Non-Hermitian swallowtail catastrophe revealing transitions among diverse topological singularities

Exceptional points are a unique feature of non-Hermitian systems at which the eigenvalues and corresponding eigenstates of a Hamiltonian coalesce. Many intriguing physical phenomena arise from the topology of exceptional points, such as bulk Fermi arcs and the braiding of eigenvalues. Here we report that a structurally richer degeneracy morphology, known as the swallowtail catastrophe in singularity theory, can naturally exist in non-Hermitian systems with both parity–time and pseudo-Hermitian symmetries. For the swallowtail, three different types of singularity exist at the same time and interact with each other—an isolated nodal line, a pair of exceptional lines of order three and a non-defective intersection line. Although these singularities seem independent, they are stably connected at a single point—the vertex of the swallowtail—through which transitions can occur. We implement such a system in a non-reciprocal circuit and experimentally observe the degeneracy features of the swallowtail. Based on the frame rotation and deformation of eigenstates, we further demonstrate that the various transitions are topologically protected. A characteristic feature of non-Hermitian systems is an exceptional point at which eigenvalues and eigenstates coalesce. They also support richer degeneracies—a swallowtail catastrophe—that reveals transitions among three different types of singularity.

Exceptional points are a unique feature of non-Hermitian systems at which the eigenvalues and corresponding eigenstates of a Hamiltonian coalesce. Many intriguing physical phenomena arise from the topology of exceptional points, such as bulk Fermi arcs and the braiding of eigenvalues.
Here we report that a structurally richer degeneracy morphology, known as the swallowtail catastrophe in singularity theory, can naturally exist in non-Hermitian systems with both parity-time and pseudo-Hermitian symmetries. For the swallowtail, three different types of singularity exist at the same time and interact with each other-an isolated nodal line, a pair of exceptional lines of order three and a non-defective intersection line. Although these singularities seem independent, they are stably connected at a single point-the vertex of the swallowtail-through which transitions can occur. We implement such a system in a non-reciprocal circuit and experimentally observe the degeneracy features of the swallowtail. Based on the frame rotation and deformation of eigenstates, we further demonstrate that the various transitions are topologically protected.
In recent years, non-Hermitian systems have attracted a great deal of interest. A main goal is to address the ubiquitous open quantum systems that undergo energy exchange with the surrounding environment via the imaginary part of their eigenenergies [1][2][3][4][5][6][7][8][9][10][11][12] . Degenerate singularities in band structures are similar to topological defects in real space. Well-known singularities in Hermitian systems are Weyl/Dirac points and nodal lines [13][14][15][16][17][18] , and their associated phenomena, such as topological edge modes 13,18 and chiral Landau levels 16 , have been fully explored. In non-Hermitian systems, the complex nature of eigenvalues results in more exotic singularities such as exceptional points, at which two or more eigenstates coalesce. Exceptional points can carry fractional topological invariants, which not only enrich the topological classes in band theory but also induce more intriguing physical consequences, such as bulk Fermi arcs 2,3 and braiding of eigenvalues 10 . In addition, the skin effect, which is associated with the point gaps in non-Hermitian bands, is also a unique feature of non-Hermitian systems [19][20][21] .
In non-Hermitian systems with parity-time (PT) symmetry or chiral symmetry, exceptional surfaces (ESs) can stably exist as singular hypersurfaces in three-dimensional (3D) parameter space, acting as boundaries between exact and broken phases [22][23][24] . Remarkably, as subspaces of the parameter space, these ESs can exhibit numerous new singularities, such as high-order exceptional points (or lines) appearing as cusps 6,9 and non-defective degeneracies that are intersections of ESs 8,11,12 . The coexistence of diverse singularities brings the possibility that these singularities can be associated with each other. However, previous works have commonly focused on a single type. The transitions among different types, as well as the underlying topological structure, remain largely unexplored. Article https://doi.org/10.1038/s41567-023-02048-w and different views of Fig. 1a in Supplementary Video 1). The ESs (red surfaces) and EL3s (black lines) result from the PT symmetry 9 of the system. The pair of EL3s merges at the MP (Fig. 1a, red star), which emits the non-defectively degenerate NL and NIL (blue lines) in opposite directions (MP is a three-fold degeneracy with two linearly independent eigenstates). The NL is isolated from ESs and is a linear degeneracy between the first and second bands. In contrast, the NIL is a complete intersection of ESs 8,11,12 , which are formed by the degeneracy of the second and third bands. The common feature is that both the NL and NIL are linear crossings of eigenvalue dispersions, and both are non-defective twofold degeneracies (that is, the two degenerate eigenstates are linearly independent of each other). Owing to the two symmetries of the system in equation (2), the NL and NIL cannot be extended into a tube or cone in parameter space. Thus, their stability is symmetry protected (see Supplementary Sections 5-7 for a demonstration). Therefore, the swallowtail is an assembly of different types of singularities (ES, EL3, NIL, NL and MP), and its existence is protected by the two symmetries ( Supplementary Fig. 4 shows various structures resulting from combinations of swallowtails by changing the Hamiltonian form in Supplementary equation (13) without breaking the symmetries in equation (2)).
We next analyse the local structure of eigenvalues over the swallowtail. The EL3s are lines at which two ESs meet, forming cusps. In catastrophe theory, a cusp is formed owing to the projection of a bending curve (or surface) onto a lower-dimensional space. Figure 1b shows that such a bending process can be observed in non-Hermitian eigenvalue structures. That is, on the plane f 3 = 0.3, the red line (ES) bends in the f 1 -f 2 -Re ω space (where ω denotes the eigenvalues). Thus, swapping of eigenvalues will occur if a tracking point moves along the ES and 'jumps' through an EL3, from the ES on one side of the EL3 to the ES on the other side, as the parameters change. Here, a jump corresponds to a quotient map in mathematics (see Supplementary Section 9 for details and the discussion in Supplementary Fig. 10e1). In contrast to the EL3s, the NIL is a transversal intersection of two ESs, and the nearby eigenvalue dispersion forms a double cone (Fig. 1b, inset). The pair of EL3s and the NIL are connected by ESs, forming a loop. Tuning of the parameters (that is, to f 3 = 0.1214) can shrink the loop in a continuous way until the EL3s and NIL merge at the MP (Fig. 1c). From the other direction along the f 3 axis, the MP can also be understood as a point of collision for a ray (NL) towards a surface (ES). Before the collision, points on the NL are isolated from the ES (Fig. 1d with f 3 = 0.01). As the NL and ES share the second band (blue surface), the tuning of system parameters can make them collide, when the three eigenvalues coalesce at the MP (Fig. 1c).
To observe the exotic swallowtail configuration and investigate the topological origin of the evolution of degeneracy features in parameter space, we employ a non-reciprocal electric circuit system emulating the interaction of three modes (labelled A, B and C in Fig. 2) as a realization of the three-state non-Hermitian Hamiltonian. Benefiting from a wide range of active circuit elements such as operational amplifiers (OpAmp), a circuit system is more flexible than other platforms, which suits our needs to accurately control the gain and loss and implement non-reciprocal hoppings. The behaviour of a circuit system can be described by the Laplacian I = JV, where I is the vector of input currents, J is the admittance matrix and V is the vector of node voltages 20 . The matrix J plays the role of the Hamiltonian matrix. Its eigenvalues, namely admittance bands j, represent the energy spectra. Thus, the synthetic dimensions of the parameter space, f 1 , f 2 and f 3 , can be mapped to the tight-binding hopping parameters between each pair of circuit nodes (Fig. 2a). The circuit element structure is shown in Fig. 2b. The non-reciprocal hopping ±f 1 (resp. ±f 2 ) between A and B (resp. A and C) is implemented and precisely controlled by an impedance converter with current inversion (INIC) circuit in tandem with the capacitance element C 1 (resp. C 2 ) as in Fig. 2c. The details of the INIC are given in Supplementary Section 1.1. The pure capacitance element C 3 realizes a In Hermitian systems with PT symmetry 17,25-27 (the corresponding Hamiltonians are real Hermitian matrices), the eigenstates were previously reported to be real and orthogonal and to form the orthonormal basis of a Euclidean-like space 17 . The nodal lines in the band structure manifest as topological obstructions of the eigenstate frames, around which the eigenstates rotate in a way characterized by non-Abelian quaternion topological charges 17 , which has been experimentally observed in a recent work 18 . Here, by expanding our scope to non-Hermitian systems, in particular those with PT symmetry and an additional η-pseudo-Hermitian symmetry (η is the metric operator), the eigenstates form a Minkowski-like orthogonal basis in which the vectors inhabit a space comparable to the Riemann space used in general relativity. As a result, a more exotic and structurally much richer degeneracy morphology emerges, known as the swallowtail catastrophe in singularity theory 28 . The swallowtail is one of the elementary catastrophes in Arnold's ADE classification [28][29][30] and has been widely applied in many branches of physics and engineering, ranging from mechanics 31 to caustics of light 32 . However, it has never been studied in eigenvalue dispersions. Here, we discover for the first time that the swallowtail catastrophe, which naturally exists in the parameter space of non-Hermitian systems with PT symmetry together with a pseudo-Hermitian symmetry, encompasses degeneracy lines of three different types. In addition to a nodal line (NL) isolated from ESs (similar to the NLs in Hermitian systems), the swallowtail also has a pair of exceptional lines of order three (EL3) and a non-defective intersection line (NIL), which lie entirely on the ESs. Both the NL and NIL are lines of diabolic points with two linearly independent degenerate eigenstates. The difference is that the NL is isolated from ESs, whereas the NIL is not isolated, as it is the intersection line of ESs. Surprisingly, these seemingly independent types of degeneracy lines are stably connected at a meeting point (MP) on the swallowtail, revealing interesting transitions among them as the parameters change. By realizing such systems in a non-reciprocal circuit, we experimentally observe the degeneracy features of the swallowtail. Furthermore, transitions among different types of singularities complying with the topological constraints associated with them are demonstrated both theoretically and experimentally.
The three-state non-Hermitian Hamiltonian we consider takes the form where f 1 , f 2 and f 3 are real numbers specifying three degrees of freedom and defining a 3D parameter space. Such a Hamiltonian preserves two symmetries 1 : Here, the metric operator η = diag(-1, 1, 1), and the first relation shows that H is η-pseudo-Hermitian. The PT symmetry operator is a combination of the parity-inversion P and time-reversal T operators. If the parameters f 1 , f 2 and f 3 are momentum space coordinates, then the PT operation takes the complex conjugate of the Hamiltonian up to a unitary transformation, that is, PT(H) = U † H * U, and the requirement of a real-valued Hamiltonian in equation (1) is equivalent to insisting that the Hamiltonian preserve the PT symmetry (see Supplementary Section 3 for more details). We note that two pairs of off-diagonal entries are anti-symmetric (H 12 = -H 21 , H 13 = -H 31 ), representing non-reciprocal hopping between modes. In contrast, the remaining pair of off-diagonal entries are symmetric (H 23 = H 32 ) and represent reciprocal hopping. The degenerate surfaces and lines in the eigenvalue structure form a swallowtail (Fig. 1a, the ADE description in Supplementary Section 2 Article https://doi.org/10.1038/s41567-023-02048-w reciprocal hopping -f 3 between B and C. One can select the values of C 1 , C 2 and C 3 in the experiments to implement the required parameters f 1 , f 2 and f 3 , respectively. A photo of the printed circuit board used for the experiments is presented in Fig. 2d. By measuring the voltage response at each node to a local a.c. input, we acquire the admittance eigenvalues and eigenstates. More details on the experimental design are given in Supplementary Section 1. Figure 3a (left) shows the ESs, EL3s and NIL obtained from the experimental measurements (solid dots) along the computed intersecting curve of the swallowtail with the plane f 3 = 0.3. These singularities are extracted from the measured admittance eigenvalues marked by circles in corresponding colours in Fig. 3a (right), which are functions of f 1 , along various lines f 2 = f 1 + s on the plane f 3 = 0.3. The ESs can be clearly recognized from the quadratic coalescence of two eigenvalues in the experimental results. Two ESs, one formed by the first and second bands and the other formed by the second and third bands, meet at the cusps of EL3s, each of which is experimentally observed as the merging point of all three eigenvalues. On the other hand, the NIL is the intersection of two transversal ESs, both formed by the second and third bands (Fig. 3a, left). In contrast to the quadratic coalescence above, it is observed as a linear degeneracy in the eigenvalue dispersion (Fig. 3a, right). The regions shaded in grey are PT-exact phase domains, while the unshaded regions denote PT-broken phases. From here, as f 3 decreases to 0.1214, the exact phase domain enclosed by the ESs shrinks to the MP (Fig. 3b, left), which is the coincidence point of the linear degeneracy and the quadratic coalescence of eigenvalues (Fig. 3b, right). With further lowering of f 3 to 0.01, the point on MP and ES are decoupled into an isolated point (NL) and a smooth curve (ES) as in Fig. 3c(left). Correspondingly, the measured admittance eigenvalues in Fig. 3c (right) indicate that the NL is a linear degeneracy of the first and second bands, while the ES is formed by the second and third bands. Evidently, the MP plays a pivotal role in linking all these degeneracy lines. To more directly observe how the degeneracy lines and surfaces are connected at the MP, we further measured the eigenvalues on the plane f 1 = f 2 (Fig. 1a, yellow plane) which contains all of them. Figure 3d (left) illustrates that the NIL and NL are smoothly connected by the MP, which also serves as a tangent point to the ES. This point separates the ES into upper and lower parts, which are formed by the degeneracies of different bands (Fig. 3d, right).
We now explain topological aspects of the above transitions among different singular lines. The swallowtail affords several transition processes among symmetry-protected degeneracies (Supplementary Sections 5 and 6). Here, we focus on the most interesting transition, that is, from the pair of EL3s to the NIL and NL. Our goal is to demonstrate that the pair of EL3s is topologically equivalent to the NIL and NL. Let us consider a loop encircling the pair of EL3s (l α in green on the plane f 3 = 0.3 in Fig. 4a(left)) and a loop that encloses both of the NIL and the NL (l β in yellow on the plane f 1 + f 2 = 0.3 in Fig. 4b  (left)). Both loops inevitably cut through the ESs, as the EL3s and NIL are hypersurface singularities. Such an approach employs mathematical notions of intersection homotopy 33 . It is different from the usual homotopical descriptions using encircling loops along which all the Hamiltonians are gapped (see Supplementary Section 6 for details). The two loops share the same starting point (SP, purple dots) so that a direct comparison can be performed. The equivalence between l α and l β is manifested by observing the eigenframe rotation and deformation processes. The concept of frame rotation has been used to label different NLs in multiband Hermitian systems with PT symmetry 17,18 , in which the eigenstates form orthogonal bases of a Euclidean-like space. Here, in our non-Hermitian system, Euclidean-like geometry is no longer applicable. The symmetries in equation (2) require that the eigenstates satisfy the orthogonality relation Article https://doi.org/10.1038/s41567-023-02048-w where the superscript 'T' denotes transposition. Since η has the same form as the Minkowski metric and the Hamiltonian is PT symmetric (equation (2)), the eigenstates φ m are analogous to the frame fields in general relativity 34 , replacing Euclidean-like geometry with Riemannian-like geometry. Hence, the eigenstates will undergo Lorentz-like transformations as the parameters vary (see Supplementary Sections 5 and 6 for details), which induce both frame rotation and frame deformation.
The trajectories of the eigenvalues along the loops l α and l β are shown in Fig. 4a (centre) and 4b (centre), respectively. The corresponding evolutions of eigenstates are indicated by the trajectories of the ball markers in Fig. 4a (right) and 4b (right), where the three axes denote the three components of the eigenstates. The experimental and theoretical results are shown in the upper and lower panels, respectively.
The three eigenstates φ 1 , φ 2 and φ 3 are marked with red, blue and black, respectively, colours corresponding to those of the eigenvalues with which they each associate. The increase in the markers' size denotes the evolution process as the parameters vary along each loop in the indicated direction. The eigenstates (according to normalization of Supplementary equation (27)) need to be rescaled to place the tip of the vector on the complex unit sphere. Since we gauge the initial eigenstates to be real at the SP, the initial and final imaginary parts of the eigenstates are all zero. Thus, the evolution of the imaginary parts is simply an intermediate process under such normalization, which is convenient for characterizing the topology. Therefore, the topology is dominantly characterized by the evolution of the real parts of the eigenstates, which determines the rotation direction and rotation angle of the eigenframe. Along both loops, the accumulated rotation angle of Article https://doi.org/10.1038/s41567-023-02048-w φ 2 (in blue) is zero, and both φ 1 and φ 3 rotate by an angle π. That is, they each evolve from the initial states to their antipodal points (as indicated by the green radial axes), owing to the PT symmetry of the system. The results show that both loops can be viewed as topologically non-trivial as the rotation angles of the eigenframe are quantized. From the SP, we observe that φ 2 and φ 3 begin to rotate in opposite directions, which is a typical frame deformation process signifying non-Hermiticity. In contrast, for PT-symmetric Hermitian systems, the eigenstates must rotate in the same manner during a pure eigenframe rotation 17,18 . The intermediate processes along l α and l β are slightly different from each other simply because they are along different trajectories. Therefore, topologically, the rotations of the eigenstates along both loops are the same, which demonstrates that l α is equivalent to l β , and further explains why the pair of EL3s can transition to the NIL and NL via the MP (Fig. 4c). Note that the SPs of l α and l β need not be the same, so the yellow and green loops in Fig. 4c need not touch in order for them to afford the same frame rotation/deformation processes (see the criteria discussed in Supplementary Section 9). The continuous deformation from l α to l β is shown in Supplementary Video 2. The analysis indicates that the transition is topologically protected. Our method based on the Lorentz-like transformation of eigenstates also confirms that the emergence of the swallowtail is allowed by the symmetries in equation (2). To summarize, we showed that the swallowtail, which plays an important role in catastrophe theory, naturally appears in the spectra of non-Hermitian systems when we considered the evolution of eigenvalues in parameter space. In a family of three-state PT-symmetric non-Hermitian systems with an additional pseudo-Hermitian symmetry, we found degeneracies of eigenvalues in the form of EL3s, an NIL and an NL, and these seemingly unrelated types of singularities are stably connected at an MP, forming a swallowtail. Moreover, they can convert into each other as the system parameters change. From the experimental observations and theoretical analysis, we see that the transitions occur because these singular lines are topologically associated with each other. Since the symmetries of the considered Hamiltonian play an important role in the emergence of the swallowtail, exploring the generic topological classification of these symmetry-protected catastrophe singularities in the future will be worthwhile. Meanwhile, realizing such Hamiltonians in lattice systems may provide valuable platforms for investigating the bulk-edge correspondence in non-Hermitian swallowtail gapless phases. Furthermore, transitions among diverse singularities may pave a new way for the development of sensing and absorbing devices 22,35 .

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Any methods, additional references, Nature Portfolio reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at https://doi.org/10.1038/s41567-023-02048-w. blue and black, respectively. The three axes denote the three components of each eigenstate. The increase in the size of the dots denotes the directed variation in the parameters along the loops. 'Re' and 'Im' denote the real and imaginary parts of the eigenstates, respectively. c, Illustration of the transition from double EL3s to the NIL and NL in the swallowtail structure. Note that in the deformation process from l α to l β , the loop does not cut through any degeneracy lines.