Nonlinear Topological Phase Diagram

Updated 22 September 2025

Nonlinear topological phase diagrams are maps that integrate nonlinear interactions and many-body correlations to define distinct quantum, classical, or semiclassical phases.
They reveal phase boundaries and multicritical points where traditional topological invariants, like the Chern number, may remain constant despite critical nonlinear transitions.
Applications include engineered phase control in photonics, superconducting circuits, and interacting spin models, with edge mode spectroscopy and correlation function analysis as key detection tools.

A nonlinear topological phase diagram is a map of quantum, classical, or semiclassical phases in systems where topology is interwoven with nonlinear interactions or nonlinear energy landscapes. Unlike conventional phase diagrams, which typically segment phases based primarily on global topological invariants such as the Chern number, nonlinear topological phase diagrams feature rich critical structures, emergent boundary phenomena, and transitions driven by strongly nonlinear dynamics or correlations. These diagrams are central to the understanding of topological quantum systems in regimes where linear approximations break down and are crucial for advancing fields ranging from interacting spin models and soliton lattices to experimental implementations in photonics, mechanics, and superconducting circuits.

1. Definition and Key Characteristics

A nonlinear topological phase diagram delineates regions in the space of control parameters (such as couplings, amplitudes, disorder strength, or driving intensity) where the system exhibits distinct topological properties, while explicitly incorporating the effects of nonlinear interactions or genuinely nonlinear dynamical regimes. Nonlinearity, in this context, refers not only to explicit terms in the Hamiltonian or equations of motion that make them nonlinear in the fundamental fields or amplitudes, but also to emergent many-body or correlation effects that render the global phase diagram and transition structure intrinsically nonlinear.

Key defining features include:

The presence of phase boundaries both between and within topological classes. For example, distinct gapped phases may share the same Chern number or other topological invariant but are separated by transitions associated with singularities in nonlocal correlation functions, a scenario not encountered in purely linear theory (Shi et al., 2010).
Transitions may be of first or higher order (as signaled, respectively, by discontinuities or nonanalytic behavior in derivatives of the ground-state energy) and need not always be tied to closing of a topological gap quantified by an invariant.
New types of multicritical or tetra-critical points can appear at the intersection of multiple topological and correlation-distinguished phases (Ezawa, 2017).
Nonlinearity may induce novel topological phases, dynamically reconstruct otherwise trivial static states, or yield amplitude-dependent topological invariants such as nonlinear Berry or Chern numbers (Sone et al., 2023, Zhou et al., 2020).

2. Topological Invariants and Nonlinear Order Parameters

Conventional topological phase diagrams segment phases by sharp invariants (e.g., Chern number, Zak phase, $\mathbb{Z}$ or $\mathbb{Z}_2$ indices). In nonlinear analogs, this segmentation is nuanced:

Topological invariants may remain conventional in form (Chern number, winding number), but the physical mechanism for phase separation goes beyond invariant jumps. For instance, two non-Abelian phases with identical Chern numbers may be separated by a "nonlinear" QPT because of distinct nonlocal many-body correlations, with the global invariant unchanged (Shi et al., 2010).
Nonlinear extensions of topological invariants are crucial for strongly nonlinear systems. For example, a nonlinear Berry phase can be constructed for nonlinear normal modes, and its quantization is enforced by symmetries such as reflection or chiral symmetry (Zhou et al., 2020, Zhou, 19 Mar 2024). The amplitude of the nonlinear mode itself can become the control parameter for the topological phase boundary.
In fully nonlinear mean-field or Gross–Pitaevskii–type systems, new invariants such as the nonlinear Chern number are constructed by considering fixed-amplitude nonlinear Bloch states, leading to phase transitions as amplitude is varied, and extending bulk–boundary correspondence beyond the linear regime (Sone et al., 2023).

Illustratively, the Chern number in the model of (Shi et al., 2010) is given by:

$\nu = \frac{1}{\pi} \mathrm{Im} \sum_{s=1}^{6} \sum_{i=1}^3 \int_{\text{BZ}} d^2 k \left[ \frac{\partial w_s^*(\varepsilon_k^{(i)})}{\partial k_x} \frac{\partial w_s(\varepsilon_k^{(i)})}{\partial k_y} \right]$

Nonlocal correlation functions, such as

$\frac{\partial E_g}{\partial \Lambda_1} = \frac{1}{6}(C_x + C_y + \Lambda_2 C_z)$

hold central significance for diagnosing transitions within a single topological class.

3. Types and Mechanisms of Nonlinear Topological Phase Transitions

Transitions in nonlinear topological phase diagrams can be classified as follows:

Conventional topological QPTs: Separating Abelian and non-Abelian phases, typically accompanied by a change in a Chern number or similar invariant and often mediated by the closing of the bulk gap.
Nonlinear intra-class transitions: Quantum phase transitions separating two phases with the same topological invariant but differing many-body wavefunction correlations or other local observables. In these transitions, criticality arises due to nonanalyticity in nonlocal correlators rather than topological index jumps (Shi et al., 2010).
Amplitude-dependent transitions: In models with nonlinear eigenvalue problems, varying the amplitude (norm) of nonlinear Bloch states can drive phase transitions where the nonlinear Chern number changes value, with no linear analog (Sone et al., 2023).
Emergent topological transitions: For instance, the spontaneous symmetry breaking in chains of nonlinearly coupled anharmonic oscillators leads dynamically to symmetry-broken tetramer states, which reconstruct a band topology not present in the symmetry-unbroken phase, and generate interface-localized topological modes (Savelev et al., 2018).
Dynamical transitions via quench: In strongly nonlinear arrays (e.g., Cooper-pair box arrays), quench dynamics can dynamically reveal otherwise "hidden" topological phases and transitions not present in the static density of states (Ezawa, 2022).

4. Phase Diagram Structure and Critical Points

The parameter space for a nonlinear topological phase diagram can be high-dimensional, incorporating coupling ratios, nonlinearity strengths, amplitude or population imbalance, and external fields. Critical features include:

Critical curves and points inside the same topological sector: These lines emerge when correlation functions or nonlinear order parameters exhibit nonanalyticities, as in the $(\Lambda_1, \Lambda_2)$ plane of the triangle–honeycomb Kitaev model (Shi et al., 2010).
Multicritical and tetra-critical points: Points where several (typically four) phases meet, as seen in the (U/t, $\eta$ ) plane of the interacting dimerized Kitaev model, with each region labeled by a combination of topological properties, edge states, and many-body order (Ezawa, 2017).
Eutectic-like points: Points in the phase diagram where all distinct gapped phases (with $\nu=0,0,1,-1$ ) converge, analogous to multicritical points in crystallographic phase diagrams (Shi et al., 2010).

These structures signify complexity that cannot be solely resolved from topological invariants and require analysis of correlation functions and effective Hamiltonians.

5. Role of Correlation Functions and Many-Body Effects

Singularities in nonlocal correlation functions are the microscopic drivers of nonlinear quantum phase transitions—either coincident with or independent of topological invariant changes:

At critical points (e.g., $\Lambda_1=0$ ), nonlocal correlation functions such as $C_z$ may exhibit discontinuities or divergent derivatives, marking first or higher order transitions even when the Chern number is unaltered.
The Feynman–Hellmann–theorem-based relationships tie ground-state energy derivatives to explicit nonlocal spin–spin correlators, making these functions both diagnostics and mechanisms for transitions.

Moreover, in interacting and dimerized systems, the structure of boundary and edge modes is tightly entwined with many-body effects, as shown by the decay coefficients of Majorana edge operators in the dimerized Kitaev superconductor (Ezawa, 2017).

6. Implications for Classification and Control of Topological Phases

The emergence of nonlinear transitions within topological classes or amplitude-driven changes of topological invariants demonstrates that conventional bulk invariants often provide an incomplete classification. This motivates refined approaches:

Combined topological-correlation state classification: Integrating global invariants with singularities or discontinuities in nonlocal correlation functions yields a more granular and accurate phase diagram (Shi et al., 2010).
Engineered phase control: The explicit dependence of effective low-energy Hamiltonians on sign and ratios of coupling constants (e.g., $\Lambda_1$ ) allows intentional tuning between phases even without a change in invariant, enabling engineered switching among different quantum orders.
Experimental detection: These insights suggest practical detection schemes—such as edge mode spectroscopy or local measurement of spin–spin correlators—to resolve subtle phase boundaries in quantum spin liquids and engineered topological superconductors.

7. Broader Applications and Future Perspectives

Nonlinear topological phase diagrams have implications beyond model systems:

Topological quantum computation: Robustness of phases and transitions dictated by nonlinear correlations are directly relevant for manipulation and protection of Majorana-based qubits.
Materials discovery and control: The capacity to induce transitions within a topological class by varying local correlations or amplitudes broadens the phase space for topological material engineering.
Theoretical generalization: The recognition that nonlinearities (through interactions, amplitude effects, or disorder) generate multicriticality and hidden transitions suggests that future generalizations of topological classification must systematically account for nonlinear, correlation-driven, and dynamical ingredients.

Potential future research includes exploration of disorder-driven multicritical points, systematic mapping of multicriticality in higher-dimensional and interacting systems, and experimental realization of amplitude-dependent topological transitions in photonic, cold atom, or superconducting platforms.

In summary, nonlinear topological phase diagrams capture a complex landscape of phases and transitions in systems where nonlinear interactions, amplitude dependence, or many-body correlations lead to phenomena inaccessible by traditional invariants alone. Transitions between phases of identical topological class, multicritical loci, and the decisive role of correlation functions mark a departure from the linear paradigm, requiring new theoretical and experimental tools to fully characterize and exploit the rich structure of topological quantum matter (Shi et al., 2010).