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Fig. 3. Experimental observation
of helicoidal structure of
topological surface states.
(A) Schematic illustration of helicoid
surface states in an ideal Weyl
system, with four Weyl points within
the surface BZ, plotted by using
the Jacobi elliptic function. The arcs
of different colors represent the
evolution of equifrequency arcs
connecting Weyl points of opposite
Chern numbers. (B) Transmitted
near-field scanning system (setup
“a”), where the source (red) is
positioned on the bottom surface
center. (C and E) Equifrequency
contour (jE z j) measured by using
setup “a” from 12.6 to 14.0 GHz.
(D and F)Bulk(blackdashed)
and surface (magenta solid)
states simulated with, correspond-
ingly. Anticlockwise (red) and
clockwise (cyan) arrows indicate
the surface arc rotation directions
with increasing frequency Downloaded from
corresponding to positive and
negative Weyl nodes, respectively.
The central solid circle indicates
the air equifrequency contour.
The plotted range for each panel is
2
[–p/a, p/a] . http://science.sciencemag.org/

nodes, as schematically shown in Fig. 3A. It is well The helicoidal structure of the surface arcs As showninFig. 3,CandD,at13.1GHz,which
on March 1, 2018
known that the gapless surface states of Weyl was probed by using the transmitted near- is below the Weyl frequency, the Fourier trans-
crystals take the form of helicoid Riemann sur- field scanning configuration, with the excitation formation of the experimentally measured field
faces (14), where the bulk Weyl points correspond source located at the center of the bottom layer distribution shows the presence of four symmet-
to the poles and zeros adopting the sign of their of the meta-crystal stack (Fig. 3B, setup “a”), rically displaced elliptical bulk states with the
respective Chern numbers. Recently, it was shown where the detecting probe can raster-scan the same size located along the diagonal directions.
that topological surface states of double Weyl top surface so as to map out both the bulk and We clearly observed two surface arcs running
systems can be analytically expressed, across the surface modes. Another configuration (fig. S4B, across the BZ boundaries and connecting the
entire BZ, as the double-periodic Weierstrass el- setup “b”), in which the excitation source is neighboring bulk states with opposite topologi-
liptic function (22). Because the Weierstrass el- positioned at the edge or corner of the top sur- cal charges. In the vicinity of the air equifrequency
liptic function has one second-order pole and face, is also used to identify the surface states. contour (circle), there exists a surface ellipse. The
one second-order zero, it is not the most fun- These two setups provide complementary in- surface ellipse joins and reroutes the surface arc
damental expression of the Weyl surface states. formation for the observation of helicoid sur- at higher frequencies (Fig. 3, E and F). Indeed,
Here, we show that our ideal-Weyl meta-crystal face states. In all near-field measurements, we the surface ellipse and surface arcs together form
of four Weyl points has surface states whose dis- set the scanning step as 1 mm (a/3), providing the same unified helicoid surface in the disper-
persion is topologically equivalent to the argu- a large surface momentum space in the range sion of the surface states.
2
ment of Jacobi elliptic function cn(z,m)oftwo of (–3p/a,3p/a) after the Fourier transforma- With increasing frequency, the top surface arc
poles and two zeros on the complex plane. cn(z,m) tion. The helicoid structure of the surface arc that emerged from the Weyl node with positive
is a meromorphic function with periods 4K(m) was experimentally measured and numerically and negative topological charge rotates anticlock-
and 4K(1 – m), where K is the complete elliptic simulated and is presented as a series of equi- wise and clockwise, respectively. The observed ro-
integrals of the first kind. For our system, the map- frequency contours between 12.6 and 14.0 GHz tation of the helicoid surface state around a Weyl
ping is given by w(k x , k y )~arg{cn[(k x – k y )/2 + (Fig. 3, C and E, in experiment, and Fig. 3, D node can therefore be used to detect the chirality
(k x + k y )i/2, 1/2]}, as plotted in Fig. 3A. and F, in simulation). of the Weyl node (23). As mentioned above, at

Yang et al., Science 359, 1013–1016 (2018) 2 March 2018 3of 4