Hybrid exceptional point created from type III Dirac point

Degeneracy (exceptional) points embedded in energy band are distinct by their topological features. We report different hybrid two-state coalescences (EP2s) formed through merging two EP2s with opposite chiralities that created from the type III Dirac points emerging from a flat band. The band touching hybrid EP2, which is isolated, is induced by the destructive interference at the proper match between non-Hermiticity and synthetic magnetic flux. The degeneracy points and different types of exceptional points are distinguishable by their topological features of global geometric phase associated with the scaling exponent of phase rigidity. Our findings not only pave the way of merging EPs but also shed light on the future investigations of non-Hermitian topological phases.

Degeneracy points and exceptional points embedded in the energy band are distinct by their topological features. We report hybrid exceptional point formed through merging two ordinary exceptional points with opposite chiralities that created from the type III Dirac points emerging from a flat band. The hybrid exceptional point is induced by the destructive interference at the proper match between the non-Hermiticity and the synthetic magnetic flux. The degeneracy points and different types of exceptional points are distinguishable by their topological features of global geometric phase accompanied with the scaling exponent of phase rigidity. Our findings not only pave the way of creating, moving, and merging EPs but also shed light on the future investigations of non-Hermitian topological phases.
The EPs possess distinct topology from DPs [26]. In a two-level non-Hermitian system, the energy levels interchange after encircling the EP for one circle in the parameter space; the interchanged energy levels restore their original values after two circles of encircling and accumulate a geometric phase ±π; the sign of the geometric phase depends on the circling direction and the chirality of the EP is defined by the accumulated geometric phase under the counterclockwise encircling [27][28][29][30][31][32][33][34][35][36]. Dynamical encircling of EPs realizes state switch and nonreciprocal topological energy transfer [37][38][39][40]; the dynamics depends on the starting/end point [41] and the homotopy of the encircling loop [42].
The EPs are different by their ways of coalescence as well as their topological properties [43]. In one aspect, the EPs are distinct by their orders: If more than two energy levels coalesce at the EP, the EP is called a high order EP [44,45], where the excitation intensity presents the polynomial increase [46]. In another aspect, even the EPs with identical order may dramatically differ from each other on the topological aspect. For example, when encircling a high order EP of three-state coalescence (EP3) in the energy band of a square root type Riemann surface in the parameter space, two energy levels flip after encircling one circle; if an EP3 is in a cubic root type Riemann surface, when it is encircled, three circles are needed for the energy levels to restore their original values. The geometric phases associated with the two EP3s are different, which reflect the different topological features of such two types of EP3s [47]. An interesting question naturally arises: whether EP2s have different topologies? If so, how to characterize their topological properties and distinguish them?
The manipulation of Dirac points in condensed matter physics is an interesting and challenging task. The merging of Dirac points induces topological phase transition and generates new types of Hermitian degeneracy; for example, two Dirac points with opposite topological charges can merge into a semi-Dirac point with linear and quadratic dispersions along two orthogonal directions [48][49][50][51]. In parallel, the merging of EPs leads to multifarious Hermitian and non-Hermitian degeneracies. It has been demonstrated that the merging of EPs may (i) lead to the DP [51]; (ii) create the high-order EP [35,36]; and (iii) form the hybrid EP [52,53]. The hybrid EP has linear and square root dispersions along two orthogonal directions and carries integer topological charge in contrast to the ordinary EP2 that carries halfinteger topological charge [54].
The hybrid EP can be formed by merging either two or-  Table I) and indicate different topological phases of the system. The topological features of EP mergers are unveiled.

II. THE BAND STRUCTURE AND PHASE DIAGRAM OF THE THREE-BAND SYSTEM
We consider a non-Hermitian three-band system with various configurations of EPs in the gapless phase [51]. The Hamiltonian reads The investigation of H in the parameter space (h x , h y , h z ) helps us grasp its topological properties. Since the couplings h x and h y play the same role, we take h x = h y without loss of generality. To be concrete, Fig. 2(a) schematically illustrates a three-band lattice. Applying the Fourier transformation, the Bloch Hamiltonian of the three-band lattice shown in Fig. 2 where k is the momentum (see Appendix A for more details). The trajectory of (h x , h z ) forms a closed circle in the h x -h z parameter space, the topological features of the Bloch bands directly relate to the topological properties of the band touching points enclosed in the trajectory of (h x , h z ) [34]. Thus, the topological properties of the three-band lattice can be obtained by studying the band structure and topology of H in Eq. (1). In Fig. 2(a), the sublattices a and c are indirectly coupled through the sublattice b and are directly coupled through a nonreciprocal coupling Je iΦ [36,55]. The Peierls phase factor e ±iΦ can be realized in various manners [56][57][58][59][60][61][62][63][64], which induces effective magnetic fluxes in the triangles but not the square plaquettes of the lattice. The synthetic magnetic fields have been experimentally realized in coupled optical resonators [65,66]. The sublattice a (c) has gain (loss) and the sublattice b is passive. PT -symmetric systems can be investigated by employing passive system with different losses, sticking absorption material or cutting waveguide induces additional loss [67][68][69]. The flat band in non-Hermitian lattice without synthetic magnetic flux was previously proposed through engineering the gain and loss [70]; besides, it was demonstrated that proposing a spectrum that entirely constituted by flat bands is possible with non-Hermitian couplings [71]. Alternatively, the flat band in the Hermitian systems maintains in the non-Hermitian situation at the proper match of synthetic magnetic flux and non-Hermiticity [72]. The synthetic magnetic flux provides a useful resource to generate different types of EPs and motivates us to study the problem of creating, moving, and merging EPs; in particularly, the topological properties of different types of EPs.
We first consider γ = 0, the lower band gap closes and the lower two bands touch at a pair of DPs for Φ = 0, π [73] and h z = 0. The DPs are two Dirac points at the peaks of two type III Dirac cones which are the critically titled type I Dirac cones with a flat band line Fermi surface. The type III Dirac cone associated with a line Fermi surface differs from the type I Dirac cone associated with a point Fermi surface as well as the type II Dirac cone associated with two cross lines Fermi surface [74][75][76]. The band touching creates a flat band −e iΦ J due to the destructive interference at the sublattice b. The lower two bands have isotropic linear dispersion near the Dirac points, as shown in the lower panel of Fig. 3(a).
The DPs disappear when gain and loss are introduced Re(E)  Fig. 2 Fig. 2 where x is replaced by h x h y in p, q. Three types of EP2s exist in the three-band system: (i) the ordinary EP2, which is the singularity point in the Riemann surface of square root type and has chirality. (ii) EP2(T), which is a merger of two ordinary EP2s with opposite chiralities. Two relevant bands touch at the EP2(T) and their real parts are gapped in the vicinity of EP2(T), the schematic of merging is indicated in Fig. 1 (IV). (iii) EP2(I), which is also constituted by merging two ordinary EP2s with opposite chiralities. Two relevant bands intersect, the schematic of merging is indicated in Fig. 1 (III). EP2(T) has anisotropic dispersions, being linear (square root) along h x (h z ) while EP2(I) is has isotropic square root dispersions along both h x and h z . If H has chiral symmetry, the EPs of H become EP3s. Figure 3 depicts the energy bands and schematically illustrates all the seven typical EP configurations in the gapless phase. The central two ordinary EP2s in Fig. 3(b) merging into one EP2(I) in Fig. 3(c) is the merging of type Fig. 1(III); and the four ordinary EP2s in Fig. 3(b) merging into two hybrid EP2(T)s in Fig. 3(e) is the merging of type Fig. 1(IV).
The EPs may disappear when the band are gapped for Φ = 0, ±π. This happens at weak non-Hermiticity |γ/J| < | sin Φ| [yellow region in Fig. 2(b)], where three bands are gapped. The gain and loss compress the band gaps, the lower two bands touch at the EPs when |γ/J| = | sin Φ|, the appropriate non-Hermiticity awakens the destructive interference and reproduces the flat band.
The flat band energy is altered to −J cos Φ [72], and the band touching points become two EP2(T)s rather than two DPs. As the gain and loss rates γ increase, the band gaps vanish and all three bands intersect when |γ/J| 1.

III. TOPOLOGICAL CHARACTERIZATION OF BAND TOUCHING POINTS
The geometric (Berry) phase of energy band are relevant to the topological features of DPs and EPs. The generalized geometric phase for the non-Hermitian systems is defined as [77,78] where n is the band index. |ψ n (k) and |φ n (k) are the eigenstates of Hamiltonians H and H † and form a biorthonormal basis φ n (k) |ψ n (k) = δ nn . The integration is performed over a loop C in the parameter space. For the DPs and EP2s that only relevant to two bands, the irrelevant third band restores its original eigenvalue when the loop C of system parameters encircling a band touching DP or EP for one circle; the corresponding eigenstate accumulates a zero geometric phase; when the bands are tangled in the presence of Hermitian or non-Hermitian band degeneracies [79,80], the non-Abelian Berry connections A mn = φ m (k) |∇ k |ψ n (k) characterize the topological properties of energy bands [80][81][82]. The global geometric phase Θ = 3 n=1 Γ n is a topological invariant [80,83,84]. The winding number w = Θ/ (mπ) characterizes the topology of band touching points, where m is the number of relevant bands.
Moreover, the global geometric phase is unable to distinguish the topology of all different EPs and needs the assistance of phase rigidity r = | ψ * n |ψ n / ψ n |ψ n |.
The phase rigidity r describes the mixing of different states [85]. In Hermitian system with a real matrix, the phase rigidity is 1. When extended to the non-Hermitian system, the defective eigenstate is self-orthogonal and the phase rigidity at the EPs reduces to 0 [86]. The phase rigidity has a scaling law in the vicinity of EPs, The phase rigidity scaling exponent ν characterizes the response manner of energy bands when approaching the EPs along the parameter γ; while the geometric phase we discussed characterizes the topological features of energy bands around EPs in the parameter space at fixed γ. Table I summarizes the winding number and the phase rigidity scaling exponent along γ for the DP and all the types of EPs. We turn to discuss the details of topological properties of the band touching points. The discussions as follows are organized in the order of gain and loss increase. The creating, splitting, moving, and merging of band touching points and their topological features are presented. We focus on the topological features of two types of unexplored two-state coalescence: EP2(T) and EP2(I); the details on the topological features of other band touching points are provided in the Appendix. Two DPs appear at (h x , h z ) = (±1, 0) for Φ = 0 at γ = 0 as depicted in Fig. 3(a), they are type III Dirac points embedded in the flat band and appear in the region marked by the black circles in the phase diagram of Fig. 2(b). The geometric phase for each of the two degenerate bands is 0 when encircling either DP for one circle; the geometric phase for the irrelevant upper band is also 0. The phase rigidity scaling exponent is ν = 2 (see Fig. 8 in Appendix C).
The band gap between the lower two bands is closed in the presence of the flat band in Fig. 3(a). When the gain and loss are introduced (γ = 0), the energy bands become even closer and each DP splits into two ordinary EP2s with opposite chiralities at the weak non-Hermiticity |γ/J| < 1. This describes the orange region in the phase diagram of Fig. 2(b). Figure 3(b) depicts the band spectrum at the situation γ/J = √ 2/2 < 1. The four ordinary EP2s are on the h z = 0 axis, and their chiralities are opposite with respect to h x = 0 as illustrated in the middle panel. The eigenvalues of two relevant coalescence states are 4π periodic in k when encircling the ordinary EP2s, but the period of the third state is 2π. After one circle of encircling, two coalescence relevant states exchange and the third state restores its original eigenvalue; the global geometric phases Θ accumulated by the three bands equals to +π (−π) for the ordinary EP2 of +1 (−1) chirality (see Fig. 6 in Appendix B), and two circles of encircling yields a +π (−π) geometric phase for either relevant band. The phase rigidity scaling exponent that associated with the ordinary EP2s is equal to ν = 1/2 (see Fig. 9 in Appendix C).
As the gain and loss rates γ increase, the central two ordinary EP2s become closer, but the outer two ordinary EP2s become far way. At |γ/J| = 1, the central two ordinary EP2s with opposite chiralities meet and merge to an EP2(I) at (h x , h z ) = (0, 0) [52], and the system enters the region marked by the red lines in the phase diagram of Fig. 2(b) and has three EP2s [ Fig. 3(c)] with identical scaling exponent ν = 1/2 although they posses distinct topology [Figs. 5(a)-5(c)]. At the EP2(I), although three levels have identical zero energy; only two levels coalesce and they degenerate with the third level; the system is defective with one eigenstate missing. The energy levels restore their original values after encircling the EP2(I) for one circle; states switch twice for the lower two levels; the geometric phases for the two lower lev-  Fig. 3(e)]. The EPs are marked by the red crosses and the trajectory of encircling in the parameter space is represented by the blue circle (insets); the loop radius is R, centered at (v, 0). els are still opposite, being π and −π, respectively; and the winding number for the global geometric phase is 0. The global geometric phase for encircling the EP2(I) at (h x , h z ) = (0, 0) is depicted in Fig. 4(a). The real and imaginary parts of eigen energies of H k for the encircling process are depicted in Figs. 4(b) and 4(c). The other two EP2s (±3 √ 6/4, 0) are ordinary EP2s. The EP2(I) in the center is connected to the other two ordinary EP2s by two Fermi arcs [87].
In the region | sin Φ| < |γ/J| < 1, four ordinary EPs with two opposite chiralities in pairs locate on the two sides of h z = 0, respectively [ Fig. 3(b)]. Through changing Φ, each pair of EPs can merge into an EP2(T) when |γ/J| = | sin Φ| for 0 < |Φ| < π/2 [ Fig. 3(e)]; this is indicated by the blue lines in the phase diagram of Fig. 2(b). Two valleys (peaks) in the middle (lower) band; the apexes of valleys and peaks touch and two EP2(T)s are formed at the appropriate match between the effective magnetic flux Φ and non-Hermiticity γ, the flat band reappears. In contrast to two DPs, the isolated band touching points are EP2(T)s located at h 2 x = J 2 cos 2 Φ. In the parameter space, the global geometric phase for encircling EP2(T) (h x , h z ) = (1/2, 0) in the counter clockwise direction is depicted in Fig. 4(d). Figures 4(e) and 4(f) are the real and imaginary parts of eigen energies of H k . No state switch occurs when encircling the EP2(T) in the parameter space for one cycle and the global geometric phase is 0; the EP2(T) has the winding number 0 without chirality and the scaling exponent of phase rigidity is ν = 1 [ Fig. 5(d)-5(f)]. In the region of weak non-Hermiticity |γ/J| < | sin Φ|, the three bands are gapped without band touching.
For |γ/J| > 1, the central EP2(I) vanishes and splits into four ordinary EP2s with two +1 and two −1 chiralities in h z = 0 region and six ordinary EP2s exist, provided that H is not chiral symmetric [ Fig. 3(d)], the phase is the cyan region in the phase diagram of Fig. 2(b). Among total six ordinary EP2s, three of them in the region h x > 0 (h x < 0) have +1 (−1) chirality. The upper two bands coalesce at these four ordinary EP2s (h z = 0); and the lower two bands coalesce at the other pair of ordinary EP2s on the h z = 0 axis. The energy bands with six ordinary EP2s is shown in Fig. 3(d), Φ = π/3 is chosen in order to observe all the EP2s within the region [−2, 2] in the parameter space.
The three-band Hamiltonian H is chiral symmetric when J = 0 or Φ = ±π/2, CHC −1 = −H, where m |C| n = (−1) m+1 δ m,4−n . A zero mode flat band is formed under the chiral symmetry. The upper and lower bands in Fig. 3(f) constitute a hybrid conical surface, the projection of which on the E-h x (E-h z ) plane is a conus of square root repulsion that differs from a Dirac cone or a semi-Dirac cone [88][89][90]. At |γ/J| < 1, the spectrum is gapped and EP vanishes. At |γ/J| = 1, two EP2(T)s merge to a single EP2(T) at (0, 0); the phase is represented by the green circles in the phase diagram of Fig. 2(b). At |γ/J| > 1, the system has one pair of EP3s with opposite chiralities at (h x , h z ) = (± (γ 2 − J 2 )/2, 0) [ Fig. 3(g)]; the phase is indicated by the purple lines in the phase diagram of Fig. 2(b). After encircling an EP3 for one circle, the upper and lower bands switch and two circles is needed to restore the original eigenvalues. The global geometric phase accumulated is +3π (−3π) after encircling the EP3 of chirality +1 (−1) for one circle (see Fig. 7 in Appendix B). The scaling exponent of the phase rigidity close to EP3 is ν = 1 (see Fig. 10 in Appendix C).

IV. DISCUSSION AND CONCLUSION
The topological characterization of band touching points applies for all types of EPs that not limited to the DP, the ordinary EP2/EP3, and the hybrid EP2s presented in the lattice model of Fig. 2(a) as listed in Table I. For the EP2 merging of types Fig. 1(I) and Fig. 1(II), the geometric phase is π and the winding number is |w| = 1. Different types of EPs are distinguishable from their topological features.
Hybrid EP generally presents in the non-Hermitian systems that possessing the flat band with EPs embedded. The EP embedded in a flat band is a hybrid EP if two-dimensional parameter space is considered [71]. Notably, H can describe non-Hermitian Lieb lattice with additional gain and loss γ. The synthetic magnetic flux Je iΦ can be induced by the spin-orbital coupling [64]. The Bloch Hamiltonian of the non-Hermitian Lieb lattice has the form of h x = 2J x cos (k x /2), h y = 2J y cos (k y /2), h z = 0 [91]. Besides, hybrid EP can appear in the absence of flat band and hybrid EP of arbitrary high order is possible to be created with asymmetric couplings [53,92]. The asymmetric coupling has connection with the gain and loss in non-Hermitian systems. In practice, the gain and loss associated with the effective magnetic flux equivalently induce asymmetric coupling and nonreciprocity in the non-Hermitian systems [55,93,94].
In conclusion, we propose the hybrid EP2 through merging two ordinary EP2s with opposite chiralities, which are created from the type III Dirac points emerging from a flat band through introducing a proper gain and loss. The topology of degenerate (exceptional) point is characterized by the winding number (w) associated with global geometric phase and phase rigidity scaling exponent (ν). The topological properties of different EP2 mergers are unveiled, the change of topological features associated with the merging of EPs indicates the topological phase transition. Our findings pave the way of creating, moving, and merging EPs and are valuable for future studies on the non-Hermitian topological phase of matter. In the future, further investigations on the dynamical encircling of hybrid EPs [53], the creating, moving, and merging of high-order EPs [36,51], as well as the topological edge states [95] would be of great interest.

ACKNOWLEDGMENTS
We acknowledge the support of National Natural Science Foundation of China (Grants No. 11975128, No. 11605094, No. 11604220, and No. 11874225

APPENDIX A: TRIANGULAR LATTICE
In this section, we show the real space lattice Hamiltonian for the three-band Bloch Hamiltonian H in Eq. (1). The three-band tight-binding lattice consists of three sublattices, the sublattice a (c) has gain (loss) and the sublattice b is passive. The sublattices a and c have nonreciprocal nearest neighbour couplings. The phase factors in the couplings are opposite, indicated by the arrows. The sublattices a and c are coupled indirectly through the sublattice b and directly through a nonreciprocal coupling Je iΦ , which can be realized in cold atomic gases by inducing the spin-orbital interaction, and can be realized by optical path imbalance, dynamic modulation, and photon-phonon interaction in optics. The triangular lattice Hamiltonian H T L in the real space is given by The couplings strengths are v, R/2, and J. The nonreciprocal couplings ±iR/2 lead to an effective magnetic flux π enclosed in each square plaquette. The gain and loss rates are γ. In the experimental studies, it is not necessarily to induce the gain to balance the loss in the investigations of PT -symmetric lattices, using the loss only passive systems brings the convenience [67][68][69]. where the discrete momentum is k = 2πn/N (integer n ∈ [1, N ]); a j , b j , and c j are the annihilation operators that satisfy the periodical boundary condition a N +1 = a 1 , b N +1 = b 1 , and c N +1 = c 1 . In the momentum space, the lattice Hamiltonian is expressed as H T L = k H k with h x = v + R cos k and h z = R sin k. The Bloch Hamiltonian H k in the momentum space is a 3 × 3 non-Hermitian Hamiltonian in the form of Eq. (1) in the main text for h x = h y .

APPENDIX B: GEOMETRIC PHASE
In this section, we show the geometric phase associated with the band touching points, including the ordinary EP2 and EP3.
The global geometric phase for non-Hermitian system is Q = i 3 n=1 C φ n (k) |dψ n (k) [83,84]. The integration is performed over a loop C in the parameter space. The trajectory of (h x , h z ) forms a closed circle C in the parameter space of the h x -h z plane.
The geometric phases for the trajectory of (h x , h z ) encircling an ordinary EP2 with right chirality and encircling an EP3 with left chirality are depicted in Fig. 6 and Fig. 7, respectively. In both cases, the coalescence associated energy levels switch after encircling the EP once and restore their original values after encircling the EP twice. The accumulated global geometric phase is π for one circle of encircling the right chiral EP2 and the accumulated global geometric phase is −3π for encircling the left chi- , phase rigidity (c), and scaling law (d) at an ordinary EP2 in Fig. 3(b). The system parameters are hx = √ 2/4, hz = 0, J = 1, Φ = 0. The scaling exponents for the coalesced states (green square and blue cross) are ν = 0.5; the scaling exponent for the third state (red circle) is ν = 1.0. ral EP3. The winding number is w = Θ/ (mπ), where m is the number of coalesced levels. Therefore, the winding number for the ordinary EP2s (EP3s) is w = ±1/2 (w = ±1). The + (−) sign is for the right (left) chirality.

APPENDIX C: PHASE RIGIDITY
In this section, we show the band spectrum, the phase rigidity, and the scaling exponent as the gain and loss approaching the DP, the ordinary EP2, and the EP3, respectively.
The phase rigidity [85,86] of an energy level r = | ψ * n |ψ n / ψ n |ψ n | has a scaling law in the vicinity of EPs in the form of |r EP − r| ∝ (γ EP − γ) ν . At Φ = 0, the degeneracies are two-level diabolic point at (±1, 0) in Hermitian lattice of γ = 0. In Fig. 8, the DPs appear when γ DP = 0; two energy levels become a complex conjugation pair, the imaginary part of which changes linearly in the vicinity of DPs. The phase rigidities for all three levels are r DP = 1.0 and the scaling exponents are all identical ν = 2.0.
In Fig. 9, the energy level, phase rigidity, and scaling law are depicted for the ordinary EP2 of Φ = 0, J = 1 at (h x , h z ) = ( √ 2/4, 0). The system has one real energy and two energy levels coalesce at the EP2. The phase rigidities of the coalesced levels are r EP = 0 at the EP2 γ EP = √ 2/2, the corresponding scaling exponent is  Spectrum (a, b), phase rigidity (c), and scaling law (d) at an EP3 in Fig. 3(g). The system parameters are hx = √ 2/2, hz = 0, J = 1, Φ = π/2. The scaling exponents for all states are ν = 1.0. ν = 0.5; the third level that not participated in the coalescence has the scaling exponent ν = 1.0.
In Fig. 10, the energy level, the phase rigidity, and the scaling law are depicted for the EP3 of J = 1, Φ = π/2 at (h x , h z ) = ( √ 2/2, 0). The system has chiral symmetry, the energy levels are symmetric about 0 and exhibit a square root dependence on the non-Hermiticity as ± 2 − γ 2 . The phase rigidities of three eigenstates equal to r EP = √ 2/2 at the EP3 γ EP = √ 2. The scaling exponents are ν = 1.