Nonlinear Frequency-Momentum Topology and Doubling of Multifold Exceptional Points

Even in the linear limit, the topology of multifold (also called higher-order) exceptional points across the Brillouin zone has lacked a general characterization, leaving the doubling theorem essentially limited to two-fold exceptional points. Here, we establish the doubling theorem of $n$-fold exceptional points [EP$n$s ($n=2,3,\ldots$)] for systems where nonlinearity enters through eigenvalues. To this end, we introduce new topological invariants, termed frequency-momentum winding numbers, which characterize nonlinear EP$n$s in $m$-band systems throughout the Brillouin zone for arbitrary $n$ and $m$ ($m\geq n$). These invariants enable a unified proof of the doubling theorem in the absence of symmetry and under several symmetry constraints, including parity-time ($PT$) and charge-conjugation-parity symmetries. Furthermore, even in the linear limit, the frequency-momentum winding number indicates $\mathbb{Z}$ topology of $PT$-symmetric EP$2$s which is beyond the previously reported $\mathbb{Z}_2$ topology. The frequency-momentum winding numbers can also be extended to a class of coupled resonators in which nonlinearity enters via the eigenvectors, whereas the spectrum is determined by a nonlinear scalar equation for the frequency.