Probing the Berry curvature and Fermi arcs of a Weyl circuit

The Weyl particle is the massless fermionic cousin of the photon. While no fundamental Weyl particles have been identified, they arise in condensed matter and metamaterial systems, where their spinor nature imposes topological constraints on low-energy dispersion and surface properties. Here we demonstrate a topological circuit with Weyl dispersion at low momentum, realizing a 3D lattice that behaves as a half-flux Hofstadter model in all principal planes. The circuit platform provides access to the complete complex-valued spin texture of all bulk and surface states, thereby revealing not only the presence of Weyl points and the Fermi arcs that connect their surface projections, but also the Berry curvature distribution through the Brillouin zone and the associated quantized chiral charge of the Weyl points. This work opens a path to exploration of interacting Weyl physics in superconducting circuits, as well as studies of how manifold topology impacts band topology in three dimensions.

Figure 1

A Weyl circuit. (a) Topology of the tunneling connectivity of a minimal lattice model exhibiting Weyl points. The unit cell consists of two sites, with A sites shown in blue and B sites shown in orange. Zero-phase tunneling is represented as a cyan solid line and $\pi$-phase tunneling as red dashed line. The lattice vectors are $\widehat{z}$, $\widehat{u}\equiv \frac{1}{\sqrt{2}}(\widehat{x}+\widehat{y})$, and $\widehat{v}\equiv \frac{1}{\sqrt{2}}(\widehat{x}-\widehat{y})$. (b) To realize such a lattice with circuit components, all lattice sites are replaced by inductors, and all tunnel couplings by a pair of capacitors. A zero-phase tunnel coupling capacitively connects the positive end of one inductor to the positive end of its neighbor, and the negative end to the negative end of the neighbor. A $\pi$-phase tunnel connection capacitively couples the positive end of an inductor to the negative end of its neighbor, and vice versa. (c) Shows numerically calculated band structure at $k_y=0$; apparent are four Weyl points in the first Brillouin zone.

Figure 2

Measured band structure of the Weyl circuit. In a configuration with periodic boundary conditions physically imposed along all three axes, it is possible to fully reconstruct the bulk band structure by exciting a single site and measuring the complex response at all sites as a function of frequency. A 3D spatial Fourier transform of the resulting response yields the dispersion shown in (a) in the vicinity of the Weyl point at $\mathit{k}=(\pi /2,0,\pi /2)$. The response below (red) and above (purple) the Weyl energy forms near-spherical shells in k space, made slightly square by the finite ($8\times 8\times 8$) system size. These shells form a conical (linear) dispersion around the Weyl point highlighted by the (red to purple) envelope surrounding the data. At the energy of the Weyl point (green), the response is localized exclusively to the momentum of the Weyl point. The equienergy surfaces are plotted over the full Brillouin zone in (b), for the band above the Weyl frequency [color scale same as (a)]. The equienergy surfaces are aspheric at frequencies far from the Weyl points, a generic result arising from the eventual merger of the Weyl cones into a single band. Also plotted (arrows) is the reconstructed spin texture of the upper band Bloch functions near the Weyl points; at $\mathit{k}=(\pi /2,0,\pi /2)$ the spin points everywhere inwardly—a radial hedgehog defect, while the other three Weyl points exhibit hyperbolic hedgehog defects, from which we can deduce the chirality of the Weyl points: $\chi =-\text{sgn}\left(k_x{k}_z\right)$. In (c) we plot a slice of the measured dispersion in the $k_v-k_u$ plane at the $k_z=\pi /2$ of two Weyl points. In (d) we plot a slice in the $k_x-k_z$ plane at the $k_y=0$ of all Weyl points. Also shown, as arrows, are the projections of all the spin texture into the planes.

Figure 3

Measured Berry curvature, Berry flux, and surface states. (a) shows the measured and interpolated Berry curvature (arrows), and associated Berry flux (blue$\to$ orange is negative $\to$ positive), over the full Brillouin zone, for the lower band. Divergence of the Berry curvature is strictly zero except at four topological defects located at the Weyl points, carrying $\pm 1$ chiral charge, with $+1$ corresponding to sources of Berry flux (orange), and $-1$ to sinks (blue), evident from the flow of the Berry curvature. Slices along $k_v-k_u$ and $k_x-k_z$ are shown in (b) and (c), respectively, highlighting the structure of this flow. (d) and (e) show the measured surface states for even and odd numbers of layers in the $v$ direction, respectively. States residing on the top surface are depicted in shades of red proportional to excitation amplitude, while those on the bottom surface are depicted in shades of blue. Only selected frequency planes are shown here and within each plane is a $8\times 8$ pixels surface Brillouin zone, limited by the system size as also shown in the Supplemental Material. It is apparent that, in the $\nu =f_{\mathrm{Weyl}}=290\text{kHz}$ plane, the surface projection of the Weyl points (outlined in dashed white squares) are connected by lines (“Fermi arcs”) of surface states; for an even number of layers, the Fermi arcs are within the Brillouin zone and overlap each other on both top and bottom surfaces, while for an odd number of layers the Fermi arc on the top surface (red) passes around the Brillouin zone, exploiting its toroidal topology—a consequence of the gauge difference between even and odd layers (see Supplemental Material for details [39]).