Sound of Fermi arcs: a linearly dispersing gapless surface plasmon mode in undoped Weyl semimetals

Using Green's function of semi-infinite Weyl semimetals, we present the quantum theory of the collective charge dynamics of Fermi arcs. We find that the Fermi arc plasmons in undoped Weyl semimetals are linearly dispersing gapless plasmon modes. The gaplessness comes from proper consideration of the deep penetration of surface states near the end of the Fermi arcs into the interior of the Weyl semimetal. The linear dispersion—rather than square root dispersion of pure 2D electron systems with extended Fermi surface—arises from the strong anisotropy introduced by the Fermi arc itself, due to which the continuum of surface particle-hole (PH) excitations in this system will have a strong resemblance to a stack of one-dimensional (1D) electron systems. This places the Fermi arc electron liquid in between the 1D and 2D electron liquids. Contamination with the bulk PH excitations in the doped case gives rise to the damping of the Fermi arc sounds.

Figure 1

PHC in (a) 1D and (b) 2D electron gases. (c) PHC of a momentum space stack of 1D systems. The scale $2k_F$ is separation of left and right movers in momentum space. (d) PHC of Fermi arc. Panel (d) coincides with low-$q$ part of (c).

Figure 2

(a) Schematic representation of the semi-infinite WSM. Color intensity encodes the exponential decay of the surface states. The layers represent the discretization of the $z$ coordinate used in numerical computation. (b) The Fermi arc on the surface of a Weyl semimetal that connects the projection of two Weyl nodes on the surface. The gray curve denotes the penetration depth of the Fermi arc states into the interior of the WSM.

Figure 3

Finite size scaling analysis of the relative slope difference ${\gamma }_N$ (left) and the gap ${\mathrm{\Delta }}_N$ (right) of the lowest Fermi arc plasmon modes. The log-log plot indicates that upon increasing $N$ the gap and slope differences quickly vanish.

Figure 4

Lowest surface plasmon mode of Fermi arc states for a grid of $N=50$ layers with penetration depth of $db=25$. The polar angle is fixed by $\phi =\pi /6$. The dashed line emphasizes the extrapolation to $q\to 0$ limit.

Figure 5

The lowest surface plasmon frequency (dashed blue) and $sin\phi$ function (solid red) as a function of the polar angle $\phi$ for $N=50$ and $db=25$ for a fixed $q=0.01b$ wave vector magnitude.

Figure 6

The electrostatic potential profile for spin $\uparrow$ and $\downarrow$ (degenerate) Fermi arc electrons as a function of distance $zb$ from the surface for $\phi =\pi /6, \left|q\right|=0.1$, and $N=150$. The modes are localized on the surface.

Figure 7

The red plot is the real part of the eigenvalues $\varepsilon \left(\omega \right)$ of the dielectric matrix. The blue plot is the imaginary part of the loss function ${\varepsilon }^{-1}\left(\omega \right)$ as a function of frequency $\omega$ for a mesh with $N=50$ layers and the horizontal momentum $\mathit{q}=0.1$.