Bulk-boundary correspondence is a defining principle of topological materials. In Weyl semimetals, it is visualized by surface Fermi arcs connecting projections of bulk Weyl points with opposite chirality. Most studies focus on high-symmetry facets, where the projected bulk and surface Brillouin-zone periodicities are naturally commensurate and a single bulk-zone projection is sufficient. On low-symmetry facets, however, these periodicities may not match within a conventional first-zone construction. Weyl-point projections and surface Fermi arcs can therefore appear to repeat differently, raising a central question: does bulk-boundary correspondence fail, or is the conventional projection incomplete?

Recently, Dr. Cong Li at the Institute of Physics, Chinese Academy of Sciences, together with Dr. Zhilong Yang at the University of Science and Technology Beijing, Profs. Youguo Shi, Quansheng Wu and Tao Xiang at the Institute of Physics, and Prof. Oscar Tjernberg's team at KTH Royal Institute of Technology, investigated the low-symmetry (103) surface of NdAlSi. Oriented single crystals were precision-cut, laser pre-cut and cleaved in situ to expose well-defined terraces. High-resolution ARPES at Diamond Light Source and MAX IV was combined with density functional theory and surface Green's-function calculations.

ARPES revealed a strong dependence on local surface quality. Disorder suppresses coherent surface states and exposes bulk-projected bands, whereas flat terraces display sharp surface states and an additional, shorter-period replica modulation (Fig. 1). Photon-energy scans from 125 to 195 eV showed that bulk Fermi-surface slices translate systematically as the momentum perpendicular to (103) changes, in close agreement with theory (Fig. 2). Successive bulk Brillouin zones therefore project as laterally phase-shifted replicas, so a first-zone analysis is incomplete.

The team resolved this mismatch with a multi-Brillouin-zone projection framework governed by a least-common-multiple (LCM) rule. For NdAlSi (103), adjacent projections carry a 2π/3 phase offset and close only after three steps (Nmin = 3), equivalently requiring a three-cell surface supercell. The resulting in-plane repeat period is one third of the conventional surface Brillouin-zone period, matching ARPES and producing a momentum-space Moiré-like interference pattern (Fig. 3). The mismatch is thus caused by incomplete projection rather than a breakdown of bulk-boundary correspondence.

Crucially, this process does more than fold the spectrum. Fermi arcs from phase-shifted projections overlap and hybridize, reconstructing the boundary states and producing replica-modulated, loop-like connectivity absent on high-symmetry facets. LCM-guided calculations reproduce the reduced periodicity and dominant replicas (Fig. 4), while the detailed arc-to-loop connectivity remains sensitive to termination mixing, reconstruction or disorder, and surface-bulk hybridization. This separates the universal commensuration constraint from facet-specific boundary physics.

This work resolves the apparent bulk-boundary correspondence paradox on low-symmetry Weyl surfaces and provides an operational route for modeling complex boundary spectra. The study, entitled "Topological Fermi Arc Interference on Low-Symmetry Weyl Surfaces", was published online in Nature Communications on 13 August 2026. Cong Li and Zhilong Yang contributed equally and are co-corresponding authors; Oscar Tjernberg is also a corresponding author.

The study was supported by the Swedish Research Council, the Knut and Alice Wallenberg Foundation, the National Natural Science Foundation of China, the Informatization Plan of the Chinese Academy of Sciences, the National Key R&D Program of China, Vinnova, and Formas.

Fig. 1. Preparation of the low-symmetry NdAlSi (103) surface, bulk-to-surface projection geometry, and ARPES Fermi-surface maps.

Fig. 2. Photon-energy-dependent bulk electronic structure of NdAlSi (103) and comparison with DFT calculations.

Fig. 3. Resolution of the apparent bulk-boundary correspondence paradox through multi-Brillouin-zone projection and the least-common-multiple criterion.

Fig. 4. Comparison of surface states, surface bulk-projected states, and least-common-multiple-guided calculations for the NdAlSi (103) surface.

Article:

Nature Communications, published online 13 August 2026
DOI: https://doi.org/10.1038/s41467-026-76639-5

Contact:

Cong Li
Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences
Email: licong@iphy.ac.cn

Keywords:

Weyl semimetal, low-symmetry surface, Fermi arc, bulk-boundary correspondence, ARPES, multi-Brillouin-zone projection, LCM commensuration

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 may not coincide within a conventional first bulk Brillouin zone projection. Here we study the (103) surface of the Weyl semimetal NdAlSi and identify Fermi arc interference in the boundary spectrum. Angle-resolved photoemission spectroscopy uncovers loop-like Fermi-arc connectivity and characteristic replica modulations that are not observed on high symmetry surfaces. Crucially, the topological surface states themselves are reconstructed because Fermi arcs from phase-shifted bulk-zone projections overlap and hybridize, producing connectivity patterns unique to low-symmetry facets. We show that these emerge from incomplete bulk projection and multi-cell interference governed by a least-common-multiple framework. Least-common-multiple guided density functional theory and Green's-function calculations reproduce the reconstructed periodicity and dominant replica structure in the spectra, providing a broadly applicable commensuration guideline. These findings resolve the apparent bulk-boundary correspondence paradox on low-symmetry surfaces and provide an operational route to model and interpret boundary spectra on complex facets.