Coalescence of exceptional points and phase diagrams for one-dimensional $\mathcal{P}\mathcal{T}$-symmetric photonic crystals

Coalescence of exceptional points and phase diagrams for one-dimensional $P T$ -symmetric photonic crystals

Kun Ding, Z. Q. Zhang, and C. T. Chan

Phys. Rev. B 92, 235310 – Published 31 December 2015

Abstract

Non-Hermitian systems with parity-time- ( $P T$ )- symmetric complex potentials can exhibit a phase transition when the degree of non-Hermiticity is increased. Two eigenstates coalesce at a transition point, which is known as the exceptional point (EP) for a discrete spectrum and spectral singularity for a continuous spectrum. The existence of an EP is known to give rise to a great variety of novel behaviors in various fields of physics. In this work, we study the complex band structures of one-dimensional photonic crystals with $P T$ -symmetric complex potentials by setting up a Hamiltonian using the Bloch states of the photonic crystal without loss or gain as a basis. As a function of the degree of non-Hermiticity, two types of $P T$ symmetry transitions are found. One is that a $P T$ -broken phase can reenter into a $P T$ -exact phase at a higher degree of non-Hermiticity. The other is that two EPs, one originating from the Brillouin zone center and the other from the Brillouin zone boundary, can coalesce at some $k$ point in the interior of the Brillouin zone and create a singularity of higher order. Furthermore, we can induce a band inversion by tuning the filling ratio of the photonic crystal, and we find that the geometric phases of the bands before and after the inversion are independent of the amount of non-Hermiticity as long as the $P T$ -exact phase is not broken. The standard concept of topological transition can hence be extended to non-Hermitian systems.

Received 26 September 2015

DOI:https://doi.org/10.1103/PhysRevB.92.235310

Physics Subject Headings (PhySH)

Phase diagrams

Photonic crystals Topological materials

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Kun Ding, Z. Q. Zhang, and C. T. Chan^*

Department of Physics and Institute for Advanced Study, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong

^*Corresponding author: phchan@ust.hk

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Vol. 92, Iss. 23 — 15 December 2015

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Figure 1
(a) Schematic picture of the unit cell of a $P T$ -symmetric photonic crystal, and the profile of real/imaginary parts of the dielectric constants $[Re (ɛ) / Im (ɛ)]$ in one unit cell. (b) Calculated band structure for the photonic crystal without loss or gain [ $Im (ɛ) = 0$ , solid blue line] with system parameters $d_{a} = 0.42 Λ, d_{b} = 0.58 Λ, ɛ_{a} = 3.8, ɛ_{b} = 1$ , and $μ_{a} = μ_{b} = 1$ . The solid red circle marks an even mode $[u_{n k} (x) = u_{n k} (- x)]$ , and the solid gray marks an odd mode $[u_{n k} (x) = - u_{n k} (- x)]$ . Band numbers are indicated on each band.
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Figure 2
Complex band structures at $Γ (= Γ_{a} = Γ_{b}) = 0.1$ (a),(b), $Γ (= Γ_{a} = Γ_{b}) = 0.5$ (c),(d), $Γ (= Γ_{a} = Γ_{b}) = 1.0$ (e),(f), and $Γ (= Γ_{a} = Γ_{b}) = 2.0$ (g),(h). All the other parameters are the same as shown in Fig. 1. The top four panels (a),(c),(e),(g) plot the real parts of the eigenfrequencies, and the bottom four panels (b),(d),(f),(h) plot the imaginary parts of the eigenfrequencies. Solid lines are calculated by Hamiltonian ((2.5)) and all the dots are calculated by full wave simulations, in which magenta dots label $P T$ -exact states, and green/gray/yellow dots label $P T$ -broken states. The letters M, S, N, and $R$ denote the EPs.
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Figure 3
(a) Real parts and (b) imaginary parts of eigenfrequencies for the states at $k Λ / 2 π = 0$ on the fourth, fifth, and sixth bands as functions of $Γ$ , in which diamond dots are calculated by the full Hamiltonian and solid lines are obtained by using the $3 \times 3$ reduced Hamiltonian.
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Figure 4
Trajectories of the exceptional points in $(k, Γ)$ space for the parameters given in Fig. 1: (a) $M$ (blue line), $S$ (red line), $R$ (yellow line), and (b) $N$ (magenta line). The gray region stands for the $P T$ -broken phases, and the white region for the $P T$ -exact phases.
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Figure 5
Phase rigidities of the states on the second band (red dots), third band (blue dots), and fourth band (dark-cyan dots) as functions of Bloch $k$ for (a) $Γ (= Γ_{a} = Γ_{b}) = 1.0$ , and (b) $Γ (= Γ_{a} = Γ_{b}) = 2.0$ . The inset shows a log-log plot of phase rigidity versus $| k - k_{0} |$ for the exceptional points $S$ (red line) and $R$ (blue line), respectively.
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Figure 6
The central panel (b) shows the trajectory of the exceptional point $S$ in $(d_{a} / Λ, Γ)$ space (red lines) for parameters $ɛ_{a} = 3.8, ɛ_{b} = 1$ , and $μ_{a} = μ_{b} = 1$ . The light gray region stands for the $P T$ -broken phase, and the blue/dark-gray regions are the $P T$ -exact phases. Typical complex band structures before and after the band inversion are shown in (a) and (c) for two points shown by white stars. The calculated Zak phase for each band is also shown. The yellow regions in (a) and (c) highlight the concerned gap. The orange arrow line in (b) shows the topological transition from blue region to dark-gray region.
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Physical Review B

covering condensed matter and materials physics

Coalescence of exceptional points and phase diagrams for one-dimensional $P T$ -symmetric photonic crystals

Kun Ding, Z. Q. Zhang, and C. T. Chan

Phys. Rev. B 92, 235310 – Published 31 December 2015

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Physical Review B

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Coalescence of exceptional points and phase diagrams for one-dimensional PT-symmetric photonic crystals

Kun Ding, Z. Q. Zhang, and C. T. Chan

Phys. Rev. B 92, 235310 – Published 31 December 2015

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Coalescence of exceptional points and phase diagrams for one-dimensional $P T$ -symmetric photonic crystals