Defect Modes of Defective Parity-Time Symmetric Potentials in One-Dimensional Fractional Schrödinger Equation

Defect modes of defective parity-time symmetry in one-dimensional fractional Schrödinger equation are theoretically investigated. Both positive and negative defect modes are extremely sensitive to the defect strength and amplitude of periodic potential. Their eigenvalues increase almost linearly with the defect strength and continuously jump among neighboring gaps in the entire defect strength domain. There exists a phase transition point where two adjacent defect mode curves merge into a single one. Below which the eigenvalues are real, and their profiles have the same symmetry as the parity-time symmetric potential. Above the transition point, the defect modes give rise to complex eigenvalues, and the above symmetric property is destroyed. The transition point also grows linearly with the defect strength. As defect strength increases, much more positive defect mode branches merge together and more phase transition points appear in each gap. However, only one phase transition point exists for negative defect modes, and moves into the higher gaps.