The Parity-Time (PT) symmetric systems present exotic wave phenomena in many fields such as optics, microwaves, and acoustics. Such systems can be realized in a coupled-resonators by designing the individual resonant elements to have the same characteristic frequency with one having energy loss and the other having equaled energy gain at resonance. The most intriguing feature of the PT-symmetric systems is the formation of an exceptional point singularity (EP)—a branch-point singularity where the eigenmodes and eigenvalues of the system coalesce—around which the interacting modes exhibit a square-root behavior. When the parameters of the Hamiltonian, such as the magnitude of the balanced gain and loss and the coupling between those two resonant elements, are appropriately varied, a phase transition between the unbroken and the broken PT-symmetry phases occurs at the EP.

Here, we describe the mechanisms of the emergence of EPs by numerically studying two classes of PT-symmetric elastodynamic metamaterial—a quasiperiodic elastodynamic metamaterial inspired by the Aubry-Harper model and a geometric fractal tree elastodynamic metamaterial made of H-shaped motifs. They are created by introducing equal-magnitude gain and loss to the fractal components on both sides. Each resonant beam element on the gain component has a corresponding resonant beam element on the loss counterpart with the same resonant frequency and mode. As the fractal generation is increased, this metamaterial exhibits richer behavior as new resonance modes emerge in a scale-free fractal nature.

We use a steady-state-dynamic finite element approach in *ABAQUS SIMMULIA* to model the fractal metamaterials. The metamaterial is harmonically excited by a sinusoidal prescribed displacement at the left end and the corresponding sinusoidal reaction force at its fixed right end is measured. By varying the balanced gain/loss intensity over three orders of magnitude (structural damping coefficient: 0.001 to 1), we study the interactions of resonant modes and the formation of EPs in each generation of the fractal metamaterials.

These elastodynamic metamaterials exhibit a cascade of EPs that occur at different balanced gain/loss intensity. The embodiment of fractality in PT-symmetric metamaterials offers the potential to create numerous EPs in a scale-free fashion, which is similar to the scale-free nature of their fractal frequency spectra. The scale-free emergence of numerous EPs is seen in both metamaterials, showing an intimate relation between the scale-free distribution of critical gain/loss intensities and the spectral fractal dimension of the corresponding Hermitian spectra (no gain no loss). This relation implies the existence of an underlying universal route for the creation of EPs in systems with a fractal spectrum. By developing a coupled-mode-theory-based model utilizing on-site resonant modes that follow Fibonacci rules, we generalize the behavior of these PT-symmetric elastodynamic metamaterials and establish the relation of EPs to the fractal dimension of the frequency spectrum of the underlying Hermitian systems.


Emergence of Exceptional Points in PT-Symmetric Fractal Elastodynamic Metamaterials

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Aerospace, Bioengineering, Applied Mechanics, Design Engineering Division, Advanced Energy Systems, Manufacturing Engineering Division, Noise Control and Acoustics, Nondestructive Evaluation, Diagnosis, and Prognosis (NDPD), Nuclear Engineering Division, Process Industries Division, Fluids Engineering, Heat Transfer, Materials, Microelectromechanical Systems, Safety Engineering and Risk Analysis, Computers & Information in Engineering


Paper Proceedings: https://asmedigitalcollection.asme.org/IMECE/IMECE2020/volume/84478

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