Doubling theorem under PT symmetry with nonlinear Kramers pairs

Establish whether a doubling theorem holds for n-fold exceptional points in the nonlinear eigenvalue equation F(ω,k)|ψ⟩=0 when the system has parity–time symmetry satisfying U_PT U_PT* = −1 (i.e., nonlinear Kramers pairs), specifically determining whether every n-fold exceptional point must be accompanied by another of opposite frequency–momentum winding number across the Brillouin zone.

Background

The paper proves a general doubling theorem for multifold exceptional points in nonlinear eigenvalue problems by introducing frequency–momentum winding numbers. This result covers cases without symmetry and with several symmetries, including PT symmetry when the antiunitary operator satisfies U_PT U_PT* = 1.

However, when U_PT U_PT* = −1, the antiunitary symmetry enforces nonlinear Kramers pairs: if |ψ_n⟩ has eigenvalue ω_n, then U_PT K |ψ_n⟩ (K is complex conjugation) is another eigenstate with eigenvalue ω_n*. The resulting degeneracy obstructs a sign change of det F(ω,k) near exceptional points, preventing a straightforward application of the established arguments for doubling.

Consequently, whether a doubling theorem persists in this Kramers-degenerate setting remains unresolved. The authors also note that a similar argument applies to CP† symmetry since it reduces to PT symmetry, but they explicitly formulate the open question for the U_PT U_PT* = −1 case.

References

While we have focused on the cases of $U_{\mathrm{PT}U*_{\mathrm{PT}=1$ so far, the doubling theorem for cases of $U_{\mathrm{PT}U*_{\mathrm{PT}=-1$ remains an open question due to nonlinear Kramers pairs. If $|\psi_n\rangle$ is an eigenstate with eigenvalue $\omega_n$, then $U_{\mathrm{PT}\mathcal{K}|\psi_{n}\rangle$ is also an eigenstate with eigenvalue $\omega*_n$ where $\mathcal{K}$ denotes the complex conjugate operator.

— Nonlinear Frequency-Momentum Topology and Doubling of Multifold Exceptional Points  (2604.00366 - Yoshida, 1 Apr 2026) in Section: Nonlinear Kramers pairs (main text)