Physics Tomorrow Letters

Non-Hermitian physical systems exhibit a unique type of singularity, known as exceptional points (EPs), where distinct eigenstates of the Hamiltonian coalesce with each other [1–7]. EPs have garnered significant interest due to their potential applications in diverse fields such as optics, acoustics, and open quantum systems [8–15]. As a system undergoes an adiabatic deformation that encircles an EP or EPs, the eigenstates exchange in a nontrivial manner. This eigenstate switching effect allows for the classification of EPs based on the conjugacy class of the permutation group [16]. Concurrently, the study of Hermitian topological phases has focused on the classification of topological invariants associated with the Berry phase [17–20]. The quantized Zak phases in the Su–Schrieffer–Heeger (SSH) model are one representative example [21–26]. In EPs, the wave functions can accompany an additional geometric (Berry) phase shift of π after the encircling of an EP. This connection between EPs and nontrivial Berry phases has been both theoretically and experimentally explored [27–31] and suggests a more complex structure for EPs. As we show below, these two seemingly unrelated phenomena—namely the nontrivial Berry phase and EPs—are intimately tied together. In this work, we demonstrate that the EPs are characterized by exceptional classifications, a scheme we propose to incorporate the information of both eigenstate switching and the additional Berry phase. In addition, by viewing one-dimensional (1D) systems as adiabatic deformations encircling EPs, we achieve a full characterization of 1D non-Hermitian topological systems. This characteristic reveals topological phase transitions between different phases, where the phase transitions accompany the EPs. Our identification of this exceptional class lays the groundwork for further exploration of the rich physics of non-Hermitian systems. Classification scheme – We consider a general N-state non-Hermitian system with two external parameters. When encircling the system’s EPs through adiabatic deformation, the eigenstates exhibit the exchange effect, which can be represented by a permutation of the N states [32, 33]. The cyclic structure of such permutations can be formally mapped by the conjugacy class [σ] (with representative permutation σ), which forms a product of cycles. Specifically, we can use the following notation to represent the permutation properties of the EPs, as in [16, 34]:

Each cycle is represented in the form c nc , where c indicates the cycle length (number of encirclings required to return to the initial state), and the superscript nc ∈ {0, 1, · · · , N} denotes the number of c-cycles in [σ]. For instance, in a two-state system, there exist two possible exchanges of eigenstates after the encircling of adiabatic deformations, represented by the conjugacy classes [e] = 12 , [σ] = 21 , where e represents the identity permutation and σ denotes a transposition. Furthermore, in addition to the conjugate classification there exist Berry phases of the wave functions. The complex Berry phase [35–38] can be defined as

Here, C is a closed path in M × R where M is the parameter space and R is the complex Riemann surface of the energy, λ is a parameterization of the path C, and ϕ and ψ respectively are the left and right eigenstates of the Hamiltonian H(λ). Due to the complex nature of the energy of non-Hermitian Hamiltonians, we will concentrate on a two-dimensional parameter space (or, codimension two). Since the double encircling of a single EP in parameter space induces a nontrivial π Berry phase for the states (see Section I in the Supplemental Material (SM)), the conjugate classifications of the EPs are further subclassified depending on the presence of the Berry phase. We refer to these finer classifications of the conjugacy class as the exceptional class. In the following discussion, we use the notation ¯c for the c-cycles with π Berry phases. A constraint arises from the consistency between the switching effect and the Berry phase: the sum of the Berry phases of the cycles in the conjugacy classes should be 0 and π for even and odd permutations, respectively [31]. Note that the parity of permutations remains invariant under conjugation. As an example, consider a two-state system having two conjugacy classes, [e] = 12 , [σ] = 21 . Under the consistency constraint, the exceptional classes are

where the sums of the Berry phases are 0 (mod 2π) for 1 2 and ¯1 2 , and π for ¯2 1 . This is consistent with the parities of e (even) and σ (odd). See Fig. 1. Note that ¯1 2 can only appear in systems with multiple EPs. This classification scheme can be generalized to N-state systems. Using signed holonomy matrices, the classes can be obtained systematically. We note that some conjugacy classes are connected by gauge transformation and should be identified; details are in Section II of SM. Non-reciprocal SSH model – In the following, we apply the exceptional classification framework to 1D systems. As an example, we consider the non-reciprocal Su– Schrieffer–Heeger (SSH) model [39–43], where the Hamiltonian is given as ........ cont.