ALT Scientific illustration of the signature of the fractal Hofstadter butterfly in the measured electrical conductance of a graphene-hexagonal boron nitride heterostructure.

ALT A two-by-two grid of illustrations that demonstrate the difference between the quantum metric and Berry curvature. Each panel maps a two-dimensional, two-level system onto the Bloch sphere, with the Bloch sphere on the right side, and two mirrored, rounded cones representing the two-level system on the left side. The left panels show a visual representation of how the quantum metric measures the distance between two states on the Bloch sphere, and the right panels show a visual representation of Berry curvature corresponding to a mapped area on the sphere. The top two panels are atomic insulators, and their cones are separated by a small distance; the bottom two panels are topological insulators, and their cones slightly overlap.

ALT A schematic demonstrates an unconventional superconductor robust against disorder. Two purple spheres represent Cooper pair electrons scattering off an impurity represented by a black sphere on a square lattice. The lattice is separated into blue and orange regions, which are conventional and sign-changing, respectively. Arrows representing the scattering are smaller in the orange region than the blue region.

ALT A schematic of the shot-noise measurement shows a DC current injected along the edge of a circuit, propagating through various circuit components before current fluctuations are measured at an amplifier contact.

ALT The phase diagram of a Haldane chain with temperature on the Y-axis and magnetic field on the X-axis. A flattened, upside-down parabola covers the Bose-Einstein condensate phase — the low-temperature area between two critical magnetic fields — the border of which marks a phase transition. Another parabolic shape starts and ends at the same spots on the magnetic field scale, but goes slightly higher in temperature, indicating the Tomonaga-Luttinger liquid phase. A “V” shape originating at the two critical magnetic field points and bordering the Tomonaga-Luttinger liquid phase shows quantum-critical regions. At high temperatures, the system is paramagnetic.

ALT A 3D rendering of three stacked quantum Hall layers linked by a coherent anyon-exciton composite, with fractionalized quasiparticles following a curved path beside the layers.

ALT A purple displacement vector points from an electron to a hole at the center of a hexagonal lattice, and blue vector indicates the particles' upward momentum. Red arrows at each of the four corners of the image show the Berry curvature.

ALT Diagram of a quantum dot network: orange spheres represent quantum dot spins in optical cavities, connected by dashed lines showing spin-spin entanglement, generating a 3D lattice of entangled photons (blue spheres) linked by wavy lines representing photonic entanglement links.

ALT A graph showing the electronic states at a material’s surface. The x-axis shows momentum along a high-symmetry path on the surface, and the y-axis is labeled E/J. One pair of red and blue lines (with colors depicting spectral weight in the spin-up and down channels, respectively) wiggles across the upper portion near a dashed yellow line that goes straight across at 0.8 E/J, and another pair wiggles across the lower portion roughly centered on -1 E/J, showing surface spin splitting.

ALT At the top, panel (a) displays a schematic diagram of the THz emission spectroscopy setup. On the left are two coiled red and blue arrows, which represent LCP and RCP, respectively. In the middle is a rectangular solid split into three vertical material layers: PtTe2 (red, left), Cr5Te6 (blue, middle), and (0001)-AI2O3 (grey, right). At the right are overlapping blue and red wave-like lines, which represent the THz emission. Panel (b) shows a molecular, structural zoomed-in view of the heterostructure interface between the PtTe2 and Cr5Te6 layers in panel (a). In this model, Pt atoms are orange, Te atoms are blue, Cr atoms are yellow, and intercalated Cr atoms are red.

ALT A schematic shows altermagnetism-induced Bogoliubov Fermi surfaces for topological superconductivity with crystal anisotropy and quantum confinement as control knobs. Panel (a) shows the anisotropic surface states in an altermagnetic topological insulator (AMTI). A cube has lateral faces with green and orange double cones and a top face with overlapping red and blue ovals. Above is a 3D plot with energy Ek versus kx and ky. Panel (b) shows a quasi-1D triangular prism nanowire of the AMTI-superconductor heterostructure hosting a pair of Majorana zero modes, 𝛄1 and 𝛄2, at either end. An inset shows energy levels driven by altermagnetic order strength 𝑡𝐴𝑀 inside the superconducting gap Δ. Panel (c) shows an AMTI-superconductor hybrid structure with a vortex line hosting Majorana zero modes at the vortex core or side surfaces. An inset shows the energy spectrum with the evolution of Bogoliubov Fermi surfaces to discrete surface energy levels as a function of chemical potential 𝜇.

ALT False-color scanning tunneling conductance map showing a triangular lattice of bright yellow spots (superconducting vortices) against a dark blue background, with an inset in the top right showing a circular Fourier transform pattern with a bright central spot and faint surrounding ring.

ALT Panel (a) displays a series of three graphs plotting electronic densities of the six orbital (c1, c2, c3, c4, f1, f2). The charts correspond to ρ1(r) = ρ2(r) (left), ρ3(r) = ρ4(r) (middle), and ρf,1(r) = ρf, 2(r) (right). A light hexagon depicts the unit cell centered at the AA site, with its edges corresponding to the BA and AB sites. Each region of the chart is colored according to a color scale bar on the right, ranging from dark blue for the lowest values to bright red for the highest. Panel (b) displays a similar trio of graphs with the same color scale bar, which display the real-space Berry curvatures for the six orbitals of the topological heavy fermion model. The charts correspond to B2(r) = –B1(r) (left), B4(r) = –B3(r) (middle), and Bf,2(r) = –Bf,1(r) (right).