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The search for artificial structure with tunable topological properties is an interesting research direction of today’s topological physics. Here, we introduce a scheme to realize topological nodal states with a three-dimensional periodic inductor-capacitor (LC) circuit lattice, where the topological nodal line state and Weyl state can be achieved by tuning the parameters of inductors and capacitors. A tight-binding-like model is derived to analyze the topological properties of the LC circuit lattice. The key characters of the topological states, such as the drumhead-like surface bands for nodal line state and the Fermi arc-like surface bands for Weyl state, are found in these systems. We also show that the Weyl points are stable with the fabrication errors of electric devices.

Recently, there is great interest in realizing topological states in various platforms. Topological states, including the quantum Hall states, quantum spin Hall states, Dirac states, Weyl states, and nodal line states, have achieved significant progresses in electronic materials [

From symmetry considerations, three types of nodal line states have been proposed in a great number of literatures [

The design scheme of realizing the nodal line state and Weyl state in LC circuit lattice is shown below. We consider a honeycomb lattice consisting of capacitors and inductors in

Schematic setup of the three-dimensional LC circuit lattice. (a) Honeycomb layers consisting of inductors and capacitors stack along

Here, we study the resonance condition of the circuit lattice, where a nonzero distribution of potential satisfies Kirchhoff’s law. We follow the method given in Ref. [

The closed loops of bands crossing points can be protected by the coexistence of space inversion symmetry and time-reversal symmetry [

(a) Nodal line (in red) in the BZ and its projection on the (001), (010), and (100) planes (in grey). The parameters are set as

The topological properties of a nodal line can be inferred from the winding number

Starting with nodal line states, Dirac states or Weyl states are possible to emerge by introducing symmetry breaking terms [

(a) Four Weyl points in the Brillouin zone and their projections on (001), (010), and (100) direction.

As a result of the nonzero Chern numbers, topologically protected gapless chiral surface states emerge in the band gap away from the Weyl points. An example of the nontrivial surface band dispersions is shown in Figure

In the above discussions, the inductance and capacitance in the circuit lattice are proposed as a set of precise values, but the fabrication error of the electronic devices brings about certain range of tolerance values. In this paragraph, we discuss whether the fabrication errors influence the results given above. For the nodal line state, maintaining intrinsic space inversion symmetry is a necessary condition. When the fabrication errors are taken into consideration, intrinsic space inversion symmetry is too demanding to preserve; therefore nodal line becomes unstable. However, the existence of the Weyl points does not require extra symmetries except the discrete translation invariant symmetry. It is expected to be more stable than the nodal line state under the perturbation of the fabrication errors. In order to investigate the stability of the Weyl points in the circuit lattice, we employ a

(a) The band gap,

To summarize, we report that the topological nodal line state and Weyl state can be realized in a three-dimensional classical circuit system. We derived a two-by-two tight-binding-like model to investigate its topological nature. Based on this model, we show that the nodal line structure is protected by the inversion symmetry, which can be achieved by setting

In this section, we present the details of the methods used to calculate the band structure of the circuit lattice. We start with deriving the tight-binding-like model given in (

For the single layer honeycomb circuit lattice given in Figure

The authors declare that they have no financial conflicts of interest.

Rui Yu conceived the research. Rui Yu, Kaifa Luo, and Hongming Weng performed the theoretical analysis and the calculations. All authors contributed to the manuscript writing.

The authors thank Hua Jiang, Yuanyuan Zhao, and Ang Cao for their very helpful discussions. This work was supported by the National Key Research and Development Program of China (nos. 2017YFA0303402, 2017YFA0304700, and 2016YFA0300600) and the National Natural Science Foundation of China (nos. 11674077, 11422428, 11674369, and 11404024). Rui Yu acknowledges funding from the National Thousand Young Talents Program. The numerical calculations in this work have been done on the supercomputing system in the Supercomputing Center of Wuhan University.