Ideal type-II Weyl points in topological circuits

Abstract Weyl points (WPs), nodal degenerate points in three-dimensional (3D) momentum space, are said to be ‘ideal’ if they are symmetry-related and well-separated, and reside at the same energy and far from nontopological bands. Although type-II WPs have unique spectral characteristics compared with type-I counterparts, ideal type-II WPs have not yet been reported because of a lack of an experimental platform with enough flexibility to produce strongly tilted dispersion bands. Here, we experimentally realize a topological circuit that hosts only topological bands with a minimal number of four ideal type-II WPs. By stacking two-dimensional (2D) layers of inductor-capacitor (LC) resonator dimers with the broken parity inversion symmetry (P), we achieve a strongly tilted band structure with two group velocities in the same direction, and topological surface states in an incomplete bandgap. Our results establish an ideal system for the further study of Weyl physics and other exotic topological phenomena.


Supplementary Information for "Ideal type-II Weyl points in
topological circuits" 1 respectively, where Ca = C + δC and Cb = C -δC are the grounded capacitors in the 2 inductor-capacitor (LC) resonators, and a is the spacing between the nearest-neighbor resonator nodes [2][3][4][5]. In the frequency domain, the above equations change to the following normalized forms where κ x = C 1 /C and ν x = C 2 /C are the hopping strengths in the x direction, κ y = C 3 /C is the hopping strength in the y direction, κ z = C 4 /C and ν z = C 5 /C are the hopping strengths in the z direction, δ C = δC/C, and 2 0 2 is the normalized frequency with 2 For a topological circuit with periodic boundaries, we assume the Bloch solutions V (x,y,z) = ψa,b exp (ikxx + ikyy + ikzz) for the white and black nodes, respectively, where ψa,b is the wavefunction. Then the corresponding Hamiltonian is We confirm that the Hamiltonian of this two-band system satisfies time-reversal symmetry (T) rather than parity (P) inversion since TH(k)T -1 = H(-k) and PH(k)P -1 ≠ H(-k). After diagonalization, the eigenvalues (normalized eigenfrequencies) are which are related to the eigenfrequencies by Eq. (S5).

Realization of ideal type-II WPs
in c, and Ca ≠ Cb, C1 ≠ C2 and C4 ≠ C5 in d.
We consider a topological circuit as shown in Fig. 1a with Ca = Cb and C1= C2 = C3 = C4 = C5. For an isolated layer in the x-y plane, there are two bands that are degenerate at the boundary of the square two-dimensional (2D) BZ and exhibit a quadratic degeneracy at the corners [6]. First, we stack identical layers along the z direction. In the Hamiltonian, both d2 and d3 are zero. The band degenerate points (BDPs) satisfy cos (kxa) + cos (kya) = 0 and they are independent on kz. The BDPs form a square tube with rotational axis along the kz direction, as shown in Fig. S1a. Second, in order to break P of the circuit, we break the mirror symmetry in the x direction Mx := x → -x by setting C1 ≠ C2, which leads to the splitting of the square degeneracy in the kx-ky plane [6,7]. In the three-dimensional (3D) BZ, the BDPs form a pair of lines along the kz direction with kx = 0 and ky being determined by the sum of C1 and C2, and they have linear dispersion in the kx and ky directions. Here we set C1 + C2 = C3, so that the degenerate lines are projected to (0, ±2/3)π/a, as shown in Fig. S1b. Third, to isolate the BDPs, we need to break the line degeneracy. This is achieved by setting C4 ≠ C5. In this case, there are two 3D degenerate points located at (0, ±2/3,0)π/a, as shown in Fig

k·p model
There are four ideal type-II WPs at (0, ±2/3, ±1/2)π/a in the first BZ. Around the type-II WPs, the circuit Hamiltonian can be reduced to the k·p model. By taking the type-II WP located at (0, 2/3, 1/2)π/a as an example, the reduced Hamiltonian is   The lossy effect of the topological circuit is mainly due to the serial resistance of inductors. If the inductors have a low Q factor (large serial resistance), resonant peaks in the transmission spectrum may merge together. Thus, to observe clear dispersion bands, it is critical to use high Q inductors. As an example, we do simulations for the case in Fig. 3d under Q = 75 @100 kHz (Rs = 8.4 Ω) and Q = 120 @100 kHz (Rs = 5.2 Ω), respectively. For simplicity, we neglect the dispersion of serial resistance. As shown in Fig. S4, the resolution of dispersion bands is improved by increasing the Q factor.
Following this guideline, it may be possible to further improve the resolution of experimental band structures by using customized inductors with ultrahigh Q factors.
Several techniques can be employed during the material selection, structure design, and fabrication process. First, Litz can be used as the inductance coil to reduce the energy loss [10]. Second, Ni-Zn ferrite can be selected as the magnetic core of inductors [11].
Finally, the shape, and the number of wraps and layers of inductance coil can be optimized.

Group velocities of the surface states
The propagation directions of the surface states are determined by the tilting direction of type-II WPs [7]. For a fixed kz, the group velocities of the surface states have the same sign. However, with a single frequency excitation in circuits, the surface states with both positive and negative kz are excited. Therefore, bidirectional propagation of the surface states was experimentally observed, as shown in Fig. S5.
Considering the periodic boundary condition in the q direction, the transmission distribution in the q direction is symmetric with respect to the input port. Here, we discuss the adjustability of our ideal type-II Weyl circuits. First, according to Eq. S2, the frequency of ideal type-II WPs is adjustable by using different inductors in the grounded LC resonators. This adjustability is challenging in other systems. Second, the group velocities near type-II WPs are easily adjustable due to the configurable circuit components. As an example, Fig. S5 shows the dependence between the coupling capacitance in the z direction and the group velocities near a type-II WP at (0, type-II WPs to type-I WPs [12]. For clarity, Fig. S5c shows that the group velocity vg,2 at the type-II WP changes in a wide range with the fixing of C4 and adjusting of C5. The above discussion implies that our topological circuits provide a clean and adjustable platform to observe ideal type-II WPs.