Complex energy structures of exceptional point pairs in two level systems

TL;DR

The paper addresses how exceptional points (EPs) in non-Hermitian two-level systems carry topological information via vorticity, and shows that EPs organize into two robust pair types: type-I (opposite vorticities, often sharing branch cuts and merging to Dirac points) and type-II (identical vorticities, no shared cuts, forming vortex points). It develops a braiding framework using real and imaginary branch-cut exchanges, and extends the analysis to triples and higher-order EP configurations, detailing their cumulative vorticities. A photonic-crystal model with lossy materials demonstrates multiple EPs and their branch-cut connections, validating the theoretical classification and illustrating transitions to Dirac and vortex points at critical parameters. The findings advance understanding of non-Hermitian topology by providing concrete building blocks for EP networks and a practical platform for exploring topological energy-band structures.

Abstract

We investigate the topological properties of multiple exceptional points in non-Hermitian two-level systems, emphasizing vorticity as a topological invariant arising from complex energy structures. We categorize EP pairs as fundamental building blocks of larger EP assemblies, distinguishing two types: type-I pairs with opposite vorticities and type-II pairs with identical vorticities. By analyzing the branch cut formation in a two-dimensional parameter space, we reveal the distinct topological features of each EP pair type. Furthermore, we extend our analysis to configurations with multiple EPs, demonstrating the cumulative vorticity and topological implications. To illustrate these theoretical structures, we model complex energy bands within a two-dimensional photonic crystal composed of lossy materials, identifying various EP pairs and their branch cuts. These findings contribute to the understanding of topological characteristics in non-Hermitian systems.

Figures (6)

• Figure 1: (a) Real parts and (b) imaginary parts of complex energy $E_{\pm}$ of Eq. (\ref{['eq:oneEP']}) when $z_1 = 0$. (c) Arguments of complex energy difference $E_{+}-E_{-}$. The argument increases by $\pi$ after a one-cycle counterclockwise journey on an encircling loop (black circle) since the white line in the arguments represents $\pm \pi /2$. The vorticity of an EP at $z=0$ is $+1/2$. (d) Schematic diagram of an EP (black circle) and real (green straight line) and imaginary (orange dashed line) branch cuts on 2D parameter space.
• Figure 2: (a) Real parts and imaginary parts of complex energy $E_{\pm}$ of Eq. (\ref{['eq:twoEP_1']}), arguments of complex energy difference $E_{+}-E_{-}$, and schematic diagram of EPs (black circle) and real (green lines) and imaginary (orange dashed lines) branch cuts on 2D parameter space when $z_1 = -1.0 - 0.5 i$ and $z_2 = 1.0 + 0.5 i$. The color scale of arguments is the same as Fig. \ref{['fig1']}(c). The vorticity of each EP are $\pm 1/2$ and the total vorticity of the two EPs is $0$. (b) Real parts and imaginary parts of complex energy $E_{\pm}$ of Eq. (\ref{['eq:twoEP_2']}), arguments of complex energy difference $E_{+}-E_{-}$, and schematic diagram when $z_1 = -1.0 - 0.5 i$ and $z_2 = 1.0 + 0.5 i$. The vorticity of each EP are $1/2$ and the total vorticity of the two EPs is $1$.
• Figure 3: (a) Real parts and imaginary parts of complex energy $E_{\pm}$ of Eq. (\ref{['eq:threeEP_21']}), arguments of complex energy difference $E_{+}-E_{-}$, and schematic diagram of EPs (black circle) and real (green lines) and imaginary (orange dashed lines) branch cuts on 2D parameter space when $z_1 = -1.0 - 0.5 i$, $z_2 = 1.0 + 0.5 i$, and $z_3 = 0.4 - 0.4 i$. The color scale of arguments is the same as Fig. \ref{['fig1']}(c). The vorticity of each EP are $\pm 1/2$ and the total vorticity of the three EPs is $1/2$. (b) Real parts and imaginary parts of complex energy $E_{\pm}$ of Eq. (\ref{['eq:threeEP_30']}), arguments of complex energy difference $E_{+}-E_{-}$, and schematic diagram when $z_1 = -1.0 - 0.5 i$, $z_2 = 1.0 + 0.5 i$, and $z_3 = 0.4 - 0.4 i$. The vorticity of each EP are $1/2$ and the total vorticity of the three EPs is $3/2$.
• Figure 4: Exceptional points (black circles), branch cuts (green lines and orange dashed lines), and selected encircling loops (blue and red closed loops) in the cases of (a) type-I and (b) type-II EP pairs and (c) type-{2,1} and (d) type-{3,0} EP triples. The parameters change on the encircling loops from the initial parameters (blue and red circles) counterclockwise. The loop crosses real and imaginary branch cuts are described by operations, $\mathcal{O}_R$ and $\mathcal{O}_I$, respectively.
• Figure 5: (a) The lowest four energy bands in (b) a photonic crystal made of two lossy triangular cavities with a complex refractive index $n$. (c) The arguments of complex energy difference between third and fourth energy bands. The color scale of arguments is the same as Fig. \ref{['fig1']}(c). It is noted that the discontinuity of arguments results from those of imaginary parts of third and fourth energy bands, defined by the order of real parts of energies. The black and red circles represent EPs with positive and negative vorticities, respectively. The green straight and orange dashed lines do simplified real and imaginary branch cuts, respectively.