Square-root higher-order Weyl semimetals

Article https://doi.org/10.1038/s41467-022-33306-9 Square-root higher-order Weyl semimetals Received: 7 February 2022 Lingling Song 1 , Huanhuan Yang 1 , Yunshan Cao1 & Peng Yan 1 Accepted: 5 September 2022 1234567890():,; 1234567890():,; Check for updates The mathematical foundation of quantum mechanics is built on linear algebra, while the application of nonlinear operators can lead to outstanding discoveries under some circumstances, such as the prediction of positron, a direct outcome of the Dirac equation which stems from the square-root of the Klein-Gordon equation. In this article, we propose a model of square-root higher-order Weyl semimetal (SHOWS) by inheriting features from its parent Hamiltonians. It is found that the SHOWS hosts both “Fermi-arc” surface and hinge states that respectively connect the projection of the Weyl points on the side surface and arris. We theoretically construct and experimentally observe the exotic SHOWS state in three-dimensional (3D) stacked electric circuits with honeycomb-kagome hybridizations and double-helix interlayer couplings. Our results open the door for realizing the square-root topology in 3D solid-state platforms. Nearly all the operators encountered in quantum mechanics are linear (or antilinear) operators, such as the rotation, translation, parity, time reversal, etc, which allows us to construct the mathematical basis of quantum mechanics formulated on linear algebra. Square-root operator is one of the few exceptions. Historically, Paul Dirac derived the Dirac equation through a square-root operation on the Klein-Gordon (KG) equation to describe all spin-21 massive particles that inherit the Lorentz-covariance of the parent KG equation1–3. The approach has inspired Arkinstall et al.4 to propose the concept of square-root topological insulator (TI) by taking the nontrivial squareroot of a tight-binding (TB) Hamiltonian in periodic lattices. The most appealing feature of square-root TI is that it inherits the nontrivial nature of Bloch wave function from its parent Hamiltonian. The square-root TI was subsequently observed in a photonic cage5. Recently, the square-root operation has been applied to higher-order topological insulators (HOTIs) that allow topologically robust edge states with codimension larger than one6–16. Besides the gapped solution, e.g., the electron-positron pair, the Dirac equation allows another crucial gapless or massless solution called Weyl fermion17 that plays an important role in quantum field theory and the standard Model. Although not yet observed among elementary particles, Weyl fermions are shown to exist as collective excitations in Weyl semimetals18–21. For the conventional Weyl semimetal, the three-dimensional (3D) topology features two-dimensional (2D) gapped surface states that connect the projection of Weyl points on the surface18–21. Very recently, higher-order Weyl semimetal was reported which supports both the 2D surface Fermi arcs and the one-dimensional (1D) hinge state22–29. It is thus intriguing to ask if the square-root operation can apply to semimetals30 or higher-order semimetals, and particularly how to realize these exotic states in experiments. In this article, we propose a TB model of the square-root higherorder Weyl semimetal (SHOWS) by a vertical stacking of 2D squareroot HOTIs with interlayer couplings in a double-helix fashion. It is found that the SHOWS hosts both 2D surface arc states and 1D hinge states with the topological feature being fully characterized by the quantized bulk polarization or edge invariant. We construct the TB model in stacked honeycomb-kagome (HK) hybridized inductorcapacitor (LC) circuit networks. By performing both the impedance and voltage measurements, we identify the fingerprint of the SHOWS by directly observing the Weyl points, the “Fermi arc” surface states, and the hinge states. It is revealed that both the surface states and the hinge states ideally connect the projections of the Weyl points on side surface and arris respectively, consistent with theoretical calculations. Results Model Figure 1a shows the lattice structure of the proposed model, the square of which can be viewed as the direct sum of a stacked honeycomb and a breathing kagome lattices (Fig. 1b and the analysis in 1 School of Electronic Science and Engineering and State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and e-mail: Technology of China, Chengdu 610054, China. Nature Communications | (2022)13:5601 1 Article https://doi.org/10.1038/s41467-022-33306-9 E D 41 5 2 3 2 2 3 z y 1 5 4 x G F H K+′ G+ G+ ′ G+″ ″ K+ K′ G′ K′— K— Fig. 1 | 3D stacked HK TB model. a Illustration of an infinite 3D stacked HK TB model. The unit cell including five nodes is represented by the dashed black rhombus. The intralayer hoppings are ta (green) and tb (blue) in the x − y plane, whereas the interlayer double-helix hopping is tz (brown). Insets: Right view of the cell. b The equivalence between the squared Hamiltonian of the HK circuit and its Supplementary Note 1). The TB Hamiltonian is given by aym cn + aym d n + aym en   X y y y bm c n + bm d n + bm e n + tb hm,ni ð1Þ  X y y y bm cn + bm d n + bm en + H:c:, + tz where a† (a), b† (b), c† (c), d† (d), and e† (e) are the creation (annihilation) operators on the site 1-5, respectively, 〈m, n〉 and 〈〈m, n〉〉 label the nearest-neighbor and next-nearest-neighbor coupling, respectively, and ta, tb, and tz are the hopping parameters. H.c. represents the Hermitian conjugate. In Fig. 1a, the nearest-neighbor sites of node 1(2) are nodes 3,4,5 with ta, tb being the intralayer hopping parameters. The next-nearest-neighbor sites of node 2 are nodes 2, 3 and 4 in the adjacent layer with tz being the interlayer hopping parameter. Without loss of generality, we assume all hopping paramaters are positive. In momentum space, the Hamiltonian can be expressed as H= Φyk O3,3 ! , ð2Þ where O2,2 and O3,3 are the 2 × 2 and 3 × 3 zero matrix, respectively, and Φk is the 3 × 2 matrix 0 ta B Φk = @ t a ta 1 t b + 2t z cosðk  a3 Þ C ½t b + 2t z cosðk  a3 Þeika1 A: ½t b + 2t z cosðk  a3 Þeika2 Nature Communications | (2022)13:5601 parents. c The bulk dispersion along the kz direction with (kx, ky) = (4π/3, 0). The dashed blue line indicates the position of the degenerate points. d The first Brillouin zone and the distribution of the Weyl points. The hallow and solid circles represent the Weyl point with the charge ± 1. e Bulk polarization p1 as a function of kz. For TB calculations in c and e, we set ta = 0.5, tb = 1, and tz = 0.5. pffiffi Herepffiffik = (kx, ky, kz) is the wave vector, and a1 = 21 x^ + 23 ^y, a2 =  1^ 3^ ^ 2 x + 2 y and a3 = z are three basic vectors. By taking the square of the original Hamiltonian (2), we can conveniently obtain the dispersion relation of ½H2 (see Supplementary Note 1) E k = 0 and hhm,nii O2,2 Φk G— G″— (...truncated)