Hidden Moiré Topology of Low-Symmetry Weyl Surfaces

Abstract

Topological materials are defined by the correspondence between bulk topology and boundary states, yet this correspondence becomes enigmatic on low-symmetry surfaces where bulk and surface periodicities are inherently mismatched. Here we reveal a hidden moiré topology emerging on the (103) surface of the Weyl semimetal NdAlSi. Angle-resolved photoemission spectroscopy uncovers closed Fermi-arc loops and momentum-space moiré modulations, phenomena unanticipated in conventional topological theory. We show that these emerge from incomplete bulk projection and multi-cell interference governed by a least-common-multiple framework. Least-common-multiple guided DFT and Green's-function calculations quantitatively reproduce the observed spectra, establishing the universality of this commensuration rule. These findings transform a long-standing paradox of bulk-boundary correspondence into a new paradigm of momentum-space moiré reconstruction, bridging crystalline and quasicrystalline topologies and opening routes to flat-band engineering on complex surfaces.

Figures (4)

• Figure 1: Crystal structure and electronic projections of NdAlSi (103) surface. (a) Crystal structure of NdAlSi with the (103) surface cut. The differently colored cuts represent the potential cleavage planes. The rightmost panel shows one possible cleavage plane. (b) BZ construction and projection onto the (103) surface. The red (blue) dots corresponding to Weyl points with chirality $+1$ ($-1$). (c) Periodicity mismatch between the SBZ and the first bulk BZ projection. The red (blue) dots corresponding to Weyl points with chirality $+1$ ($-1$). (d) XRD confirming the exposed (103) surface. (e, f) Fermi surfaces measured at 150 eV on uneven (disordered) (e) and flat (well defined) (f) regions of the (103) surface. Purple lines denote the (103) theoretical SBZ, whose size is determined by the (103) cross section of half a primitive unit cell. Since the body-centered lattice halves the real-space periodicity along the [001] direction, the corresponding BZ along $k_{z}$ is doubled. For consistency, the (103) SBZ is likewise defined by the cross section of half a primitive unit cell (see Section 4 of the Supplementary Materials for the details). Green lines denote the experimentally observed periodicity extracted from (f); pink lines denote the first bulk BZ projected onto the (103) surface.
• Figure 2: Bulk electronic structures of the (103) surface. (a) DFT calculated 3D Fermi surface of NdAlSi. (b) The (103) surface and its perpendicular cross-section in momentum space are expected to be probed with photon energies ranging from approximately 125 eV to 205 eV. (c) DFT calculated bulk Fermi surface of the momentum-space slices along the [103] direction ($k_{\perp}$). (d) DFT-calculated bulk Fermi surface of the plane [cyan plane in (b)] perpendicular to the (103) surface [purple plane in (b)]. (e) Photon-energy dependent ARPES spectral intensity map at the Fermi level along the $k_{y}$ direction on a relatively strong disordered (103) surface, where both surface states and SBPSs are nearly suppressed. (f-i, n-q) Photon energy dependent bulk Fermi surface mappings (125-195 eV) revealing $k_{\perp}$ dispersion. (j-m, r-u) Corresponding DFT calculations. Each slice produces a shifted replica of the bulk Fermi surface within the SBZ, reflecting the intrinsic periodicity of bulk states along the [103] direction; the green, orange, and pink boxes highlight features that are shifted as $k_{\perp}$ changes. The features do not appear strictly identical here because only discrete $k$-slices are shown, but denser sampling would reveal exact repetitions.
• Figure 3: Resolving the BBC paradox on the (103) surface. (a) Projection of Weyl points onto the (103) surface considering only the first bulk BZ, showing a mismatch with the surface periodicity. (b-c) Inclusion of the second-order (b) and third-order (c) bulk BZs along $k_{z}$ progressively restores the periodicity and brings the bulk projection into agreement with the surface states. Red (blue) dots denote Weyl points of chirality $+1$ ($-1$). (d-e) Comparison between the bulk projection Weyl points and the measured surface states demonstrates that multiple BZs are required to recover the correct periodicity. (e) is the magnified view of (d). The red and cyan lines represent surface Fermi arcs, which hybridize to form SFALs as indicated by the black arrows. (f-g) General scheme for different surfaces: for the (001) surface (f), the first bulk BZ is sufficient, whereas for the (103) surface (g), more bulk BZ projections are need to generate a superlattice periodicity consistent with the SBZ. The Fermi surface in (g) is an enlarged view of the red dashed boxed region in (d). Purple lines denote the (103) theoretical SB; green lines denote the newly formed (103) SBZ, corresponding to the SBZ observed in experiment; pink lines denote the first bulk BZ projected onto the (103) surface. The connection of the surface Fermi arcs in panel (g) is not the actual connectivity, but rather a schematic illustration indicating that Fermi arcs from different bulk BZ may hybridize upon projection onto the (103) surface (see Section 6 of the Supplementary Materials for details).
• Figure 4: Comparison of experimental and DFT results for surface states and SBPSs on NdAlSi (103) surface. (a) Fermi surface of the NdAlSi (103) surface measured on a flat region with the photon energy of 41 eV, revealing surface states. (b) Surface projected DFT calculation of the (103) Fermi surface for a Si atom terminated surface cleavage at the Si-Al layer (see Section 5 in the Supplementary Materials for details). (c) Impurity-induced disorder filters out surface statesCLi_PNAS2025_OTjernberg, exposing SBPS. (d) DFT calculation of the corresponding SBPSs on the (103) surface. Green lines indicate the experimental SBZ, consistent with Fig. \ref{['1']}e. (e-f) Band dispersions measured along momentum Cut1 [red line in (a), (e)] and Cut2 [orange line in (a), (f)]. (g-h) Surface projected DFT band structures for the Si-terminated (103) surface along momentum Cut1 (g) and Cut2 (h). (i-j) Band dispersions along the same cuts measured on the disordered surface, where comparison with calculations (k-l) confirms the assignment of surface states and SBPSs.