Topological Invariant and Anomalous Edge States of Strongly Nonlinear Systems

The advent of topological band theory has led to the burgeoning field of “topological phases of matter” which manifest exotic properties, such as surface conduction of electronic states, and wave propagation insensitive to backscattering and disorder Haldane (1988); Fu et al. (2007); Hasan and Kane (2010); Ryu et al. (2010). In classical structures Kane and Lubensky (2014); Süsstrunk and Huber (2015); Hadad et al. (2018); Li et al. (2018); Duncan et al. (2017); Zhou and Zhang (2020); Vila et al. (2019); Prodan and Prodan (2009), enormous efforts have been devoted to topological states that emulate their quantum analogs and enable many pioneering applications Ma et al. (2018); Smirnova et al. (2020); Li et al. (2014); Nash et al. (2015); Zhou et al. (2018); Luo et al. (2021); Zhou et al. (2019); Lo et al. (2021); Zheng et al. (2020).

To date, most of the studies of classical structures have been limited to linear topological band theory, with a few exceptions in the weakly nonlinear regime Hadad et al. (2016, 2018); Chaunsali and Theocharis (2019); Pal et al. (2018) where perturbation theory is available. In 1D problems, the topological invariant called Berry phase Zak (1989) is quantized by symmetries expressed as matrix operators. Due to bulk-boundary correspondence, topologically protected evanescent modes emerge on system boundaries. Although varied topological physics has been explored in linear systems, nonlinear dynamics are more ubiquitous in nature, such as biochemical processes Van Kampen (1992), fluid dynamics Meerson (1996), and metamaterials Chen et al. (2014); Zhou et al. (2020), etc. They give rise to rich properties like bifurcation Crawford (1991), instability, solitons Su et al. (1979), and chaos Eckmann and Ruelle (1985); Altmann et al. (2013). The question naturally arises: can topological invariants and phases be extended to nonlinear systems?

In this paper, we present a systematic study of topological attributes in 1D generalized nonlinear Schrödinger equations beyond Kerr-nonlinearities Hadad et al. (2016). The nonlinear parts of interactions are comparable to the linear ones and perturbation theory breaks down, which we designate the “strongly nonlinear regime”. We limit our considerations within the amplitude range Narisetti et al. (2010); Zaera et al. (2018) that chaos does not occur. Consequently, nonlinear bulk modes Vakakis et al. (2001); Fronk and Leamy (2017) are remarkably distinct from sinusoidal waves (e.g., fig.1(b) and fig.D6(c)). We develop the proper definition of Berry phase in nonlinear bulk modes. By adopting a symmetry-based analytic treatment, we demonstrate the quantization of Berry phase in reflection-symmetric systems, regardless of the availability of linear analysis. The emergence of nonlinear topological edge modes is associated with a quantized Berry phase that protects them from defects. Finally, instead of exponentially localizing on lattice boundaries, topological edge modes exhibit anomalous behaviors that decay to a plateau governed by the stable fixed points of nonlinearities.

Generalized nonlinear Schrödinger equations are widely studied in classical systems like nonlinear optics Smirnova et al. (2020); Scalora et al. (2005) and electrical circuits Hadad et al. (2018). Their equations of motion are summarized as the general form in Eqs.(1) below. We study nonlinear bulk modes, from which we define Berry phase and demonstrate its quantization in reflection-symmetric models.

The considered model is a nonlinear SSH Su et al. (1979) chain composed of $N$ classical dimer fields $\Psi_{n}=(\Psi_{n}^{(1)},\Psi_{n}^{(2)})^{\top}$ ($\top$ is matrix transpose) coupled by nonlinear interactions, as represented pictorially in Fig.1(a). The chain dynamics is governed by the 1D generalized nonlinear Schrödinger equations,

subjected to periodic boundary condition (PBC), where $\epsilon_{0}\geq 0$ is the on-site potential, and $f_{i}(x,y)$ for $i=1$ and $i=2$ stand for intracell and intercell nonlinear couplings, respectively. $f_{i}(x,y)$ are real-coefficient general polynomials of $x$, $x^{*}$, $y$, and $y^{*}$ ($*$ represents complex conjugate), which offer time-reversal symmetry Hasan and Kane (2010). Given a nonlinear solution $\Psi_{n}(t)$, time-reversal symmetry demands a partner solution $\Psi_{n}^{*}(-t)$, as demonstrated in App. B1. For systems such as those with Bose-Einstein condensates Leggett (2001), $|\Psi(\vec{r},t)|^{2}$ corresponds to a particle number density and third-order nonlinearities are thus limited to $|\Psi|^{2}\Psi$ to enforce particle number conservation; in our case the fields do not correspond to particle densities and more general nonlinearities are thus permitted.

In linear regime, the polynomials are approximated as $f_{i}(x,y)\approx c_{i}y$ ($c_{i=1,2}>0$) to have “gapped” two-band models when $c_{1}\neq c_{2}$. The bulk mode eigenfunctions are sinusoidal in time, and Berry phase is quantized by reflection symmetry. In the “strongly nonlinear regime” where nonlinear interactions become comparable to the linear ones, nonlinear bulk modes are significantly different from sinusoidal waves (e.g. figs.1(b), D6(c)), and the frequencies naturally deviate from their linear counterparts. The nonlinearities become increasingly important as the bulk mode amplitude rises. Hence, the frequency of a nonlinear bulk mode is controlled both by wavenumber and amplitude. We thus define nonlinear band structure Hadad et al. (2018, 2016) $\omega=\omega(q\in[-\pi,\pi],A)$ as the frequencies of nonlinear bulk modes for given amplitude $A$. We consider the simple case that nonlinear bulk modes are always non-degenerate (i.e., different modes at the same wavenumber have different frequencies) unless they reach the topological transition amplitude when the nonlinear bands merge at the band-touching frequency. Hence, given the amplitude, frequency, and wavenumber, a nonlinear bulk mode is uniquely defined. Extended from gapped linear models, the lattice is a “gapped two-band nonlinear model”. In what follows, we define Berry phase for nonlinear bulk modes of the upper-band by adiabatically evolving the wavenumber across the Brillouin zone.

The considered nonlinear bulk mode is spatial-temporal periodic. It takes the traveling plane-wave ansatz,

where $\omega$ and $q$ are the frequency and wavenumber, respectively. $\Psi_{q}^{(j=1,2)}(\theta)$ are $2\pi$-periodic wave components, where the phase conditions are chosen by asking ${\rm Re\,}\Psi_{q}^{(j)}(\theta=0)=A$, and $A\overset{\rm def}{=}\max({\rm Re\,}\Psi_{q}^{(j)})$ is the amplitude. This is analogous to the phase condition ${\rm Re\,}\Psi(t=0)=\max({\rm Re\,}\Psi(t))$ adopted in Schrödinger equation in order to have the eigenfunctions $\Psi(t)=|\Psi|e^{-\mathrm{i}\epsilon t/\hbar}$. Following this condition, $\phi_{q}$ in Eq.(2) characterizes the relative phase between the two wave components. Nonlinear bulk modes are not sinusoidal. They fulfill $\mathrm{i}\partial_{t}\Psi_{q}=H(\Psi_{q})$, where $H(\Psi_{q})$ is the nonlinear function determined by Eqs.(1) and is elaborated in App. A. Given the band index and the amplitude $A$ of a nonlinear bulk mode, we find that $\omega$, $\phi_{q}$, and the waveform are determined by the wavenumber $q$.

We adopt the ansatz in Eq.(2) based on a number of reasons. First, typical studies on weakly nonlinear bulk modes Vila et al. (2019); Fronk and Leamy (2017); Pal et al. (2018); Zhou et al. (2020); Narisetti et al. (2010); Zaera et al. (2018); Vakakis et al. (2001) reveal that the dynamics of all high-order harmonics are controlled by the single variable $\theta=\omega t-qn$: $\Psi_{q}^{(j)}=\sum_{l}\psi_{l,q}^{(j)}e^{-\mathrm{i}l(\omega t-qn)}$, where $\psi_{l,q}^{(j)}=(2\pi)^{-1}\int_{0}^{2\pi}e^{\mathrm{i}l\theta}\Psi_{q}^{(j)}d\theta$ is the $l$-th Fourier component of $\Psi_{q}^{(j)}$. Second, numerical experiments such as shooting method (see figs.1(b), D6(a,c), and Refs. Renson et al. (2016); Ha (2001); Peeters et al. (2008)) manifest non-dispersive, plane-wave like bulk modes in the strongly nonlinear regime. Finally, it is demonstrated in App. C3 that the analytic solutions of nonlinear bulk modes at high-symmetry wavenumbers are in perfect agreement with Eq.(2).

We realize the adiabatic evolution of wavenumber $q(t^{\prime})$ traversing the Brillouin zone from $q(0)=q$ to $q(t)=q+2\pi$, while the amplitude $A$ remains unchanged during this process. According to the nonlinear extension of the adiabatic theorem Xiao et al. (2010); Liu et al. (2003); Pu et al. (2007); Liu and Fu (2010), a system $H(\Psi_{q})$ initially in one of the nonlinear modes $\Psi_{q}$ will stay as an instantaneous nonlinear mode of $H(\Psi_{q(t)})$ throughout this procedure, provided that the nonlinear mode $\Psi_{q}$ is stable Pu et al. (2007) within the amplitude scope of this paper. The stability of nonlinear bulk modes is confirmed in App. C via the algorithm of self-oscillation Fronk and Leamy (2017); Vila et al. (2019); Pal et al. (2018). Therefore, the only degree of freedom is the phase of mode. At time $t$, the mode is $\Psi_{q(t)}(\int_{0}^{t}\omega(t^{\prime},q(t^{\prime}))dt^{\prime}-\gamma(t))$, where $\gamma(t)$ defines the phase shift of the nonlinear bulk mode in the adiabatic evolution. The dynamics of $\gamma$ is depicted by $(d\gamma/dt)(\partial\Psi_{q}/\partial\theta)=(dq/dt)(\partial\Psi_{q}/\partial q)$. After $q$ traverses the Brillouin zone, the wave function acquires an extra phase $\gamma$ dubbed Berry phase of nonlinear bulk modes,

where $j,j^{\prime}=1,2$ denote the two wave components, and the mathematical derivations are in App. A. In general, $\gamma$ is not quantized unless additional symmetry properties are imposed on the model, which we will discuss below. We note that the eigenmodes of linear problems are sinusoidal in time, which reduces Eq.(3) to the conventional form Xiao et al. (2010) $\gamma=\oint_{\rm BZ}\mathrm{d}q\,\mathrm{i}\langle\Psi_{q}|\partial_{q}|\Psi_{q}\rangle$.

Now we demonstrate that Berry phase defined in Eq.(3) is quantized by reflection symmetry. The model in Eqs.(1) respects reflection symmetry, which means that the nonlinear equations of motion are invariant under reflection transformation,

Given a nonlinear bulk mode $\Psi_{q}$ in Eq.(2), reflection transformation demands a partner solution $\Psi_{-q}^{\prime}=(\Psi^{(2)}_{q}(\omega t+qn),\Psi^{(1)}_{q}(\omega t+qn-\phi_{q}))^{\top}$ that also satisfies the model. On the other hand, a nonlinear bulk mode of wavenumber $-q$ is by definition denoted as $\Psi_{-q}=(\Psi_{-q}^{(1)}(\omega t+qn),\Psi_{-q}^{(2)}(\omega t+qn+\phi_{-q}))^{\top}$. Since there is no degeneracy of nonlinear bulk modes, $\Psi_{-q}^{\prime}$ and $\Psi_{-q}$ have to be identical, which imposes the constraints

Thus, the Fourier components of nonlinear bulk modes satisfy $\psi_{l,q}^{(2)}=\psi_{l,-q}^{(1)}$. This relationship, together with Eqs.(5), is the key to quantizing the Berry phase in Eq.(3) (details in App. B2),

where $\phi_{q=0,\pi}$ are the relative phases of the upper-band nonlinear modes at high-symmetry points. They are determined by comparing the frequencies $\omega(\phi_{q}=0)$ and $\omega(\phi_{q}=\pi)$ for $q=0$ and $\pi$. $\gamma=\pi$ if $\omega(\phi_{0}=0)$ and $\omega(\phi_{\pi}=\pi)$ belong to the same band, whereas $\gamma=0$ if they are in different bands. Interestingly, $\gamma$ encounters a topological transition induced by the critical amplitude $A=A_{c}$ if frequencies merge at $\omega(\phi_{\pi}=0,A_{c})=\omega(\phi_{\pi}=\pi,A_{c})$. This transition is exemplified by the minimal model of nonlinear topological lattice in Sec.III. It is worth emphasizing that despite all the discussions of nonlinear Schrödinger equations and the quantization of Berry phase, the model is purely classical in the sense of $\hbar$ being zero.

Having established quantized Berry phase, we now search additional properties for vanishing on-site potential, $\epsilon_{0}=0$. The model’s linear limit respects charge-conjugation symmetry Ryu et al. (2010); Kane and Lubensky (2014), which demands that the states appear in $\pm\omega$ pairs, and the topological mode have zero-energy. To have $\pm\omega$ pairs of modes in the nonlinear problem, we require the parity of interactions to satisfy $f_{i}(-x,y)=-f_{i}(x,-y)=f_{i}(x,y)$. Consequently, the system is invariant under the transformation $(\Psi^{(1)}_{n}(\omega t),\Psi^{(2)}_{n}(\omega t))\to(-\Psi^{(1)}_{n}(-\omega t),\Psi^{(2)}_{n}(-\omega t))$. Given a nonlinear mode $\Psi_{\omega}$ defined in Eq.(2), this transformation demands a partner solution $\Psi_{-\omega}=(-\Psi^{(1)}_{q}(-\omega t-qn),\Psi^{(2)}_{q}(-\omega t-qn+\phi_{q}))^{\top}$. Therefore, nonlinear modes always appear in $\pm\omega$ pairs. Similar to charge-conjugation symmetric models in linear systems Ryu et al. (2010), the frequencies of nonlinear topological modes are zero, which is illustrated in the following minimal model.

We now clarify the nonlinear extension of bulk-boundary correspondence Chaunsali and Theocharis (2019); Shang et al. (2018) by demonstrating topological edge modes in the minimal model that respects time-reversal symmetry, where the couplings are specified as

with $c_{i},d_{i}>0$ for $i=1,2$. This interaction offers numerically stable nonlinear bulk and topological edge modes and can be realized in passive photonic and active electrical circuit metamaterials (Sec.IV and App. E). We are interested in attributes unique to nonlinear systems, in particular the topological phase transition induced by bulk mode amplitudes. Thus, the parameters yield $c_{1}<c_{2}$ and $d_{1}>d_{2}$ ($c_{1}>c_{2}$ and $d_{1}<d_{2}$) to induce topological-to-non-topological phase transition (non-topological-to-topological transition) as amplitudes increase. We abbreviate them as “T-to-N” and “N-to-T” transitions, and they are converted to one another by simply flipping intracell and intercell couplings. In the remainder of this paper, a semi-infinite lattice subjected to open boundary condition (OBC) is always considered whenever we refer to topological edge modes.

We first study the case $c_{1}<c_{2}$ and $d_{1}>d_{2}$, in which a T-to-N transition occurs. Fig.1(d) numerically illustrates nonlinear band structures and topological transition by considering $\epsilon_{0}=1.5$, $c_{1}=0.25$, $c_{2}=0.37$, $d_{1}=0.22$, and $d_{2}=0.02$. Given that Berry phase $\gamma(A=0)=\pi$, the lattice is topologically nontrivial in the linear limit. As amplitudes rise, the topological invariant $\gamma(A<A_{c})=\pi$ cannot change until it becomes ill-defined when the nonlinear bandgap closes at the transition amplitude $A_{c}$. The bandgap reopens above $A_{c}$, allowing the well-defined Berry phase to take the trivial value $\gamma(A>A_{c})=0$, as depicted in the inset of Fig.1(d). $A_{c}$ is numerically computed by solving the bandgap-closing equation $\omega(\phi_{\pi}=0,A_{c})=\omega(\phi_{\pi}=\pi,A_{c})$. We propose a convenient approximation Detroux et al. (2014) $f(\Psi_{n^{\prime}}^{(j^{\prime})},\Psi_{n}^{(j)})\approx(c_{i}+\frac{3}{4}d_{i}A^{2})\Psi_{n}^{(j)}$ to estimate the transition amplitude $A_{c}\approx\sqrt{-4(c_{2}-c_{1})/3(d_{2}-d_{1})}$. The good agreement between this approximation and the numerical solutions is shown in App. C. We highlight that $A_{c}^{2}\max(d_{1},d_{2})/\max(c_{1},c_{2})\approx 0.5$, which demonstrates the comparable nonlinear and linear interactions in the strongly nonlinear regime.

Bulk-boundary correspondence has been extended to weakly nonlinear Newtonian Chaunsali and Theocharis (2019) and Schrödinger Shang et al. (2018) systems by showing topological boundary modes guaranteed by topologically non-trivial Berry phase. In the strongly nonlinear problem, we utilize analytic approximation and numerical experiment, to doubly confirm this correspondence by identifying nonlinear topological edge modes. In the former, the lattice is composed of $N=45$ unit cells with OBCs on both ends to mimic semi-infinite lattice, and the parameters are carried over from Fig.1. The topological mode and frequency are denoted as $\Psi_{n}=(\Psi_{n}^{(1)},\Psi_{n}^{(2)})^{\top}$ and $\omega_{\rm T}$, respectively. Analogous to linear SSH chain Su et al. (1979), the analytic scheme is to approximate $\Psi_{n}^{(1)}\gg\Psi_{n}^{(2)}$, which is numerically verified in Fig.2(d). We make one further approximation to truncate the equations of motion to fundamental harmonics. Therefore, the nonlinear topological edge mode is approximated as $\Psi_{n}\approx(\psi_{1,n}^{(1)},0)^{\top}e^{-\mathrm{i}\epsilon_{0}t}$, where $\psi_{1,n}^{(1)}$ are the fundamental harmonic components. By doing so, we find $\omega_{\rm T}=\epsilon_{0}$, and

From Eq.(III), the semi-infinite lattice hosts topological evanescent modes when $|\Psi_{1}^{(1)}|<\sqrt{-4(c_{2}-c_{1})/3(d_{2}-d_{1})}\approx A_{c}$, whereas no such mode exists for $|\Psi_{1}^{(1)}|>\sqrt{-4(c_{2}-c_{1})/3(d_{2}-d_{1})}\approx A_{c}$. In App. D, the frequency and analytic expression are applied in weakly nonlinear regime, and they are perfectly in line with method of multiple-scale Fronk and Leamy (2017); Narisetti et al. (2010); Zaera et al. (2018); Snee and Ma (2019). The numerical scenario is accomplished by applying a Gaussian profile signal $S_{n}=\delta_{n1}Se^{-\mathrm{i}\omega_{\rm ext}t-(t-t_{0})^{2}/\tau^{2}}(1,0)^{\top}$ on the first site, where the carrier frequency $\omega_{\rm ext}=\epsilon_{0}=1.5$, $T=2\pi/\omega_{\rm ext}$, $\tau=3T$ controls Gaussian spread, and $t_{0}=15T$ denotes trigger time. Figs.2(b) and (f) together verify bulk-boundary correspondence Chaunsali and Theocharis (2019); Shang et al. (2018) by identifying the presence and absence of topological boundary excitations below and above the critical amplitude $A_{c}$, respectively. In fig.2(d), the flattened part near the lattice boundary is the manifestation of nonlinearities.

One may find it unusual that the frequencies of topological modes $\omega_{\rm T}=\epsilon_{0}$ are independent of amplitudes, although this result is in agreement with Ref. Hadad et al. (2018, 2016); Chaunsali and Theocharis (2019) in weakly nonlinear regime. Here we propose an explanation for this intriguing result. Because the evanescent mode fades to zero in the bulk, the “tail” of this mode eventually enters into small-amplitude regime where nonlinearities are negligible and linear analysis becomes effective. Linear topological theory Su et al. (1979) demands the tail of the mode to be $\omega_{\rm T}=\epsilon_{0}$, which in turn requires the frequency of the nonlinear topological mode to be independent of the amplitude.

Topological protection is featured in multiple aspects. As visualized in Fig.1(d), the frequencies of topological modes stay in the bandgap and are distinct from nonlinear bulk modes. The appearance and absence of these modes are captured by the topological invariant that cannot change continuously upon the variation of system parameters. Lastly, topological modes are insensitive to defects, which is numerically verified in App. D.

When $\epsilon_{0}=0$, the model manifests nonlinear bulk modes in $\pm\omega$ pairs. Topologically protected nonlinear boundary modes do not oscillate in time, in contrast to the $\epsilon_{0}\neq 0$ systems. Thus, we obtain exact solutions of nonlinear topological modes via the recursion relation, $f_{1}(\Psi_{n}^{(1)},\Psi^{(2)}_{n})+f_{2}(\Psi_{n}^{(1)},\Psi^{(2)}_{n-1})=f_{1}(\Psi_{n}^{(2)},\Psi^{(1)}_{n})+f_{2}(\Psi_{n}^{(2)},\Psi^{(1)}_{n+1})=0$. This is the nonlinear analog of charge-conjugation symmetric systems.

In the second case of $c_{1}>c_{2}$ and $d_{1}<d_{2}$, N-to-T (non-topological-to-topological) transition occurs as amplitudes rise. We exemplify boundary excitations in Fig.3 by letting $\epsilon_{0}=8$, $c_{1}=0.37$, $c_{2}=0.25$, $d_{1}=0.02$, and $d_{2}=0.22$. A Gaussian signal is applied on the first site of the lattice, where the carrier frequency $\omega_{\rm ext}=\epsilon_{0}=8$, $T=2\pi/\omega_{\rm ext}$, Gaussian spread $\tau=10T$, and trigger time $t_{0}=25T$. In the small-amplitude regime, we consider a chain of $N=45$ unit cells. As shown in Fig.3(b), the lattice is free of topological modes for $|\Psi_{1}^{(1)}|<A_{c}=0.8944$. In the large-amplitude regime, the lattice is constructed from $N=120$ unit cells. Anomalous topological edge modes emerge when $|\Psi_{1}^{(1)}|>A_{c}$ (see figs.3(f,h)). In contrast to conventional topological modes that shrink to zero over space, $\Psi_{n}^{(1)}$ decay to the plateau $A_{c}$ governed by the stable fixed point of Eq.(III), whereas $\Psi_{n}^{(2)}$ increase to $A_{c}$ by absorbing energy Fronk and Leamy (2017) from $\Psi_{n}^{(1)}$. Theoretical analysis predicts that the plateau should extend to infinity, but the plateau is limited to reach site $60$ by the finite lifetime of topological modes due to the energy conversion to bulk modes, as elaborated in Fig.D2. Despite the huge nonlinearities ($|\Psi_{1}^{(1)}|/A_{c}\sim 10$, and $|\Psi_{1}^{(1)}|^{2}\max(d_{1},d_{2})/\max(c_{1},c_{2})\sim 10$), this mode is stable within the finite lifetime of more than 400 periods. This model serves as the combined prototype of long-lifetime, high-energy storage, long-distance transmission of topological modes, and efficient frequency converter from Gaussian inputs to monochromatic signals.

Although T-to-N and N-to-T transitions are converted to one another by choosing the unit cell, topological modes behave qualitatively different (Fig.2(d) and 3(h)) due to the distinction in the fixed points of Eq.(III). The modes converge to the stable fixed point $0$ in T-to-N transition ($A_{c}$ in N-to-T transition), but this fixed point becomes unstable in N-to-T transition (T-to-N transition).

Upon establishing nonlinear topological band theory, it is natural to ask if any realistic physical systems enjoy these unconventional properties. Classical passive and active structures are proposed here to realize the minimal model of Eq.(7), as detailed in App. E.

Topological photonics D’Aguanno et al. (2008); Christodoulides and Joseph (1988), passive system: Our theoretical prototype is readily testified in 1D array of optic lattice. Each unit cell is composed of two waveguides to guide electro-magnetic modes along the axial direction, and the permittivity and permeability are nonlinearly modulated by the fields. Hence, the adjacent electro-magnetic fields are coupled nonlinearly. It can be shown that the propagation of electro-magnetic fields along the axial $z$-direction is depicted by 4-field extension of generalized nonlinear Schrödinger equations, where the $z$-coordinate takes place of the time-like differential variable D’Aguanno et al. (2008); Christodoulides and Joseph (1988). Consequently, this photonic system realizes the minimal model of Eq.(7).

Topoelectrical circuit Hadad et al. (2018), active system: The second promising direction is to construct a ladder of cascaded diatomic unit cells composed of two LCR resonators and two capacitors $C_{j=1,2}\ll C$. The inductances are connected to external power sources which are nonlinear functions of $V_{n}^{(j=1,2)}$. The motion equation of the unit cell voltages $V_{n}^{(j=1,2)}$ are captured by Eqs.(1) and Eq.(7), and nonlinear topological attributes can be studied here.

In this paper, we extend topological band theory to strongly nonlinear Schrödinger equations beyond Kerr-type nonlinearities. The proper definition of Berry phase is carried out for nonlinear bulk modes, and its quantization is demonstrated in reflection-symmetric models. The topological invariant experiences transitions induced by mode amplitudes. These results can be extended to higher dimensional systems with arbitrarily complex unit cells, but we leave the full proof for the future.

The advent (disappearance) of topological modes is associated with a change in the Berry phase to its topological (non-topological) value. As amplitudes increase, T-to-N (topological-to-non-topological) and N-to-T (non-topological-to-topological) transitions take place for different choices of unit cells. Anomalous topological modes decrease away from lattice boundaries to a plateau controlled by the stable fixed point of nonlinearities.

A rich variety of problems can be studied following this paper, such as the nonlinear extension of topological chiral edge modes in 2D systems Haldane (1988), and higher-order topological states Benalcazar et al. (2017). Experimental characterizations of photonic, acoustic, and electrical metamaterials with built-in nonlinearities can also be studied in future.