Abstract
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.
Highlights
Exceptional points (EPs) are non-Hermitian degeneracies [1,2,3,4], at which the system Hamiltonian is defective and eigenstates coalesce [5]
Frequency sensing is enhanced because the responses of energy levels to the detuning perturbation are the square root near a two-state coalescence (EP2) and the cubic root near a three-state coalescence (EP3) in non-Hermitian systems, which is more efficient than the linear response near a diabolic point (DP) in Hermitian systems [24,25]
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]
Summary
Exceptional points (EPs) are non-Hermitian degeneracies [1,2,3,4], at which the system Hamiltonian is defective and eigenstates coalesce [5]. The EPs are distinct in 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 orders may dramatically differ from each other in the topological aspect. When encircling a high-order EP with 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-roottype Riemann surface, when it is encircled, three circles are needed to restore the energy levels to their original values. The DPs and various types of EPs possess distinct topological properties; they are well distinguished from each other under the developed topological characterization (see Table I) and indicate different topological phases of the system.
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