periodic potential, the Bloch ansatz should read Ψj(r) = eik·rψj(k, r), where ψj(k, r) is a periodic function of r whose period is equal to that of the periodic potential. Then, the wavenumber-space representation of the nonlinear eigenequation becomes fj(k, ψ(k); r) = E(k)ψj(k, r), and the nonlinear Chern number can be written as

$${C}_{{{{\rm{NL}}}}}=\frac{1}{2\uppi i}\mathop{\sum}\limits_{j}{\int}_{S}{{\rm{d}}}^{2}{{{\bf{r}}}}\int\,{\nabla }_{{{{\bf{k}}}}}\times \left[({\psi }_{j}({{{\bf{k}}}},{{{\bf{r}}}})/\sqrt{w}){\nabla }_{{{{\bf{k}}}}}({\psi }_{j}({{{\bf{k}}}},{{{\bf{r}}}})/\sqrt{w})\right]{{\rm{d}}}^{2}{{{\bf{k}}}},$$
(14)

where S represents the unit cell of the periodic system. The squared amplitude is defined as w = ∑j∫Sd2r∣ψj(k, r)∣2 in this continuum case.

Real-space description of the nonlinear QWZ model

To investigate the existence or absence of topological edge modes in lattice systems, we construct a minimal lattice model of a nonlinear Chern insulator, which we term the nonlinear QWZ model (equation (4) shows the wavenumber-space description). Its real-space dynamics is described as

$$\begin{array}{l}{\rm{i}}\frac{{\rm{d}}}{{\rm{d}}t}{{\varPsi}}_{j}(x,y)=\mathop{\sum}\limits_{l}\Big\{{({\sigma }_{z})}_{jl}\Big[2u{{\varPsi}}_{l}(x,y)+{{\varPsi}}_{l}(x+1,y)+{{\varPsi}}_{l}(x-1,y)\\\qquad\qquad\quad+{{\varPsi}}_{l}(x,y+1)+{{\varPsi}}_{l}(x,y-1)\Big]/2\\\qquad\qquad\quad+{({\sigma }_{x})}_{jl}\Big({{\varPsi}}_{l}(x+1,y)-{{\varPsi}}_{l}(x-1,y)\Big)/2i\\\qquad\qquad\quad+{({\sigma }_{y})}_{jl}\Big({{\varPsi}}_{l}(x,y+1)-{{\varPsi}}_{l}(x,y-1)\Big)/2i\Big\}\\\qquad\qquad\quad-{(-1)}^{\,j}\kappa\left(| {{\varPsi}}_{1}(x,y){| }^{2}+| {{\varPsi}}_{2}(x,y){| }^{2}\right){{\varPsi}}_{j}(x,y),\end{array}$$
(15)

where (j, l) and (x, y) represent the internal degree of freedom and location, respectively, and Ψj(x, y) is the jth component of the state vector at location (x, y). The values of \({({\sigma }_{i})}_{jl}\) are the (j, l)th component of the ith Pauli matrix. This lattice model introduces the staggered Kerr-like nonlinearity −(−1)jκ(∣Ψ1(x, y)∣2 + ∣Ψ2(x, y)∣2)Ψi(x, y) to the linear QWZ model45.

Exact bulk solutions of the nonlinear QWZ model

To obtain the phase diagram shown in Fig. 2b, we analytically solve the nonlinear eigenequation in equation (4). If we focus on special solutions where the squared amplitude ∣Ψ1(k)∣2 + ∣Ψ2(k)∣2 = w has no k dependence, the nonlinear eigenequation (equation (4)) exactly corresponds to a linear one. Therefore, by solving the corresponding linear eigenequation, we obtain the following exact bulk eigenvalues and eigenvectors as

$$\begin{array}{l}{E}_{\pm }\left({k}_{x},{k}_{y}\right)\\=\pm \sqrt{2\cos {k}_{x}\cos {k}_{y}+2\left(u+\kappa w\right)\left(\cos {k}_{x}+\cos {k}_{y}\right)+{\left(u+\kappa w\right)}^{2}+2},\end{array}$$
(16)
$$\left(\begin{array}{c}{\psi }_{1\pm }\left({{{\bf{k}}}}\right)\\ {\psi }_{2\pm }\left({{{\bf{k}}}}\right)\end{array}\right)=\frac{\sqrt{w}}{{c}_{\pm }\left({{{\bf{k}}}}\right)}\left(\begin{array}{c}u+\kappa w+\cos {k}_{x}+\cos {k}_{y}+{E}_{\pm }\left({k}_{x},{k}_{y}\right)\\ \sin {k}_{x}-{\rm{i}}\sin {k}_{y}\end{array}\right),$$
(17)

where \({c}_{\pm }\left({{{\bf{k}}}}\right)=\sqrt{{\left(u+\kappa w+\cos {k}_{x}+\cos {k}_{y}+{E}_{\pm }\left({k}_{x},{k}_{y}\right)\right)}^{2}+{\sin }^{2}{k}_{x}+{\sin }^{2}{k}_{y}}\) is a normalization constant. By using these nonlinear eigenvectors, we analytically obtain the nonlinear Chern number as

$${C}_{{{{\rm{NL}}}}}=\left\{\begin{array}{ll}1\quad &(0 < u+\kappa w < 2)\\ -1\quad &(-2 < u+\kappa w < 0)\\ 0\quad &({{{\rm{otherwise}}}})\end{array}\right.,$$
(18)

as summarized in Fig. 2b. We note that one can generally obtain exact bulk solutions if the nonlinear equation has the form in equation (5) (Supplementary Note 4).

Perturbation analysis of the bulk modes of the nonlinear QWZ model

Although we calculate the exact solutions of the bulk modes of the nonlinear QWZ model in the main text, we can also obtain the same bulk modes from the perturbation analysis or self-consistent calculations. Specifically, if we conduct the perturbation analysis described earlier, the calculation stops at the first-order perturbation and derives the same bulk modes as the exact solutions.

The zeroth-order solutions, that is, the linear solutions of the bulk modes of the QWZ model are

$$\left(\begin{array}{c}{\psi }_{1\pm }^{(0)}\left({{{\bf{k}}}}\right)\\ {\psi }_{2\pm }^{(0)}\left({{{\bf{k}}}}\right)\end{array}\right)=\frac{\sqrt{w}}{{c}_{\pm }\left({{{\bf{k}}}}\right)}\left(\begin{array}{c}u+\cos {k}_{x}+\cos {k}_{y}+{E}_{\pm }\left({k}_{x},{k}_{y}\right)\\ \sin {k}_{x}-{{i}}\sin {k}_{y}\end{array}\right),$$
(19)
$${E}_{\pm }\left({k}_{x},{k}_{y}\right)=\pm \sqrt{2\cos {k}_{x}\cos {k}_{y}+2u\left(\cos {k}_{x}+\cos {k}_{y}\right)+{u}^{2}+2},$$
(20)
$${c}_{\pm }\left({{{\bf{k}}}}\right)=\sqrt{{\left(u+\cos {k}_{x}+\cos {k}_{y}+{E}_{\pm }\left({k}_{x},{k}_{y}\right)\right)}^{2}+{\sin }^{2}{k}_{x}+{\sin }^{2}{k}_{y}},$$
(21)

where we fix the norm as |ψ1±(k)|2 + |ψ2±(k)|2 = w. Then, substituting these solutions into the state-dependent Hamiltonian of the nonlinear QWZ model, one can obtain the first-order perturbation solutions. Due to the nonlinear terms depending only on the norm of the nonlinear eigenvector, the substituted effective Hamiltonian is described as

$$\begin{array}{l}H({{{\bf{k}}}},{\bf{\uppsi}} ({{{\bf{k}}}}))=\\ \left(\begin{array}{cc}u+\kappa w+\cos {k}_{x}+\cos {k}_{y}&\sin {k}_{x}+{{i}}\sin {k}_{y}\\ \sin {k}_{x}-{{i}}\sin {k}_{y}&-(u+\kappa w+\cos {k}_{x}+\cos {k}_{y})\end{array}\right),\end{array}$$
(22)

independently of the wavenumber. The eigenvalues and eigenvectors of this Hamiltonian are the same as those of the nonlinear QWZ model (equations (16) and (17)), and thus, the first-order perturbation calculation is consistent with the exact solutions.

The self-consistent calculation is equivalent to the higher-order perturbation calculation (as shown later). However, if one substitutes the first-order solutions into the state-dependent Hamiltonian of the nonlinear QWZ model, one obtains the same effective Hamiltonian as equation (22). Therefore, the nonlinear eigenvectors and eigenvalues obtained from the self-consistent calculation are the same as those obtained from the first-order perturbation and exact solutions.

Numerical simulations of the real-space dynamics of the nonlinear QWZ model

In Fig. 2, we numerically calculate the dynamics of the nonlinear QWZ model by using the fourth-order Runge–Kutta method. We consider a 20 × 20 square lattice, where each lattice point has two internal degrees of freedom. We impose the open boundary condition in the x direction and the periodic boundary condition in the y direction. We set the time step dt = 0.005. The simulation starts from the localized initial states that have non-zero values \({{\varPsi}}_{1}(x,y)=w/\sqrt{2}\) and \({{\varPsi}}_{2}(x,y)={\rm{i}}w/\sqrt{2}\) only at x = 1. We use the parameters u = 3 and κ = 0.1 in Fig. 2c and u = −1 and κ = 0.1 in Fig. 2d. We also set the initial amplitude at the edge site as w = 1. In these figures, we plot the square root of the sum of the square of absolute values of the first and second components.

Self-consistent calculation of nonlinear band structures

To obtain the nonlinear band structures in Fig. 3, we numerically calculate the nonlinear eigenvalue problem by using the self-consistent method. We first rewrite the nonlinear dynamics as i∂tΨ = f(Ψ) = H(Ψ)Ψ, where H(Ψ) is a state-dependent effective Hamiltonian. Then, we conduct the self-consistent calculation in the following procedure. (1) We numerically diagonalize \(H(\bf{0})\) and set the initial guess of the eigenvalue and eigenvector Ψ0 and E0, respectively, by adopting a pair of the obtained eigenvalue and eigenvector of \(H(\bf{0})\). We fix the norm of Ψ0 to be ∣∣Ψ0∣∣2 = wL, where L is the system size. (2) We substitute the guessed eigenvector Ψi after i iterations into H(Ψ) and diagonalize H(Ψi). (3) We choose the obtained eigenvalue that is the closest to the previous guess Ei, and the corresponding eigenvector as the next guess Ei+1 and Ψi+1. (4) We iterate steps (2) and (3) until the distance between Ψi and Ψi+1 becomes smaller than the threshold of ∣∣Ψi+1 − Ψi∣∣ < ϵ, or the iteration reaches a fixed number. We also perform these calculations starting from all the eigenvectors of \(H(\bf{0})\) and obtain a set of nonlinear eigenvectors and eigenvalues of f(Ψ).

In Fig. 3, we consider the parametrized state-dependent Hamiltonian that corresponds to the nonlinear Chern insulator under the assumptions of the y-periodic Bloch ansatz and the open boundary condition in the x direction. The state-dependent Hamiltonian is described as

$$\begin{array}{l}H(\Psi )=\\\left(\begin{array}{cc}u+{\Delta }_{x}^{2}+\cos {k}_{y}+\kappa\Big(| {{\varPsi}}_{1}{| }^{2}+| {{\varPsi}}_{2}{| }^{2}\Big)&\Big({\rm{i}}{\Delta }_{x}+{\rm{i}}\sin {k}_{y}\Big)\\ \Big({\rm{i}}{\Delta }_{x}-{\rm{i}}\sin {k}_{y}\Big)&-\Big(u+{\Delta }_{x}^{2}+\cos {k}_{y}+\kappa\Big(| {{\varPsi}}_{1}{| }^{2}+| {{\varPsi}}_{2}{| }^{2}\Big)\Big)\end{array}\right),\end{array}$$
(23)

where Δx and \({\Delta }_{x}^{2}\) are the difference operators defined as ΔxΨ(x) = [Ψ(x + 1) − Ψ(x)]/2 and \({\Delta }_{x}^{2}{{\varPsi}} (x)={{\varPsi}} (x+1)+{{\varPsi}} (x-1)-2{{\varPsi}} (x)\), respectively. Then, we calculate the nonlinear eigenvalues of this state-dependent Hamiltonian at ky = nΔk, where n = −N, −N + 1… N − 1, N, and Δk = π/N (N = 50). Here we set the system size in the x direction as L = 10 and fix the average amplitude of the nonlinear eigenvector as ∑x,j∣Ψj(x; ky)∣2/L = w independent of wavenumber ky. We note that the self-consistent calculation is only stable at weak nonlinearity. To calculate the band structures in strongly nonlinear systems than those analysed in the main text, one should instead use the Newton method (Supplementary Methods).

Theorem of bulk–boundary correspondence in weakly nonlinear systems

We mathematically show the bulk–boundary correspondence in weakly nonlinear systems. Here we use a simple notation \(f\,(\bf{\Psi })=E\,\bf{\Psi }\) instead of fj(Ψ; r) = EΨj(r), where \(\bf{\Psi }\) is the nonlinear eigenvector whose components correspond to the state variables at locations r and internal degrees of freedom j. The claim of the theorem is as follows:

Theorem

Suppose \(f\,(\bf{\Psi })=E\,\bf{\Psi }\) is a nonlinear eigenvalue problem on a two-dimensional lattice system that satisfies the following assumptions. (1) When we rewrite the nonlinear function f as \(f\,(\bf{\Psi })=H(\bf{\Psi })\,\bf{\Psi }\), there exists a positive real number c < 1 that satisfies \(| | H(\bf{\Psi })-H(\bf{0})| | < gc/2\) for any complex vector \(\bf{\Psi }\), where g is the bandgap of \(H(\bf{0})\) and ∣∣⋅∣∣ is the operator norm. (2) There exists a positive real number c′ < 1 such that for any pairs of complex vectors Ψ and Ψ + ΔΨ with the norm w, they satisfy \(| | H(\,\bf{\Psi }+\Delta \bf{\Psi })-H(\,\bf{\Psi })| | \le g(1-c){c}^{{\prime} }{(6\sqrt{2}w)}^{-1}| | \Delta \bf{\Psi }| |\). (3) For any complex vector \(\bf{\Psi }\), one can rewrite the nonlinear function \(f(\bf{\Psi })\) as \(f(\bf{\Psi })=\tilde{H}(\bf{\Psi })\;\bf{\Psi }\), where \(\tilde{H}\) is a Hermitian matrix. (4) The nonlinear equation satisfies the U(1) symmetry, \(f({{\rm{e}}}^{{\rm{i}}\theta }\bf{\Psi })={{\rm{e}}}^{{\rm{i}}\theta }f(\bf{\Psi })\). We also assume that the number of nonlinear eigenvectors is equal to that of the linear eigenvectors of \(H(\bf{0})\). Then, the nonlinear eigenequation \(f\,(\bf{\Psi })=E\,\bf{\Psi }\) exhibits robust gapless boundary modes if and only if its nonlinear Chern number is non-zero.

We note that here we relate the average amplitude wave = ∑r,j∣Ψj(r)∣2/L to the amplitude used in the definition of the nonlinear Chern number. To prove the theorem, we first show the following proposition.

Proposition 1

In the nonlinear eigenvalue problem that satisfies the assumptions in the theorem, the self-consistent calculation converges, namely, Ei → E∞ and \({\bf{\Psi }}_{i}\to {\bf{\Psi }}_{\infty }\). Furthermore, there exists an eigenvector \({\bf{\Psi }}_{0}\) and eigenvalue E0 of \(H(\bf{0})\) that satisfy ∣∣E∞ – E0∣∣ < gc/2(1 – c′) and \(| |\;{\bf{\Psi }}_{\infty }-{\bf{\Psi }}_{0}| | < {2}^{-1/2}c{(1-2{c}^{{\prime} })}^{-1}w\).

To show this proposition, we utilize the perturbation theorem of the eigenvectors in linear systems. Suppose H is a non-degenerate Hermitian matrix and g is the minimum difference between its two eigenvalues. If ∣∣A∣∣ < g/2 in terms of the operator norm, for an arbitrary eigenvector \(\bf{\Psi }\) of H + A, there exists an eigenvector \({\bf{\Psi }}_{0}\) of H that satisfies \(| |\;\bf{\Psi }-{\bf{\Psi }}_{0}| | \le D{g}^{-1}| | A| | \,| |\;{\bf{\Psi }}_{0}| |\) (\(D=2\sqrt{2}\)). We iteratively use this theorem and evaluate the distance between the guess of eigenvectors \({\bf{\Psi }}_{i}\) and \({\bf{\Psi }}_{i+1}\) at each step.

We also show that the resulting eigenvalue and eigenvector, namely, E∞ and \({\bf{\Psi }}_{\infty }\), respectively, are indeed a pair of nonlinear eigenvalue and eigenvector of \(f(\bf{\Psi })\), which is summarized below.

Proposition 2

The converged solutions of the self-consistent calculation E∞ and \({\bf{\Psi }}_{\infty }\) satisfy \(f(\,{\bf{\Psi }}_{\infty })={E}_{\infty }{\bf{\Psi }}_{\infty }\).

We show this proposition by using simple inequalities and limit evaluations. By using these propositions, we finally show the following lemma to prove the theorem.

Lemma

For an arbitrary eigenvector of a nonlinear eigenvalue problem \((H+f(\,\bf{\Psi }))\,\bf{\Psi }=E\,\bf{\Psi }\) that satisfies the conditions in the theorem, there exists the eigenvector of \([H+(1-\epsilon )\,f(\,{\bf{\Psi }}_{\epsilon })]\,{\bf{\Psi }}_{\epsilon }=E\,{\bf{\Psi }}_{\epsilon }\) that satisfies \(| |\,\bf{\Psi }-{\bf{\Psi }}_{\epsilon }| | < Cw\epsilon\) (0 < ε < g′/g, where g′ is the minimum difference of the eigenvalues of \(H+f(\bf{\Psi })\)), where w is the norm of \(\bf{\Psi }\) and \({\bf{\Psi }}_{\epsilon }\) and C is a constant independent of the eigenvector \(\bf{\Psi }\) and constant ϵ.

This lemma indicates that in weakly nonlinear systems, one can map the nonlinear eigenvalues and eigenvectors onto those of a linear eigenvalue problem. Thus, we can show the bulk–boundary correspondence in weakly nonlinear systems.

Derivation of the nonlinear Dirac Hamiltonian and its gapless mode

We have derived the nonlinear Dirac Hamiltonian (equation (25)) as the effective theory of the low-energy dispersion of the nonlinear QWZ model around the critical amplitude. For example, if we focus on the critical amplitude wc = (−2 − u)/κ, the nonlinear band structure of the model closes the gap at (kx, ky) = (0, 0). Then, around the critical amplitude w ≈ wc and the gap-closing point (kx, ky) ≈ (0, 0), w remains the leading order term of the wavenumber-space description of the nonlinear QWZ model. Finally, by substituting the wavenumbers with the derivative, we derive the real-space description of the nonlinear Dirac Hamiltonian in equation (25) as

$${\rm{i}}{\partial }_{t}{\boldsymbol{\Psi}} ({{{\bf{r}}}})=\hat{H}({\boldsymbol{\Psi}} ({{{\bf{r}}}})){\boldsymbol{\Psi}} ({{{\bf{r}}}}),$$
(24)
$${\hat{H}}({\bf{\Psi}} ({{{\bf{r}}}}))=\left(\begin{array}{cc}m+\kappa\Big(| {{\varPsi}}_{1}({{{\bf{r}}}}){| }^{2}+| {{\varPsi}}_{2}({{{\bf{r}}}}){| }^{2}\Big)&-{{i}}{\partial }_{x}+{\partial }_{y}\\ -{{i}}{\partial }_{x}-{\partial }_{y}&-m-\kappa\Big(| {{\varPsi}}_{1}({{{\bf{r}}}}){| }^{2}+| {{\varPsi}}_{2}({{{\bf{r}}}}){| }^{2}\Big)\end{array}\right).$$
(25)

Starting from other critical amplitudes wc = −u/κ and wc = (2 − u)/κ, one can derive similar state-dependent Hamiltonians (Supplementary Note 8).

We have also analytically calculated the spatial distribution of the gapless modes of the nonlinear Dirac Hamiltonian. Here we assume the Bloch ansatz \({{\varPsi}}_{i}(x,y)={{\rm{e}}}^{{\rm{i}}{k}_{y}y}{{\varPsi}}_{i}^{{\prime} }(x)\) that is periodic in the y direction and consider the semi-infinite system that has an open boundary at x = 0 and is extended to x → ∞. We calculate the localized mode with the wavenumber ky and the eigenvector E = ky. Constructing an analogy to the linear case, one can use an ansatz \({({{\varPsi}}_{1}(x,y),{{\varPsi}}_{2}(x,y))}^{{{T}}}={{\rm{e}}}^{{\rm{i}}{k}_{y}y}\phi (x){(1/\sqrt{2},-{{i}}/\sqrt{2})}{\,}^{{{T}}}\) to describe the localized mode. Then, ϕ(x) should satisfy ∂xϕ(x) = mϕ(x) + κ∣ϕ(x)∣2ϕ(x). We can analytically obtain the solution of this equation as

$$\phi (x)={{\rm{e}}}^{{\rm{i}}\theta }\sqrt{\frac{1}{-(\kappa /m)+D{{\rm{e}}}^{-2mx}}},$$
(26)

where D is the integral constant and −(κ/m) + De−2mx must be positive.

Nonlinear Chern number of the nonlinear Dirac Hamiltonian

To calculate the nonlinear Chern number of the nonlinear Dirac Hamiltonian (equation (25)), we use the Bloch ansatz Ψi(x, y) = ψi(k)exp(i(kxx + kyy)) (without explicit (x, y) dependence because of the continuous translational symmetry). Then, we analytically obtain the nonlinear Chern number of the nonlinear Dirac Hamiltonian as

$${C}_{{{{\rm{NL}}}}}=\left\{\begin{array}{ll}\frac{1}{2}\quad &(m+\kappa w > 0)\\ -\frac{1}{2}\quad &(m+\kappa w < 0)\end{array}\right.,$$
(27)

where w = ∫SdS[∣Ψ1(x, y)∣2 + ∣Ψ2(x, y)∣2]/∣S∣ is the average squared amplitude of plane waves in this nonlinear system. We note that the nonlinear Chern number of the nonlinear QWZ model corresponds to the sum of those of the nonlinear Dirac Hamiltonians obtained from the expansion around the gap-closing points (kx, ky) = (0, 0), (0, π), (π, 0), (π, π).

Numerical calculations of the quench dynamics

We have numerically solved the nonlinear Schrödinger equation (equation (15)) with the initial condition localized at the left edge (Fig. 5a). As the initial conditions, we study the two cases. One is

$${{\varPsi}}_{1}\left(x,y\right)={\delta }_{x,1}/\sqrt{2},\qquad {{\varPsi}}_{2}\left(x,y\right)={\rm{i}}{\delta }_{x,1}/\sqrt{2},$$
(28)

which corresponds to the solution with u = −1 of the linear model (κ = 0). The other is

$${{\varPsi}}_{1}\left(x,y\right)={\left(-1\right)}^{\;y}{\delta }_{x,1}/\sqrt{2},\qquad {{\varPsi}}_{2}\left(x,y\right)={\rm{i}}{\left(-1\right)}^{\;y}{\delta }_{x,1}/\sqrt{2},$$
(29)

which corresponds to the solution with u = 1 of the linear model (κ = 0). Supplementary Note 13 shows the derivation. For the numerical calculation, we used the NDSolve function in Mathematica (version 13.3). We consider aL × L lattice (we set L = 10 in the numerical calculation) under the open boundary condition in the x direction and the periodic boundary condition in the y direction.

We solve the nonlinear Schrödinger equation (equation (15)) under the two initial conditions in equations (28) and (29). The time evolution of |Ψ1(x, y)|2 + |Ψ2(x, y)|2 is shown in Fig. 5d,e. We define the phase indicator P as

$$P\equiv \mathop{\max }\limits_{0.99T < t < T}\frac{1}{L}\mathop{\sum }\limits_{y=1}^{L}\left[{\left\vert {{\varPsi}}_{1}\left(x = 1,y\right)\right\vert }^{2}+{\left\vert {{\varPsi}}_{2}\left(x = 1,y\right)\right\vert }^{2}\right],$$
(30)

where T = 10. We have P ≈ 0 in the trivial phase (Fig. 5b), which implies the absence of localized edge states. On the other hand, we have P ≈ 1 in a topological phase (Fig. 5c), which implies the presence of a localized edge mode. To elucidate the topological phase diagram, we plot P in the u–κw plane (Fig. 5d,e). We find that the phase indicator is 1 along the blue line, where the exact solution is valid. It shows that the exact solution is realized by the quench dynamics.

To compare the phase boundary obtained from the quench dynamics and the nonlinear Chern number, we relate the bulk amplitude w with the edge amplitude wedge = 1. Then, grey lines in Fig. 5 show the phase boundary of the nonlinear Chern number. This definition of the amplitude under the open boundary condition is different from that used in the calculation of the nonlinear band structures because we should choose the proper definition to observe the bulk–boundary correspondence in each numerical method for demonstrating the nonlinear edge modes (Supplementary Methods).