Nonlinear Frequency-Momentum Topology and Doubling of Multifold Exceptional Points

Abstract

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.

Figures (4)

• Figure 1: Illustration of the doubling theorem of nonlinear EP$n$s in $2n$-dimensional $\omega$-$\bm{k}$ space. Panel (a) shows that an EP$n$ with $W_{2n-1}=1$ (red dot) is accompanied by another EP$n$ with $W_{2n-1}=-1$ (blue dot) due to the periodicity of the BZ and $\bm{k}$ independence of $F(\omega,\bm{k})$ for $|\omega|\to \infty$ [see Eqs. \ref{['eq: W no symm']} and \ref{['eq: Wtot=0 nosymm']}]. In this panel, the axes of $\mathrm{Im}\omega$ and $k_j$ ($j=3,\ldots, 2n-2$) are omitted. Panel (b) illustrates the winding structure of $\bm{d}(\omega,\bm{k})$ [Eq. \ref{['eq: dvec nosymm']}] in the vicinity of the EP$n$. The green (orange) surface separates the regions of positive and negative $d_{2n-1}$ ($d_{2n}$). The red surface denotes the region where $d_1=d_2=\ldots=d_{2n-2}=0$ holds, forming a manifold of EP$n'$ with $n'=n-1$. The vector $\bm{d}$ winds the $(2n-1)$-sphere in the vicinity of the EP$n$ emerging on the two-dimensional manifold of EP$n'$s with $n'=n-1$.
• Figure 2: (a) Winding number $W_2$ for each mesh. The winding number is $W_2=1$ ($W_2=-1$) on the surface of red (blue) cuboids. Here, we take a $10\times10\times10$ mesh. The winding number is computed based on the method proposed in Ref. Fukui_JPSJ2005. For the computation of $W_2$, each cuboid is further subdivided into a $10\times10\times 10$ mesh. (b) Eigenvalues $\omega\in\mathbb{R}$ as functions of $\bm{k}=(k_1,k_2)$. The data for $k_2=2.80176$ ($k_1=2.15984$) are represented by red (blue) points. The others are represented by gray points. The blue (black) point denotes an EP2 (EP3). Data are obtained for $(\delta_1,\delta_2)=(0.01,1)$.
• Figure S1: (a) [(c)]: Color plot of the argument $\mathrm{Arg}[d_1+\mathrm{i} d_2]/\pi$ as a function of $k$ and $\omega$ ($\omega\in\mathbb{R}$) of the Hamiltonian $H_{4\times4}$ for $\kappa=0.5$ [$\kappa=0$]. In these panels, the FM winding number takes the value $W_1=-1$ along the path enclosing points denoted by black arrows. (b) [(d)]: Eigenvalues as functions of $k$ of the Hamiltonian $H_{4\times4}$ for $\kappa=0.5$ [$\kappa=0$]. The real and imaginary parts of eigenvalues are represented by red and blue lines. The data are obtained for $V=0.3$.
• Figure S2: (a) [(c)]: Color plot of the argument $\mathrm{Arg}[d_1+\mathrm{i} d_2]/\pi$ as a function of $k$ and $\omega$ ($\omega\in\mathbb{R}$) of the Hamiltonian $H_{2\times2}$ for $m_0=0.5$ [$m_0=1.1$]. In these panels, the FM winding number takes the value $W_1=-1$ ($W_1=1$) along the path enclosing points denoted by the black (white) arrow. (b) [(d)]: Eigenvalues as functions of $k$ of the Hamiltonian $H_{2\times2}$ for $m_0=0.5$ [$m_0=1.1$]. The real and imaginary parts of eigenvalues are represented by red and blue lines.