R4000

High-quality single crystals were grown using a Bi rich melt in the ratios YbMnBi10 and EuMnBi10. The elements were heated to 1000 °C and cooled to 400 °C at 0.1 C/min, then subsequently centrifuged to remove excess Bi. The crystals exposed mirror-like portions of the surface after the cleave in ultra-high vacuum breaking z-periodicity. The structures of the compounds were solved by single crystal X-ray diffraction (Supplementary Table 3).
ARPES measurements were performed at the I05 beamline of Diamond Light Source, UK. Single crystal samples were cleaved in situ at a pressure lower than 2 × 10⁻¹⁰ mbar and measured at temperatures about 7 K. Measurements were performed using (s,p)-polarized synchrotron light from 18 to 100 eV and employing Scienta R4000 hemispherical electron energy analyzer with an angular resolution of 0.2–0.5° and an energy resolution of 3–20 meV^{39 (link)}.
Magneto-optical imaging of magnetic domains at the surface of the YbMnBi₂ crystal was performed in a magneto-optical Kerr microscope adjusted for the Voigt effect at perpendicular light incidence at room temperature. White, linearly polarized light from a LED lamp was utilized.

The angle and energy resolutions of the ARPES measurements (Scienta R4000) are set as 0.2^o and ∼10 meV. The photon energy-dependent ARPES experiments were performed with photon energy varying from 70 eV to 115 eV. The kz was calculated by . The inner potential V₀=12 eV. Unless otherwise stated, we used photon energy at 100 eV for normal ARPES measurements. The 2D and 3D carrier density can be calculated from the Luttinger area of the FSs. and . From the Ti³⁺ ratio, the carrier density can also be calculated by n_2D=ratio × a⁻² and n_3D=ratio × a⁻³ with the lattice constant α=3.9 Å.

ARPES measurements (T = 10–160 K, $h\nu$ = 45–85 eV) of in-situ cleaved single-crystal samples were performed using a Scienta R4000 hemispherical analyser at beamline 10.0.1 of the Advanced Light Source with an energy resolution between 8 and 35 meV, and an angular resolution of $0.{35}^{\circ }$ .

The ARPES experiments were performed on the Bi₂O₂Se bulk crystals at beamline I05 of the Diamond Light Source. The data were recorded using a Scienta R4000 analyzer with the total convolved energy and angle resolutions of 20 meV and 0.2°, respectively. During the experiment, the sample was maintained in the UHV system under a pressure better than 1 × 10⁻¹⁰ torr, and the sample temperature was kept at 10 K. A fresh surface of Bi₂O₂Se single crystal for the ARPES measurement was obtained by cleaving the sample in situ along its natural (001) cleavage plane.

ARPES measurements were performed at beamline 10.0.1 of the Advanced Light Source (ALS) at Lawrence Berkeley National Laboratory. The measurement pressure was kept below 3 × 10⁻¹¹ Torr at all times. The data was recorded using a Scienta R4000 analyser at variable sample temperatures. The total convolved energy and angle resolutions were 16 meV and 0.2°, respectively. The fresh surface for ARPES measurement was obtained by cleaving the thin film sample in situ as described in ref. 28 .