1. Introduction

A wide range of topological quantum effects manifest themselves in symmetries of an excitation spectrum. This relation has been explored extensively for fermionic systems, for which the single-particle Hamiltonian can obey a chiral symmetry or a charge-conjugation symmetry [1–3]. Allowing also for time-reversal invariance one can identify ten universality classes featuring a variety of topological quantum numbers. These quantum numbers determine the formation of zero modes and unidirectional transport channels at edges, surfaces, and interfaces of various dimensions [4,5]. In the fermionic case these symmetries are fundamental as they are owed to the structure of Fock or Nambu space [6], and therefore also extend to the interacting case [7]. Prime examples are superconducting systems, for which a charge-conjugation symmetry dictates that excitations ψ exp(−iωt) at a positive frequency ω are paired with excitations Xψ^* exp(iωt) at the corresponding negative frequency −ω, where X is a suitable unitary transformation. The systems can then also feature robust Majorana zero modes ψ = Xψ^* at ω = 0 [8–10], an effect which translates to the fermion parity anomaly in the full many-body theory [11–13].

Some of these symmetries also play a natural role beyond the fermionic context. For instance, time-reversal symmetry is intimately related to optical reciprocity [14], a classical concept whose modification leads to topological effects in photonic structures [15] and analogous optical [16–19], acoustic [20,21] and mechanical [22] systems. Topological effects can also be engineered into bosonic quantum systems, such as cold atomic gases [23] and exciton polaritons [24–26]. For weakly interacting bosons the excitations are again characterized by a Bogoliubov theory [27–30], and recent work has explored a variety of mechanisms to engineer topological features into this description [31–37]. The charge-conjugation symmetry can also be induced into the single-particle theory of bosonic systems, where one again can admit linear gain or loss [38–42]. The spectrum of the effective Hamiltonian then displays symmetric pairs of complex frequencies (Ω_n, $-{\mathrm{\Omega }}_n^{*}$), which can be interpreted as resonances. Topological features persist because this spectral constraint can enforce a number of purely imaginary self-symmetric resonances ${\mathrm{\Omega }}_n=-{\mathrm{\Omega }}_n^{*}$, which provide an analogue to broadened fermionic zero modes [43–47].

Five recent experiments aimed at realizing topological zero modes in polaritonic condensates [48,49] and lasers [50–52]. As these are inherently nonlinear systems, the question arises whether the notion of zero modes and topological protection persists. Here, we provide a unifying perspective on these systems based on a notion of charge-conjugation symmetry that applies directly to the time-dependent nonlinear wave equation. Our general strategy is as follows: Topological states in linear systems are protected by symmetry, but their number can change discretely in phase transitions, which are generally linked to degeneracies (such as when a band gap closes). This notion is underpinned by the continuity of the spectrum under smooth parameter changes (deformations of the system), a feature at the heart of linear spectral analysis. Analogously, we show that nonlinear systems can display dynamical solutions that are protected by symmetry. Their number can change in dynamical degeneracies, which correspond to bifurcations. This notion is then underpinned by the structural stability of dynamical solutions, which is at the heart of nonlinear dynamics. This nonlinear notion of topological states reduces to the conventional spectral notions in the linear limit. New effects also appear: We identify a new class of topological solutions that does not have a linear counterpart, and also exploit the link of bifurcations to a qualitative change of the linear stability against changes of initial conditions, which again does not feature in the linear context.

In detail, as we will see, nonlinearities in the gain or loss lead to complex-wave dynamics where the time-dependent solutions appear in pairs Ψ(t) and XΨ^*(t). In contrast to a general time-reversal symmetry, which induces a partner solution XΨ^*(−t), the solutions Ψ(t) and XΨ^*(t) both exhibit the same arrow of time. This time-preserving symmetry therefore allows for self-symmetric states—including stationary states $\mathrm{\Psi }(t)={\mathrm{\Psi }}_0=X{\mathrm{\Psi }}_0^{*}$, which we interpret as zero modes. As a notable feature without analogue in the linear case, the symmetry also protects a twisted variant of time-dependent power-oscillating states with Ψ(t + T/2) = XΨ^*(t), for which periodicity Ψ(t + T) = Ψ(t) is enforced by symmetry—suggesting that the nonlinear setting admits for a richer notion of protected states than the linear case. That these states enjoy topological protection can be further ascertained by identifying topological excitations in the stability spectrum, which we find to govern the spectral phase transitions between the different types of symmetry-protected solutions—corresponding to different topological phases of the dynamical system. We illustrate our findings for two model systems representing one-dimensional topological laser arrays with saturable gain and two-dimensional flat-band polariton condensates with density-dependent loss.

2. Results

2.1. Model and classification of states

Consider a bosonic system whose classical limit is described by a complex-wave equation

We will describe practical ways to realize such systems soon below but first discuss the consequences of the following general observation. For any solution Ψ(t) of Eq. (1), we obtain a partner solution

Let us first reflect on the possible classes of solutions admitted by Eq. (1). Because of the underlying charge-conjugation symmetry of H, the linear system possesses pairs of solutions Ψ(t) = Ψ_n exp(−iΩ_nt) and $\tilde{\mathrm{\Psi }}(t)=X{\mathrm{\Psi }}_n^{*}\text{exp}\left(i{\mathrm{\Omega }}_n^{*}t\right)$, where the latter expression corresponds to a frequency ${\tilde{\mathrm{\Omega }}}_n=-{\mathrm{\Omega }}_n^{*}$. This includes the possibility of topologically protected zero modes with purely imaginary frequency Ω_n = Ω̃_n = i Im Ω_n and time dependence Ψ(t) = Ψ̃(t) = Ψ_n exp(Im Ω_nt) [39–41, 43–47]. If for any of these states Im Ω_n is positive the linear system is unstable, but the nonlinear effects can stabilize the system. This results in stationary states Ψ(t) = Ψ_n exp(−iΩ_nt) with real Ω_n, which are self-consistent solutions of the condition [54]

The nature of a state can be assessed, e.g., by comparing the correlation functions C = |〈Ψ(0)|Ψ(t)〉|, C̃ = |〈Ψ(0)|Ψ̃(t)〉|, which coincide or alternate for self-symmetric or twisted solutions.

Based on the possible invariance of time-parameterized orbits under discrete involutions, this categorization is minimal and complete. Note that the solutions of class (a), (d) and (e) all describe orbits Ψ(t) that are invariant under the symmetry operation (2). In the self-symmetric cases (a) and (d), this invariance is local at every point along the orbit, while for the twisted states (e) the invariance occurs under a translation by T/2. The latter leaves the orbit invariant as the solution is periodic. All these solutions are furthermore symmetry-protected, in that their number can only change due to structural reconfigurations of the dynamical solutions. That such changes are indeed possible at all is necessary in order to have a setting with meaningful topological features, and not just symmetry-imposed constraints. In linear systems, the phase transitions at which the number and nature of symmetry-protected solutions changes involves degeneracies; as we will explore further in Section 2.3, for the nonlinear setting this naturally extends to bifurcations. First, however, we provide concrete model systems with the time-preserving symmetry of Eq. (1) and examples of the different classes of solutions.

2.2. Realization in lasers and condensates

Figures 1 and 2 illustrate the different types of solutions for two model systems, which are both formed of bipartite lattices (N_A sites A and N_B sites B) that give rise to a pseudospin degree of freedom. The dynamics is governed by the equations

 

Fig. 1 Illustration of the five types of modes for a topological laser array based on a Su-Schrieffer-Heeger chain with background loss γ_A = γ_B = 0.3 and various amounts of saturable gain [see Eqs. (4), (5) and (6)]. (a) Stationary symmetry-breaking mode (g_A = 0.4, g_B = 0.7). (b) Stationary self-symmetric zero mode (g_A = 0.8, g_B = 0.0). (c) Oscillating symmetry-breaking mode (g_A = 0.7, g_B = 0.4). (d) Oscillating self-symmetric mode (g_A = 0.7, g_B = 0.4 with symmetry-preserving initial conditions). (e) Twisted oscillating mode (g_A = 0.8, g_B = 0.2). The sketches at the very top symbolize the traces of the solutions in low-dimensional cross-sections of the dynamical phase space, where in (a–d) the symmetry is represented as a reflection and in (e) as a rotation. In the second row, the circles represent the resonators, where the area denotes the intensity (A and B sublattice in red and blue; in the time-dependent case, we show two circles corresponding to the largest and smallest intensity over a cycle.) The third row shows the time traces of the intensities I_A = |A|² (red), I_B = |B|² (blue), and I_tot = I_A + I_B (black). The bottom panels in (a,b) show the stability spectra of the stationary states, while in (c–e) they show the correlation functions C = |〈Ψ(0)|Ψ(t)〉| (orange), C̃ = |〈Ψ(0)|Ψ̃(t)〉| (brown) of the oscillating states.

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Fig. 2 Same as Fig. 1 but for a polaritonic flat-band condensate based on a Lieb lattice with linear gain and density-dependent loss [see Eq. (7)]. To demonstrate the generality of our findings we include 50% relative disorder in all parameters, including for the couplings around their mean value t_kl = 1, and the losses with average strength γ_A = γ_B = 0.3 (see Fig. 3 for details of the configuration). (a) Stationary symmetry-breaking mode (average gain g_A = 0.15, g_B = 0.3). (b) Stationary self-symmetric zero mode (average background loss g_A = −0.2 and gain g_B = 0.5). (c) Slowly oscillating symmetry-breaking mode (g_A = 0.35, g_B = 0.2). (d) Oscillating self-symmetric mode (g_A = 0.5, g_B = 0.3 with symmetry-preserving initial conditions). (e) Twisted oscillating mode (g_A = 0.1, g_B = 0.4).

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Fig. 3 Detailed geometry of the two illustrative models investigated in this work. (a) The model based on the SSH chain consists of a linear arrangement of 21 sites (11 A sites and 10 B sites) with alternating couplings 1, 0.7. The centre contains a defect with two consecutive couplings 0.7. This separates two configurations, denoted α and β, which can be characterized by topological features of their band structure. In the hermitian model the coupling defect induces one zero mode, which is spatially localized and exhibits an anomalous response to loss and gain of different strength on the two sublattices. For this model, we introduce nonlinearities in the form of saturable gain. (b) The model based on the Lieb lattice also consists of 21 sites, but these are arranged in two dimensions so that 12 are A sites and 9 are B sites. In the hermitian limit, there are now at least three zero modes, even in the case of disorder in the couplings. In this model we study density-dependent losses, and include disorder in the couplings t_kl as well as in the gain and loss parameters g_A,B and γ_A,B. This disorder is generated from independent random numbers r_n with a box distribution over [0.75, 1.25], so that p_n = pr_n for any model parameter p_n with average p.

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For the SSH chain, we consider nonlinearities that represent saturable gain,

For any solution with amplitudes A(t) and B(t), Eqs. (4) and (5), exhibits a partner solution with amplitudes A^*(t) and −B^*(t), so that X = 𝟙_A ⊕ (−𝟙_B) acts as a Pauli-z matrix in pseudospin space. Admitting different amounts of gain g_A, g_B and loss γ_A, γ_B on the two sublattices allows us to study the competition between topological and nontopological states, which leads to the different examples shown in the figures (for a detailed mapping of the phase space in the SSH model see [78]).

Note that the form of X in these models implies that self-symmetric states display a rigid phase-shift of i between the two sublattices. Given that Ψ̃(t) = Ψ(t), the amplitudes A(t) are real while the amplitudes B(t) are imaginary, which then persists for all times. On the other hand, symmetry-breaking stationary states with a finite frequency must have equal weight on both sublattices [40]. The different types of states can therefore also be discriminated by their distinct spatial features.

2.3. Phase transitions and topological features of the excitation spectrum

The presented classification of solutions is exhaustive in terms of symmetry-protection. To complete the analogy to topological notions in linear systems, it is essential to explore how the number of symmetry-protected solutions can change. In linear systems, this is generally connected to degeneracies, relying on the structural stability of the spectrum against parametric changes. In nonlinear systems, this is paralleled by the notion of structural stability of dynamical solutions (again against parameter changes), with qualitative changes mandated by bifurcations. In this context, we can then exploit the general link of bifurcations to a change of the linear stability (against initial conditions), which allows an emerging spectral interpretation in terms of the stability of linear perturbations. To conclude this paper, we therefore further illuminate the topological aspects of the symmetry-protected solutions by spectral signatures. Here, we in particular exploit the pinning of excitations to symmetry protected positions, in analogy to Majorana zero modes in fermionic systems with charge-conjugation symmetry [1,2] and analogous zero modes in periodically driven systems [5, 75]. These connections can be established by utilizing the link between dynamical stability of nonlinear systems and their excitation spectrum, which is addressed by Bogoliubov theory. We first present the results of this analysis, and describe the technical details in the following subsection.

The analysis amounts to identifying the eigenmodes ψ = (u, v)^T of linear perturbations ue^−iωt + v^*e^iωt, and here results in a spectrum of 2(N_A + N_B) excitation frequencies ω_m. For a stable system all of these excitations must obey Im ω_m ≤ 0. For non-self-symmetric pairs of stationary modes Ψ_n, Ψ̃_n, the time-preserving symmetry of Eq. (1) implies that their excitation spectra are identical, but does not impose any further constraints on these spectra. For self-symmetric stationary zero modes Ψ₀ = Ψ̃₀ with Ω₀ = 0, on the other hand, we can classify the perturbations into symmetry-preserving modes v₊ = u₊ and symmetry-breaking modes v₋ = −u₋. For our models of gain or loss, these perturbations fulfill the reduced eigenvalue equations

Panel (b) in Figs. 1 and 2 shows that the eigenvalues ω₋ determine the stability of the stationary zero modes while the eigenvalues ω₊ lie deep in the complex plane. Therefore, if we restrict the perturbations to preserve the symmetry, the stability of such modes is greatly enhanced. Note that both reduced spectra in these examples have an odd number of excitations. Thus, each reduced spectrum has an odd number of eigenvalues pinned to the imaginary axis. Setting u₋ = Ψ₀ we recover that one of the eigenvalues ω_−,0 = Ω₀ = 0 vanishes, in accordance with the U(1) symmetry of the wave equation.

A similar analysis can be carried out for time-dependent solutions Ψ(t), whose stability is then characterized by a corresponding quasienergy spectrum obtained from a propagator U(t) with eigenvalues λ_m = exp(−iω_mt). This again includes the U(1) Goldstone mode with λ₀ = 1, but also an additional Goldstone mode λ_T = 1 that describes the time-translation freedom of any solution Ψ(t). For general pairs of states Ψ(t), Ψ̃(t), the time-preserving symmetry again dictates that their quasienergy excitation spectra are identical. For self-symmetric states Ψ(t) = Ψ̃(t), we can once more separate perturbations that preserve or break the symmetry. This leads to time-dependent variants of Eqs. (8) and (9),

For the final case of a twisted periodic state with Ψ(t + T/2) = Ψ̃(t), the evolution operator of the perturbations factorizes as U(T) = U′² with a half-step propagator U′ = 𝒳Σ_xU(T/2), where 𝒳 = 𝟙₂⊗X while Σ_x = σ_x ⊗𝟙 interchanges u and v. The U(1) gauge and time-translation freedoms can now in principle be satisfied in two ways, corresponding to ω′_0,T T/2 = 0 or π. Inspecting the Goldstone mode ψ₀(t) = (Ψ(t), −Ψ^*(t))^T we invariably find that the twisted solutions realize ω′₀T/2 = π, while the time-translation freedom remains associated with ω′_TT/2 = 0. This separation of eigenvalues to symmetry-protected positions is again a robust spectral feature that can only change in further phase transitions.

The key observation in this respect is that these phase transitions naturally link the appearance of twisted modes to the instability of zero modes, which occurs when a pair of excitations ω̃_−,★ = − ω̃_−,★ = 2π/T crosses the real axis (see Fig. 4), and hence corresponds to a Hopf bifurcation [73, 74] into a time-dependent state that here is still protected by symmetry. At the transition, the excitations ω_n map onto the stability spectrum λ_n = exp(−iω_nT) of the emerging twisted state, where ω_−,★ and ω̃_−,★ both map onto eigenvalues λ_−,★ = λ̃_−,★ = 1. These are degenerate with the U(1) Goldstone mode λ₀ = 1. Away from the transition, these three excitations rearrange into a decaying excitation |λ_f| < 1 describing the stabilization of the oscillation amplitude of the twisted mode, and the Goldstone modes λ₀ = λ_T = 1 from U(1) gauge and time-translation invariance. These phase transitions therefore provide the mechanism by which the proposed class of nonlinear systems acquires robust dynamical features that do not have a counterpart in the linear case.

 

Fig. 4 (a) Phase transition from a stationary zero mode to a twisted oscillating state of period T as signalled by the linear stability excitation spectrum, here shown for the SSH laser array with g_A = 0.693, g_B = 0.1. At the transition the excitations match up via the relation λ = exp(−iωT). This leads to a three-fold degeneracy of marginally stable excitations with λ = 1. (b) Away from the transition (g_A = 0.8, g_B = 0.1), the excitations rearrange to describe the stabilization of the oscillation amplitude of the emerging twisted mode (λ_f), leaving a two-fold degeneracy of marginal excitations λ₀ = λ_t = 1 corresponding to U(1) and time-translation invariance. In the half-step operators, these two excitations are separated at λ′₀ = −1, λ′_t = 1 and hence structurally stabilized, which provides a signature of twisted states in terms of topological excitations.

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2.4. Detailed derivation of the symmetry-constrained excitation spectrum

We here provide the technical details on the derivation of the stability excitation spectrum for the different solutions identified in the previous subsection.

For stationary states, their stability can be analyzed by setting

The time-preserving symmetry of the nonlinear wave equation (1) induces the relation

For our models with saturable gain or density-dependent loss, this simplifies to Eqs. (8) and (9). As described there, the excitation spectrum is thus composed of two parts, eigenvalues ω₊ that display an enhanced nonlinearity and eigenvalues ω₋ that coincide with those of the nonlinear wave operator H + f, with Ψ = Ψ₀ fixed to the zero mode. According to Eq. (12), the perturbations [$e^{-i{\omega }_{-}t}{u}_{-}-e^{i{\omega }_{-}t}X{u}_{-}^{*}$] corresponding to ω₋ break the time-preserving symmetry of the zero mode. Setting u₋ = Ψ₀ we recover that one of the eigenvalues ω₋ = Ω₀ = 0 vanishes, in accordance with the U(1) symmetry of the wave equation.

For time-dependent solutions Ψ(t), their stability against perturbations u(t) + v^*(t) is governed by the time-dependent Bogoliubov equation

For general pairs of states Ψ(t), Ψ̃(t), the propagators are related as U(t) = 𝒳Ũ^*(t)𝒳 = 𝒳Σ_xŨ(t)Σ_x𝒳, which again implies that their quasienergy excitation spectra are identical. This also applies to the case of periodic solutions with Ψ(t) = Ψ(t + T), where the propagator U(T) represents a Bogoliubov-Floquet operator. Such solutions are unstable if |λ_m| > 1, hence Im ω_m > 0, apart from the two Goldstone modes that are fixed at λ₀ = λ_T = 1. For the more general case of periodic solutions with Ψ(t) = exp(iφ)Ψ(t + T), this applies when the Floquet operator is redefined as diag(e^−iφ, e^iφ)U(T).

For self-symmetric states Ψ(t) = Ψ̃(t), we have at all times U(t) = 𝒳U^*(t)𝒳 = 𝒳Σ_xU(t)Σ_x𝒳. This allows us to separate perturbations that preserve or break the symmetry, and leads to the time-dependent variants (10) and (11) of Eqs. (8) and (9). For periodic states Ψ(t) = exp(iφ)Ψ(t + T), the self-symmetry constraints the phase to φ = 0, π, hence τ = exp(iφ) = ±1, which needs to be respected in the proper definition of the Bogoliubov-Floquet operator F = τU(T) to result in two Goldstone modes pinned at λ₀ = λ_T = 1.

For the final case of a twisted periodic state with Ψ(t + T/2) = Ψ̃(t), we have automatically Ψ(T) = Ψ(0) without any additional phase [furthermore, the case Ψ(t + T/2) = −Ψ̃(t) is not separate since we can then redefine Ψ(t) → iΨ(t)]. The Floquet operator factorizes as

3. Conclusion

In summary, we showed that spectral symmetries underpinning topological quantum systems can be extended to nonlinear complex-wave equations, where they lead to robust constraints of the dynamics. Conceptually, this allowed us to identify a generalization of zero modes from the linear to the nonlinear setting, as well as symmetry-protected power-oscillating modes that do not have a counterpart in the linear case. These states furthermore support symmetry-protected excitations, which play a crucial role in their dynamical features. Practically, nonlinearities are of fundamental importance to stabilize systems with loss and gain in quasistationary operating regimes. In particular, the symmetry-protected modes can be induced into weakly interacting bosonic systems with saturable gain or density-dependent loss, such as recently realized in polaritonic condensates [24, 49] and topological lasers [50–52]. The results can also be reinterpreted in classical wave settings, as encountered, e.g., in conventional optics and acoustics.

These findings show that the notion of symmetry-protected topological states persists in nonlinear bosonic systems, and that the nonlinearities lead to an even richer phenomenology than in the original single-particle context. On the practical side, it is worthwhile to explore further extensions to include the dynamics of the medium [79], which is particularly important when one considers time-dependent solutions. More generally, it would be highly worthwhile to explore nonlinear extensions for other representatives of linear topological universality classes, for which experimental realizations and a general classification are currently unknown.