Cascades of Exceptional Points in Non-Hermitian Systems

• Cascade of exceptional points are non-Hermitian degeneracies where eigenvalues and eigenvectors coalesce, forming hierarchical structures of varying orders.
• They emerge via analytic continuation, tensor-product composition, and symmetry protection in models such as quantum chains and photonic lattices.
• Their cascaded structure enhances spectral sensitivity and topological control, enabling applications in ultrasensitive detection and frequency comb generation.

A cascade of exceptional points (EPs) refers to a structured, often hierarchical proliferation of non-Hermitian degeneracies—points in parameter space where eigenvalues and eigenvectors of a non-Hermitian Hamiltonian simultaneously coalesce. This manifests as families of EPs of different orders (EP$_n$: $n$-fold degeneracy), emerging either through analytical continuation of model parameters, tensor-product composition of subsystems, systematic symmetry constraints, or interaction-induced deformation in multi-particle and multimode systems. Cascades of EPs organize the spectral topology and dynamic response of a broad class of open quantum, photonic, and many-body systems, with quantifiable consequences for critical response, mode permutation, and robustness under perturbation.

1. Foundational Definitions and Exemplary Cascade Architectures

An exceptional point of order $n$ (EP$_n$) is a spectral singularity of a non-Hermitian operator at which $n$ eigenvalues and their corresponding eigenvectors coalesce, resulting in a single Jordan chain of length $n$ and algebraic multiplicity $n$ but geometric multiplicity one (Wiersig et al., 9 Apr 2025). Physical models often feature not just isolated EPs but entire cascades—interconnected structures of EP$_m$ with $m<n$, terminating or accumulating at higher-order singularities.

A paradigmatic realization is found in the Baxter–Fendley free-parafermion quantum chain, where analytic continuation of a magnetic-field–like parameter $\lambda$ into the complex plane yields a cascade of $N(L-1)$ EPs (in a $Z_N$-symmetric model of $L$ sites) (Henry et al., 2023). The positions $\lambda_\mathrm{EP}$ are determined by transcendental and algebraic conditions, explicitly:

$$\begin{aligned} & \sin\!\left[(L+1)k\right] + \lambda^{-N/2} \sin(Lk) = 0\,, \ & \sin\!\left[(2L+1)k_{\mathrm{EP}}\right] - (2L+1) \sin k_{\mathrm{EP}}= 0\,, \ & \lambda_{\mathrm{EP}}^{N} = \left[ -\frac{ \sin\!((L+1)k_{\mathrm{EP}}) }{ \sin(Lk_{\mathrm{EP}}) } \right]^2\,. \end{aligned}$$

As $L\rightarrow\infty$, these EPs become dense along the unit circle in $\lambda$, converging at the quantum critical point.

A further class of cascades arises in composite non-Hermitian systems via tensor-product composition. Here, assembling $k$ uncoupled subsystems with individual EPs of orders $n_i$ produces a single EP of order $N=1+\sum_{i=1}^k (n_i-1)$ in the total system without requiring any explicit intersystem coupling (Wiersig et al., 9 Apr 2025). This process is algebraically regulated by the nilpotency properties of the Jordan block structure.

Another avenue for cascade formation leverages strict symmetry protection. In interacting fermion systems, PT- or chiral symmetry can stabilize entire "fans" of EPs in multiparameter spaces even under interactions, resulting in cascades that display distinct creation, migration, and annihilation phenomena as the system parameters evolve (Schäfer et al., 2022).

2. Algebraic and Geometric Structure of Cascaded EPs

The algebraic manifestation of a cascade is seen in the progression of codimension and degeneracy: as one moves from EP$_2$ lines or surfaces, to EP$_3$ arcs, to isolated EP$_4$ points, the underlying polynomial constraints become increasingly restrictive. For an $n$-band non-Hermitian Hamiltonian,

$$\det\bigl(H-E\mathrm{I}_n\bigr) = 0\,, \ \partial_E^{m} \det\bigl(H-E\mathrm{I}_n\bigr) = 0\ \forall m<n-1$$

characterize an EP$_n$. Symmetry constraints (e.g., pseudo-Hermiticity: $H = \eta H^\dagger \eta^{-1}$) can substantially reduce the number of real equations required for an EP$_n$, thereby controlling the codimensionality and enabling higher-order EPs to emerge with lower parameter tunings (Montag et al., 4 Aug 2025).

In multiband lattice models, cascades can manifest as closed loops or nested surfaces of EPs in momentum space, connected and organized by underlying chiral or mirror-time symmetry (Xiao et al., 2020). For example, in the non-Hermitian Lieb lattice, an astroid-shaped EP$_2$ loop emerges, punctuated by EP$_3$ cusps, which itself splits and collapses in a sequence as the non-Hermiticity parameter changes.

Geometric and topological cascades arise when the branch-point structures associated with multiple EPs interlace. Encircling combinations of EPs in a two-dimensional parameter space yields a sequence of permutations and geometric phase accumulations, systematically mapped by the Riemann surface topology of eigenvalues (Ryu et al., 29 Aug 2024, Bhattacherjee et al., 2018). For systems with three modes and three EPs, all five permutation-cycle classes of $S_3$ can be realized by different loop configurations, each corresponding to a characteristic mode exchange and phase structure.

3. Dynamical, Spectral, and Topological Consequences

The dynamical implications of cascading EPs are profound. Spectral sensitivity—quantified by the response strength $\mathcal{R}$ or $\sigma_\mathrm{EP}$—scales with the EP order. For an order-$n$ EP, generic perturbations (e.g., $\varepsilon V$) induce eigenvalue splittings $\Delta E \sim (\varepsilon \mathcal{R})^{1/n}$, leading to dramatically enhanced response in higher-order EPs (Wiersig et al., 9 Apr 2025, Wiersig, 2023). When moving along an exceptional manifold toward a higher-order singularity, $\sigma_\mathrm{EP}$ diverges, evidenced both in analytic residue calculus and in the blow-up of the Petermann factor.

The cascade also determines the phase topology of the system. Encircling one EP yields standard square-root permutation and phase exchange; encircling two in succession produces a three-cycle (cubic-root) permutation, and vertically stacks multiple such operations following the algebra of the braid group (Ryu et al., 29 Aug 2024, Bhattacherjee et al., 2018). In composite systems, cascading induces the phenomenon of "EP-induced disentanglement": after sufficiently long non-unitary evolution under a composite EP, any initial state—including highly entangled ones—evolves to the unique product-form eigenvector of the EP (Wiersig et al., 9 Apr 2025).

Topological invariants associated with cascaded EPs, such as winding numbers derived from resultants or discriminants, characterize the robustness of cascaded structures. In complex momentum space, discriminant number (e.g., for the Lieb model) protects discrete EPs against annihilation unless colliding with an oppositely charged EP (Xiao et al., 2020).

4. Hierarchical and Period-Doubling Cascades

Structured schemes for engineering cascades are exemplified by hierarchical constructions in photonic systems. By coupling two PT-symmetric microrings each supporting an EP$_N$, and introducing non-uniform unidirectional (chiral) coupling, one can systematically realize an EP$_{2N}$. Breaking uniformity in the coupling matrix ensures coalescence into a single Jordan block of maximal order (Zhong et al., 2020). This principle confers exceptional fabrication robustness: only a single parameter constraint is generically violated by random disorder, guaranteeing persistence of the higher-order EP.

Another nonlinear expression of cascades is found in exceptional-point lasers: as the pump parameter is increased, the frequency combs generated near an EP undergo successive period-doubling bifurcations, accumulating to a temporal Feigenbaum cascade (Gao et al., 24 Jan 2025). Each period-doubling transition is marked by the crossing of Floquet exponents at $\Re\omega_F = \omega_d/2^n, \Im\omega_F = 0$, producing nested branches of multistable comb-generation regimes.

5. Cascade Phenomena in Interacting and Disordered Systems

The fate of cascades under interaction and disorder reveals further phenomenology. In interacting fermion systems with preserved PT or chiral symmetry, a cascade is visible as "exceptional fans" in the space of twist angle and interaction strength, with lines of EPs inherited from single-particle degeneracies and new, interaction-induced EPs emerging at critical interaction strengths (Schäfer et al., 2022). Pairwise annihilation of lines of EP$_2$ can produce isolated EP$_3$s at their intersection, observing a direct dynamical cascade in the non-Hermitian many-body spectrum.

In band-structured systems, symmetry breaking, such as next-nearest-neighbor hopping that destroys chiral symmetry, can collapse extended EP cascades (loops or surfaces) into isolated and topologically protected EPs, demonstrating the sensitivity of cascade structure to perturbations and the central role of symmetry protection (Xiao et al., 2020, Montag et al., 4 Aug 2025).

6. Experimental Realizations and Applications

Photonic, circuit, and quantum optical systems provide fertile ground for realizing and probing cascades of exceptional points. Hierarchical coupling in PT-microrings, as verified by full Maxwell simulations, demonstrates fabrication-resilient EP$_4$ singularities useful for ultrasensitive detection schemes (Zhong et al., 2020). Nonlinear dynamics in EP-lasers produces multistable and period-multiplied frequency combs potentially relevant for integrated frequency metrology (Gao et al., 24 Jan 2025). Microcavities hosting multiple modes and engineered boundary deformations exhibit not only all topological classes of mode exchange via cascaded EPs but also practical protocols for accessing these features through parameter-space loop manipulations (Ryu et al., 29 Aug 2024).

In non-Hermitian quantum spin chains and many-fermion models, cascades provide analytic access to the physical precursors of phase transitions, critical behavior, and spectral non-analyticity in open-system quantum critical points (Henry et al., 2023, Schäfer et al., 2022).

7. Open Problems and Generalizations

A natural generalization of cascade concepts involves their extension to arbitrary composite systems, open quantum evolutions (Liouvillian cascades), and the systematic exploration of resultant topology in higher dimensions and symmetry classes (Wiersig et al., 9 Apr 2025, Montag et al., 4 Aug 2025). The unifying role of the Jordan chain, nilpotent algebra, and resultant invariants in organizing cascaded EP structures underlies ongoing developments in the classification and engineering of robust and tunable non-Hermitian singularity networks. Notably, the scaling relations in spectral response and the emergence of higher-order dynamics without explicit intersystem coupling indicate robust routes to high-sensitivity devices and topological control in complex open systems.

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References:

• "Exceptional Points in the Baxter-Fendley Free Parafermion Model" (Henry et al., 2023)
• "Higher-order exceptional points in composite non-Hermitian systems" (Wiersig et al., 9 Apr 2025)
• "Realization of geometric phase topology induced by multiple exceptional points" (Ryu et al., 29 Aug 2024)
• "The analytically tractable zoo of similarity-induced exceptional structures" (Montag et al., 4 Aug 2025)
• "Symmetry protected exceptional points of interacting fermions" (Schäfer et al., 2022)
• "Realization of Third Order Exceptional Singularities in a Three level non-Hermitian System: Towards Cascaded State Conversion" (Bhattacherjee et al., 2018)
• "Bi-stability and period-doubling cascade of frequency combs in exceptional-point lasers" (Gao et al., 24 Jan 2025)
• "Moving along an exceptional surface towards a higher-order exceptional point" (Wiersig, 2023)
• "Hierarchical construction of higher-order exceptional points" (Zhong et al., 2020)
• "Exceptional points make an astroid in non-Hermitian Lieb lattice: evolution and topological protection" (Xiao et al., 2020)