Photonic Weyl degeneracies in magnetized plasma

Results

Hamiltonian formalism of magnetized plasma

Unlike artificial metamaterials that could possess highly non-local response caused by either non-local modes in ‘meta-atoms’ or Brillouin zone boundary, the cold magnetized plasma’s optical response can be considered to be local as long as the wavelength of interest is far greater than the mean spacing of the charged particles in the plasma. To start, we develop the Hamiltonian formalism of magnetized plasma based on a previous work by Raman et al.29 (see Supplementary Note 1for detailed derivation):

where ωp=Ne2/m is plasma frequency, ωc=eB/m is cyclotron frequency,V is the electron’s velocity field, . The damping frequency Γ can be neglected as it can be orders of magnitude less than the plasma frequency in a gaseous plasma30. Each element in the matrix is a 3-by-3 matrix, c is the speed of light in vacuum, I is unit matrix, and the 3-by-3 matrix Δ=[σy, 0; 0, 1] represents the coupling between Vx and Vyinduced by Lorentz force. In realistic cold plasma, presence of ions can also be formulated into the Hamiltonian, which results in nearly negligible effect to the dispersion close to the electron plasma frequency (Supplementary Fig. 1).

Dimension of the Hamiltonian in equation (1) is 9-by-9, which results in an overall of nine dispersion curves—four dispersion curves in the positive frequency regime, four dispersions in the negative frequency regime and one zero frequency mode. Since the dispersion curves in the positive and negative frequencies are linked to each other through the transformation: ω→−ω and [E, H, V]T→[E*, −H*, V*]T , we therefore only need to consider the dispersion curves in the positive frequency regime, which are calculated for ωc=1.2ωp and plotted in Fig. 1a. Band structures obtained in the range ωc<ωp is given in Supplementary Fig. 2.

Figure 1: Dispersion relation of a magnetized plasma.

(a) Three-dimensional energy band of the magnetized plasma for ωc=1.2ωp, ‘Weyl degeneracies’ are highlighted by the coloured spots (red for +1 and blue for −1). Each slice of this band structure at some definite kz is also a cross-section plot of an arbitrary plane across the momentum space’s centre in x–y plane. The straight line at ω=ωp, Kx=Ky=0 is the longitudinal plasma mode. The bands are numbered from bottom to top, and the fourth band is not shown here for conciseness. (b) Plot of equation (7) in one quadrant of the parameter space. The outer Weyl points (yellow) goes to infinity when ωc=ωp, while the inner Weyl points (grey) annihilate at the origin. The inset shows the energy bands of the effective Hamiltonian of the plasmon Weyl points. Intersection between the grey planes that pin a definite frequency, and the energy band is a straight line at ω=ωp, while is parabola at shifted frequencies. (c) ‘Fermi surfaces’ and Berry curvature plot when shifted frequency δωequals −0.01(grey, corresponds to the first band) and 0.01(black, corresponds to the second band), respectively. The colours indicate normalized intensity of Berry curvatures.