Triple exceptional point with unitary paths of unfolding in a three-site fermionic Swanson-like model

Abstract: A fermionic three-site generalization of the popular bosonic Swanson model is studied as providing an exactly solvable five-parametric example of the quantum-mechanical unitary-evolution process leading to an ultimate loss of the observability and fall in an exceptional-point singularity (EP3). The instant of degeneracy is found to have an explicit one-parametric form. Its unitarity-compatible vicinity (i.e., the corridor of access to EP3) is also specified in closed form. The exact, numerical-error-independent solvability is found essential due to another, avoided, false energy-level crossing which is found to occur not too far from the true EP3 singularity.

• The paper presents a rigorous analytic construction and localization of an order-three exceptional point (EP3) in a fermionic Swanson-like model.
• It employs unitary perturbative unfolding to reveal controlled stability corridors, preserving quasi-Hermiticity under finely tuned parameters.
• Detailed eigenvector and numerical analyses demonstrate the system’s Jordan block structure and its sensitivity to parameter perturbations.

Triple Exceptional Point with Unitary Paths of Unfolding in a Three-Site Fermionic Swanson-like Model

Introduction and Formal Framework

The paper presents a rigorous analysis of non-Hermitian degeneracies in a family of fermionic Swanson-like quantum Hamiltonians, focusing specifically on the emergence and fine structure of an order-three exceptional point (EP3) in a three-site system. The work builds on the framework of quasi-Hermitian quantum mechanics (QHQM), where the Hamiltonian $H$ is not necessarily Hermitian with respect to the initial Hilbert space but admits a similarity transformation to a Hermitian form in a suitably defined physical space. The study utilizes the construction of exactly solvable finite-dimensional models to enable analytic treatment of high-order EPs, which are otherwise challenging due to the mathematical complexity typical for $N > 2$ cases.

Model Construction and Analysis

The foundational model is a fermionic generalization of the Swanson oscillator. In its two-site version, truncation yields a $4 \times 4$ matrix reducible to $2\times2$ non-Hermitian blocks. The spectrum and the onset of non-diagonalizability are controlled by the interaction parameters, with the EP2 (order-2 exceptional point) identified at the coalescence of eigenvalues and eigenvectors.

Transitioning to the three-site case, the Hamiltonian includes three on-site energies and intersite (pairing-like) couplings, systematically generalized to allow independent control of coupling constants. The many-body Fock space truncation produces an $8 \times 8$ matrix, decomposed (after block-diagonalization) into two $3 \times 3$ non-Hermitian matrices, which constitute the core dynamic subsystem for the analysis. The parameterization is further refined to expose explicit dependence of the spectrum only on the products of the basic couplings, facilitating direct manipulation to achieve EP3 conditions.

Localization and Structure of the EP3

The triple degeneracy (EP3) requires the secular equation to possess a triple root, producing constraints that reduce the effective parameter count. The closed-form localization of the EP3 is achieved by expressing two coupling products, $A$ and $B$, as explicit functions of a single auxiliary parameter $u$. The resulting functions $A^{(EP3)}(u)$ and $N > 2$0 are smooth, non-intersecting, and analytically traceable, allowing for controlled passage through the EP3 manifold.

Eigenvector analysis confirms the Jordan block structure at the EP3, exhibiting explicit parallelization of all eigenvectors in the degenerate limit. The transition matrix relating the physical Hamiltonian to its canonical Jordan form is constructed and shown to depend on auxiliary parameters, reflecting the enlarged structure and accessibility of the degeneracy in parameter space compared to EP2.

Perturbative Unfolding and Stability Corridors

The critical examination centers around the effects of infinitesimal perturbations near the EP3. The generic case demonstrates that arbitrary perturbations induce complexification of at least some eigenvalues in the immediate vicinity of the EP3, signaling loss of unitarity and physical instability. However, when the perturbation structure is fine-tuned---with specific scaling of certain matrix elements as functions of the unfolding parameter $N > 2$1---the energy branches can be chosen to remain real, preserving quasi-Hermiticity and physical interpretability in a sharply defined "corridor of stability."

The energy spectrum in this controlled scenario evolves with $N > 2$2, and the roots of the reduced secular equation are required to be real—which imposes explicit conditions on the admissible perturbation parameters. The work formalizes how the stability corridor is characterized by the positivity of certain parameters and strict inequalities, making the safe navigation of parameter space around the EP3 an issue of algebraic geometry in the space of couplings.

Numerical Aspects and Practical Considerations

A discussion of the practical difficulties of numerically identifying and tracing high-order EPs in models of this type is included. Specifically, the paper demonstrates via analytic solutions and detailed graphical studies that apparent real level crossings near EP3 can be misleading due to limitations in numerical precision. Proper discrimination requires tracking the algebraic structure of the secular solutions, with the use of inverse representations (Sturmian eigenvalues) facilitating analytic and computational clarity. The local unfolding of the spectrum follows $N > 2$3, highlighting the nontrivial nature of the triple root.

Implications and Future Perspectives

The paper establishes the explicit construction, localization, and perturbative analysis of an EP3 in an exactly solvable fermionic model. The identification of unitary corridors delineates the conditions under which the system retains an interpretable quantum evolution near high-order non-Hermitian degeneracies. This has significant implications both for the theoretical understanding of exceptional points in finite quantum systems and for the practical design and control of quasi-Hermitian and $N > 2$4-symmetric systems where access to and traversal of EP3-type singularities are physically relevant, e.g., in engineered photonic devices and ultracold atomic platforms.

The results contribute to clarifying the topological and analytical landscape of higher-order EPs, providing a template for future studies in more complex or physically realistic settings, including open quantum systems, driven-dissipative models, and nonlinear extensions. The analytic tractability and the exposure of stability corridors strengthen the theoretical toolkit for perturbative and nonperturbative analysis in non-Hermitian quantum mechanics, with potential cross-fertilization into mathematical domains such as spectral theory and algebraic geometry.

Conclusion

The study delivers a comprehensive formal treatment of an EP3 in a three-site fermionic Swanson-like model, encompassing analytic localization, eigenvector structure, perturbative unfolding, and physical stability conditions. The work underscores the critical role of parameter constraints in sustaining unitary, physically interpretable behavior near high-order exceptional points and provides a concrete methodological advance in the analytic study of non-Hermitian quantum models (2606.03789).