Square-root higher-order Weyl semimetals

The work explores how square-root operations on Hamiltonians—akin to Dirac’s square-root of the Klein–Gordon equation—can generate novel topological semimetal phases. Prior square-root constructions have yielded topological insulators and higher-order topological insulators. The research question is whether a square-root operation can produce a higher-order Weyl semimetal that simultaneously exhibits surface Fermi-arc states and hinge states, and how to realize such a phase experimentally. The study proposes a square-root higher-order Weyl semimetal (SHOWS), predicts its coexistence of 2D surface arcs and 1D hinge states connecting Weyl point projections, and establishes its topological characterization via bulk polarization and edge invariants. The importance lies in extending square-root topology to gapless 3D systems and demonstrating a feasible solid-state platform using topolectrical circuits.

Square-root topology was introduced to derive nontrivial tight-binding models from parent Hamiltonians and has been realized in photonic and electrical systems. Higher-order topological insulators supporting states of codimension >1 have also been generalized to square-root settings. Weyl semimetals host Fermi-arc surface states, and higher-order Weyl semimetals were recently proposed and observed in various platforms, exhibiting both surface arcs and hinge states. Square-root topological semimetals have been theoretically discussed. This work builds on these advances by constructing a square-root higher-order Weyl semimetal, linking its properties to those of its parent lattices (honeycomb and breathing kagome) and demonstrating it experimentally in 3D topolectrical circuits.

• Theoretical model: Construct a 3D tight-binding (TB) lattice formed by vertically stacking 2D square-root HOTIs with honeycomb–kagome (HK) hybridization and double-helix interlayer couplings. The Hamiltonian includes intralayer nearest-neighbor and next-nearest-neighbor hoppings (ta, tb) and interlayer couplings (tz). In momentum space, H(k) has off-diagonal block structure with a 3×2 Φk matrix encoding tb and tz via k·a vectors (a1= (1,1,1)/3, a2= (−1,1,1)/3, a3= (0,−2,1)/3). Squaring H(k) yields analytically tractable bands: Ek = 0 and tb^2+tz^2 ± sqrt[(tb^2−tz^2)^2 + 4 tz^2 tb^2 |Δ(k)|^2], with Δ(k)=(1+e^{ik·a1}+e^{ik·a2})/3; original energies are ±√Ek. Band touchings occur at K1 = (4π/3, 0, ±kzw), with kzw = arccos[(ta tb)/(2 tz)] when ta tb < 2 tz. Time-reversal partners are at G1 = (−4π/3, 0, ±kzw). Low-energy expansion confirms linear Weyl cones. Topological charges ±1 at Weyl nodes are computed from Berry curvature, showing monopole-like singularities.
• Topological characterization: For systems with C3 symmetry, compute bulk polarization pn(k) via pn = (1/2π) arg θ(k) along lines (kx, ky)=(4π/3,0) as kz varies. For the first band, p1 = 1/3 for |kz|<|kzw| and 0 for |kz|>|kzw|, indicating a higher-order topological phase between the Weyl points. Edge topological invariants corroborate phase transitions at kz = ±kzw. Bulk–hinge correspondence is established.
• Circuit realization: Map the TB model to a 3D stacked LC circuit network (topolectrical circuit) with HK hybridization and double-helix interlayer couplings. Kirchhoff’s law I_a(ω)=∑_b J_ab(ω) V_b(ω) defines the circuit Laplacian J, which under resonance conditions reproduces the TB Hamiltonian with one-to-one correspondence of couplings: −ω0 CA ↔ ta, −ω0 CB ↔ tb, −ω0 CZ ↔ tz. Component choices: CA=0.5 nF, CB=1.0 nF, CZ=0.5 nF; LA=30 µH, LB=7.5 µH, LC=18 µH; an additional grounding inductor LG=22 µH shifts hinge modes to zero admittance to facilitate impedance probing without altering eigenmodes. Geometry: a 10-layer stacked triangular-prism structure with zigzag edges (N=2860 nodes), enabling uniform onsite admittance in the breathing kagome block.
• Measurement protocols: Theoretical spectra computed for ideal components (no loss/disorder) for Fig. 2; subsequent simulations include 2% random disorder reflecting component tolerances. Experimental hardware includes printed circuit boards connected by flexible flat cables; resonant frequency f≈755 kHz (from f=1/(2π√(3 CA LA))). Instruments: Keysight E4990A impedance analyzer, GW AFG-3022 function generator (drive vs(t)=5 sin(ωt) V), and Keysight MSOX3024A oscilloscope. Impedance Z(ω) and node voltages measured at representative hinge, surface, and bulk nodes; spatial impedance/voltage maps obtained. For hinge-state dispersion along kz, drive a hinge node and record voltage vs position z; Fourier transform v(ω,z) to extract projected dispersion. For Fermi arcs, scan frequency to target Weyl-node admittances j=±0.004082 Ω^{-1}, corresponding to f≈835 kHz and 1441 kHz; due to component-frequency dependence, detailed Fermi-arc mapping is performed at f≈835 kHz; geometry is a slab with open boundaries along y and periodic along x,z. Experimental color maps are compared with theoretical equal-admittance contours and Weyl-node projections (charges ±1).

• Existence of square-root higher-order Weyl semimetal (SHOWS): The TB model exhibits Weyl points at K1=(4π/3,0,±kzw) and their time-reversal partners, with kzw=arccos[(ta tb)/(2 tz)] (for ta tb < 2 tz). Low-energy theory shows linear crossings; Berry curvature reveals monopole singularities with topological charges ±1 at the nodes.
• Higher-order topology between Weyl points: Quantized bulk polarization for the first band p1=1/3 for |kz|<|kzw| and p1=0 otherwise, indicating a higher-order phase supporting hinge states; edge invariant analysis yields the same transition points at kz=±kzw.
• Coexisting boundary states: The SHOWS supports both 2D Fermi-arc surface states and 1D hinge (prismatic) states that connect projections of the Weyl points on side surfaces and arrises (hinges), respectively, demonstrating bulk–hinge correspondence.
• Circuit realization and observations: In 3D stacked HK LC circuits (10 layers, 2860 nodes), impedance spectra display in-gap, threefold-degenerate hinge modes (due to C3 and generalized chiral symmetries), alongside bulk/surface bands. Spatial distributions confirm hinge, surface, and bulk localization. Projected hinge-state dispersion along kz experimentally connects two Weyl points at resonant frequencies around 835–860 kHz and 1441 kHz, matching simulations.
• Fermi arcs: Numerical and experimental Fermi-arc surface dispersions observed; due to component frequency response, full experimental mapping is presented at f≈835 kHz, showing arcs that terminate at the projected Weyl points with opposite charges.
• Distinctive feature vs conventional higher-order Weyl semimetals: SHOWS exhibits Weyl pairs at both positive and negative energies arising from the square-root construction, and inherits exotic boundary modes from its parent sublattices.

The findings confirm that a square-root operation applied to appropriate parent lattices can generate a higher-order Weyl semimetal phase. The quantized bulk polarization and edge invariants identify a topological phase between Weyl nodes, where robust hinge states coexist with surface Fermi arcs. Experimental topolectrical circuits faithfully implement the TB model and reveal the predicted hinge dispersions and Fermi-arc contours, thereby validating the bulk–hinge correspondence. Compared to conventional higher-order Weyl semimetals, the SHOWS inherits boundary characteristics from its parent Hamiltonians and exhibits Weyl pairs at both positive and negative energies, illustrating a fundamental difference induced by square-root topology. The approach suggests tunability via interlayer coupling (e.g., breathing tz) to potentially realize third-order states and indicates that square-root methodologies can broaden the topological classifications and realizations of gapless systems.

This work introduces and realizes a square-root higher-order Weyl semimetal by stacking square-root HOTI layers with double-helix interlayer couplings. The model hosts Weyl nodes with opposite topological charges and supports both Fermi-arc surface states and hinge states that connect the Weyl node projections. Topological characterization via bulk polarization and edge invariants, together with topolectrical-circuit experiments (impedance and voltage mapping), demonstrate bulk–hinge correspondence and directly observe the hallmark boundary states. The square-root framework enables Weyl pairs at positive and negative energies and transfers exotic boundary modes from parent sublattices. Future directions include: tuning interlayer couplings (e.g., breathing tz) to realize third-order corner states; searching for solid-state materials supporting SHOWS; leveraging hinge/surface states for robust wave/charge transport and multiplexing; integrating topolectrical circuits in CMOS platforms; extending to higher-dimensional (e.g., 4D) and Floquet square-root phases; and generalizing to 2^n-root insulators and semimetals.

• Experimental frequency constraints: Component values vary significantly with frequency, limiting detailed measurements near f≈1441 kHz; comprehensive Fermi-arc mapping was conducted at f≈835 kHz only.
• Component tolerances and losses: Practical LC elements have ~2% tolerances and losses; simulations incorporate 2% disorder to match experiments, but residual discrepancies may persist.
• Finite-size implementation: The circuit uses a 10-layer, finite geometry (N=2860 nodes), which may introduce finite-size effects compared to the ideal infinite lattice.
• Ideal assumptions in part of theory: Some calculations (e.g., Fig. 2) assume lossless, disorder-free components, which deviate from experimental realities.