Dataset DOI: 10.5061/dryad.9zw3r22wc

Associated publication

"Observation of quantum vortex core fractionalization and skyrmion formation in a superconductor", Science.

Description of the data and file structure

This Dryad dataset Raw_Data.zip contains the source data associated with the main-text and supplementary figures of the above Science article. The analysis and simulation code associated with this study Code.zip is deposited separately in a linked Zenodo software record under the MIT License.

The uploaded Dryad dataset includes two types of materials:

1. Experimental measurements — source data obtained from scanning tunneling microscopy/spectroscopy and related physical experiments.
2. Computational and theoretical results — numerical data obtained from simulations, theoretical calculations, or first-principles calculations and used for plotting the corresponding manuscript figures.

No manuscript figure image files are included in this Dryad dataset. The files deposited here are the numerical source data underlying the corresponding figures.

File structure

The uploaded data are organized by figure number. The dataset contains folders for the main-text figures and supplementary figures:

• Fig1: source data for Fig. 1 in the main text.
• Fig2: source data for Fig. 2 in the main text.
• Fig3: source data for Fig. 3 in the main text.
• Fig4: source data for Fig. 4 in the main text.
• FigS1–FigS26: source data for Figs. S1–S26 in the Supplementary Materials.

Each figure folder contains the numerical data used to generate the corresponding figure or figure panels. The data are provided in TXT and/or Excel formats. File names, sheet names, and column headers indicate the corresponding panel and plotted quantities where applicable.

Files and variables

The data files are organized by figure number. File names indicate the corresponding figure or figure panel. In general:

• files labeled with Fig correspond to main-text figures;
• files labeled with FigS correspond to figures in the Supplementary Materials;
• TXT files contain tabulated numerical data;
• Excel files contain source data in spreadsheet form, with sheet names and/or column headers indicating the plotted quantities.

Common variables and units include:

• Bias or Bias_mV: sample bias voltage, in millivolts (mV);
• dI/dV: differential conductance, in arbitrary units (a.u.);
• ZBC: zero-bias conductance, in arbitrary units (a.u.);
• DOS: density of states, in arbitrary units (a.u.);
• LDOS: local density of states, in arbitrary units (a.u.);
• Distance: distance along a line cut, in nanometers (nm);
• x, y: spatial coordinates, typically in nanometers (nm), unless otherwise specified;
• Temperature: temperature, in kelvin (K);
• Magnetic field or B: magnetic field, in tesla (T);
• E or Energy: energy, typically in millielectronvolts (meV), unless otherwise specified;
• E/Delta or E_over_Delta: energy normalized by the superconducting gap;
• FFT: fast Fourier transform;
• QPI: quasiparticle interference;
• STS: scanning tunneling spectroscopy;
• STM: scanning tunneling microscopy;
• VBS: vortex bound state;
• CDW: charge-density wave;
• DW: domain wall;
• NV: composite or normal vortex, depending on the file context;
• FV: fractional vortex;
• TRSB: time-reversal-symmetry breaking.

Additional information is provided in the file names, sheet names, and column headers.

Notes on data processing

Some plotted spectra or maps in the associated article include standard visualization procedures, such as normalization, vertical offsetting of spectra for clarity, background subtraction, interpolation for display, or adjustment of color-scale limits. These procedures are used only for visualization and do not change the underlying physical interpretation.

For some zero-bias conductance maps shown in the article, the scalar color-bar limits in the plotted figures were multiplied by 1000 relative to the uploaded source data to make the labels in the figures more concise. The uploaded files contain the original numerical values.

Blank cells or NaN values, where present, indicate missing, masked, or non-applicable data points.

Figure-by-figure documentation

The descriptions below provide context for the source data in each figure folder. They are intended to allow users to understand the deposited files without having the manuscript open simultaneously. For exact experimental conditions and full interpretation, please consult the associated article and Supplementary Materials.

Main-text figures

Fig. 1. Topographic characterization of K-terminated surface.
This folder contains source data for Fig. 1C–F.

Fig1C.txt contains the two-dimensional STM topographic data shown in Fig. 1C. The topographic height values are given in angstroms (Å). The spatial field of view is x = 0–71.82 nm and y = 0–71.82 nm.

Fig1D.txt contains the STM topographic data for the √2 × √2 K-terminated surface shown in Fig. 1D. The topographic height values are given in picometers (pm). The spatial field of view is x = 0–7.5 nm and y = 0–7.5 nm.

Fig1E.txt contains the STM topographic data for the “1 × 1” K-terminated surface shown in Fig. 1E. The topographic height values are given in picometers (pm). The spatial field of view is x = 0–7.6 nm and y = 0–7.6 nm.

Fig1F.txt contains large-range dI/dV spectra acquired on the √2 × √2 and “1 × 1” K-terminated surfaces. The first column gives the sample bias in millivolts (mV). The second and third columns give the normalized dI/dV values measured on the √2 × √2 and “1 × 1” K-terminated surfaces, respectively.

Fig. 2. Enhanced superconductivity and VBS.
This folder contains source data for Fig. 2A–H.

Fig2A.txt contains dI/dV spectra acquired on the “1 × 1” K-terminated surface and on the cleaved Ba0.23K0.77Fe2As2 surface. The first and second columns give the sample bias in millivolts (mV) and the normalized dI/dV spectrum measured on the “1 × 1” K-terminated surface, respectively. The third and fourth columns give the sample bias in millivolts (mV) and the normalized dI/dV spectrum measured on Ba0.23K0.77Fe2As2, respectively.

Fig2B.txt contains temperature-dependent dI/dV spectra acquired on the “1 × 1” K-terminated surface. The first column gives the sample bias in millivolts (mV). The following columns give the dI/dV spectra measured at different temperatures: 0.3 K, 1.8 K, 4.0 K, 6.0 K, 8.0 K, 10.0 K, 12.0 K, 14.0 K, and 16.0 K.

Fig2C.txt contains a series of high-resolution dI/dV spectra acquired on the “1 × 1” K-terminated surface at 20 mK. The first column gives the sample bias in millivolts (mV). The other columns give dI/dV spectra measured at different spatial positions. The column labeled avg gives the averaged dI/dV spectrum.

Fig2D.txt contains the high-resolution zero-bias conductance (ZBC) map around a vortex shown in Fig. 2D. The image size is 25 nm × 25 nm. ZBC denotes the dI/dV signal measured at zero bias. The matrix values represent the measured ZBC intensity at each spatial pixel.

Fig2E1.txt and Fig2F1.txt contain line-cut dI/dV intensity plots measured along the As–As and Fe–Fe directions, respectively. The horizontal axis is sample bias in millivolts (mV), ranging from -10 to 10 mV. The vertical axis is distance in nanometers (nm), ranging from 0 to 14.8828 nm.

Fig2E2.txt and Fig2F2.txt contain the extracted spatial distributions of vortex bound states (VBSs) along the As–As and Fe–Fe directions, respectively.

Fig2G.txt and Fig2H.txt contain the corresponding waterfall plots of the line-cut spectra shown in Fig. 2E and Fig. 2F, respectively. The first column gives the sample bias in millivolts (mV). Each subsequent column gives a dI/dV spectrum measured at a different spatial position along the line cut.

Fig. 3. Quantum vortex core fractionalization and characteristics of fractional vortices.

This folder contains source data for Fig. 3A–G.

Fig3A1.txt to Fig3A6.txt contain zero-bias conductance (ZBC) maps recorded during a temperature-cycling process under an applied out-of-plane magnetic field of 2 T. The corresponding temperatures are 4.2 K, 1.8 K, 2.5 K, 3.0 K, 3.5 K, and 4.2 K, respectively, with the first and last maps corresponding to the initial and final 4.2 K states in the temperature cycle. The matrix values represent the measured ZBC intensity at each spatial pixel. The image size is 125 nm × 125 nm. For the plotted figures, the scalar color-bar limits were multiplied by 1000 relative to the uploaded source data to make the labels in the figures more concise.

Fig3B_S1_N1_1.txt to Fig3B_S1_N1_5.txt, Fig3B_S1_F4_1.txt to Fig3B_S1_F4_5.txt, and Fig3B_S1_F6_1.txt to Fig3B_S1_F6_5.txt contain temperature-dependent ZBC maps recorded in the vicinity of vortex cores #N1, #F4, and #F6, respectively, during the temperature-cycling process. For vortex cores #N1 and #F4, the spatial field of view is x = 0–38.807 nm and y = 0–21.966 nm. For vortex core #F6, the spatial field of view is x = 0–53.452 nm and y = 0–30.021 nm. The matrix values represent the measured ZBC intensity at each spatial pixel. For the plotted figures, the scalar color-bar limits were multiplied by 1000 relative to the uploaded source data to make the labels in the figures more concise.

Fig3C_S1_F4.txt, Fig3C_S1_F6.txt, and Fig3C_S1_N1.txt contain the spatial dependence of ZBC along the red arrow line cuts crossing vortex cores #F4, #F6, and #N1, respectively, during the same temperature-cycling process. The horizontal axis is distance in nanometers (nm).

Fig3D_S2_F1_1.txt to Fig3D_S2_F1_3.txt contain ZBC maps recorded around vortex core #S2_F1 in sample S2 under different temperature cycle. The corresponding temperatures are 1.7 K, 4.3 K, and 30 mK, respectively. The spatial field of view is x = 0–37.441 nm and y = 0–20.543 nm. The matrix values represent the measured ZBC intensity at each spatial pixel. For the plotted figures, the scalar color-bar limits were multiplied by 1000 relative to the uploaded source data to make the labels in the figures more concise.

Fig3E1.txt and Fig3E2.txt contain line-cut dI/dV intensity plots measured across a normal vortex and an anomalous vortex, respectively. The horizontal axis is sample bias in millivolts (mV), ranging from -10 to 10 mV.
The vertical axis is distance (nm), ranging from 0 to 15.875 nm.

Fig3F.txt contains dI/dV spectra taken at the core centers of multiple normal vortices and anomalous vortices. The horizontal axis is sample bias in millivolts (mV), ranging from -10 to 10 mV.

Fig3G.txt contains the number of experimentally observed vortices under different magnetic-field conditions, compared with the theoretically expected number of vortices calculated under the assumption of conventional flux quantization.

Fig. 4. Numerical simulation results and vortex phase diagram.
This folder contains source data for Fig. 4A–E.

Fig4A.txt contains the zero-bias conductance (ZBC) map measured at 4.2 K under an applied out-of-plane magnetic field of 2 T. The ZBC map was corrected by subtracting the zero-field map after topographic correction to remove impurity-related contributions to the image contrast. The matrix values represent the measured ZBC intensity at each spatial pixel. The spatial field of view is x = 0–183 nm and y = 0–183 nm. For the plotted figure, the scalar color-bar limits were multiplied by 1000 relative to the uploaded source data to make the labels in the figure more concise.

Fig4B.txt contains the theoretically calculated density of states (DOS) at zero energy for integer vortices and fractional vortices. The matrix values represent the calculated DOS at zero energy at each spatial pixel.

Fig4CFCore.txt, Fig4CFDW.txt, Fig4CN0.txt, and Fig4CNCore.txt contain calculated DOS spectra for the fractional vortex core (F_core), domain wall (F_DW), vortex-less system (N_0), and normal vortex core (N_core), respectively.

Fig4DE.txt contains numerical data for Fig. 4D and Fig. 4E. The columns x and y give the spatial coordinates. The columns N1, N2, and N3 give the densities of the three superconducting components. The total density of superconducting components plotted in Fig. 4D is Ψ†Ψ = N1 + N2 + N3. The column dQ gives the skyrmionic charge-density distribution plotted in Fig. 4E. Additional columns provide the calculated magnetic-field distribution, phase differences, current components, and pseudospin components used in the theoretical analysis.

Supplementary figures

Fig. S1. Two types of ordered K-terminated surface.

This folder contains source data for Fig. S1A–C.

FigS1A.txt contains the STM topographic image of a freshly cleaved KFe2As2 single-crystal sample, revealing two types of ordered K-terminated surfaces. The topographic height values are given in angstroms (Å). The spatial field of view is x = 0–100 nm and y = 0–100 nm.

FigS1B.txt contains the height profile measured along the white dashed line in Fig. S1A. The first column gives the distance in nanometers (nm). The second column gives the topographic height in angstroms (Å).

FigS1C.txt contains ln(I)–Δz spectroscopy measurements acquired on the two terraces. The first column gives the measured ln(I) data for the “1 × 1” termination, with the corresponding Δz axis defined by a linear scale from -3.5294 Å to 3 Å. The third column gives the measured ln(I) data for the √2 × √2 (rt2) termination, with the corresponding Δz axis defined by a linear scale from -3.7451 Å to 3 Å. The second and fourth columns give the corresponding linear fitting results for the “1 × 1” and √2 × √2 (rt2) terminations, respectively.

Fig. S2. Evidence of electron doping effect on the “1 × 1” K-termination.

This folder contains source data for Fig. S2B and Fig. S2C.

FigS2.xlsx contains multiple sheets with source data for quasiparticle interference (QPI) and ARPES comparisons.

The sheets FigS2B and FiS2C contain quasiparticle-interference energy-versus-momentum (E–k) cuts. In these two-dimensional datasets, the vertical axis corresponds to energy and the horizontal axis corresponds to momentum. For FigS2B, the energy range is y = -20 to 20 meV and the momentum range is x = -0.74409 to 0.73464 Å⁻¹. For FigS2C, the energy range is y = -25 to 25 meV and the momentum range is x = -0.40212 to 0.39898 Å⁻¹.

The sheet ARPES_FigS2B contains the extracted ARPES dispersion used for comparison with Fig. S2B. The first column gives the momentum coordinate, and the second column gives the energy in millielectronvolts (meV).

The sheet Fit_ARPES_FigS2C contains the fitted QPI result and the extracted ARPES result used for comparison in Fig. S2C. The first and second columns give the fitted QPI dispersion, with the first column corresponding to momentum and the second column corresponding to energy in millielectronvolts (meV). The third and fourth columns give the extracted ARPES dispersion, with the third column corresponding to momentum and the fourth column corresponding to energy in millielectronvolts (meV).

Fig. S3. AC susceptibility measurement of the KFe2As2 bulk materials.

This folder contains source data for Fig. S3.

FigS3.xlsx contains AC susceptibility data measured on KFe2As2 bulk single-crystal materials. The first column gives the temperature in kelvin (K). The second and third columns give the normalized real and imaginary components of the AC susceptibility, denoted as chi_norm' and chi_norm'', respectively.

Fig. S4. DFT calculation.

This folder contains source data for Fig. S4.

Ba0p23K0p77Fe2As2_band.txt, KFe2As2_band.txt, and KFe2As2_K-terminated-surface_band.txt contain calculated electronic band structures for Ba0.23K0.77Fe2As2, KFe2As2, and the K-terminated surface of KFe2As2, respectively.

In each file, the data begin from the second line. The even-numbered lines give the coordinates of the corresponding k point in reciprocal space, i.e., in the Brillouin zone. The odd-numbered lines contain the band-energy data: the first column gives the k-path coordinate used as the horizontal axis in the band-structure plot, and the second through last columns give the energies of the calculated bands at that k point, used as the vertical-axis values.

Fig. S5. High resolution dI/dV spectra obtained on the “1 × 1” K-terminated surface.

This folder contains source data for Fig. S5.

FigS5.txt contains high-resolution dI/dV spectra obtained on the “1 × 1” K-terminated surface and the √2 × √2 K-terminated surface. The first and second columns correspond to the “1 × 1” K-terminated surface, with the first column giving the sample bias in millivolts (mV) and the second column giving the corresponding dI/dV spectrum. The third and fourth columns correspond to the √2 × √2 K-terminated surface, with the third column giving the sample bias in millivolts (mV) and the fourth column giving the corresponding dI/dV spectrum.

Fig. S6. Multiple superconducting gaps and vortex bound states.

This folder contains source data for Fig. S6A, Fig. S6C, and Fig. S6D.

FigS6A.txt contains the spatially averaged high-resolution dI/dV spectrum acquired on the “1 × 1” K-terminated surface at 20 mK. The first column gives the sample bias in millivolts (mV). The second column gives the corresponding dI/dV spectrum.

FigS6C1.txt and FigS6D1.txt contain line-cut dI/dV intensity plots measured along the As–As and Fe–Fe directions, respectively. The horizontal axis is sample bias in millivolts (mV), ranging from 0 to 10 mV. The vertical axis is distance in nanometers (nm), ranging from 0 to 14.8828 nm.

FigS6C2.txt and FigS6D2.txt contain the extracted spatial distributions of vortex bound states (VBSs) along the As–As and Fe–Fe directions, respectively.

Fig. S7. Multi-peak fitting of dI/dV spectra near the vortex-core edge.

This folder contains source data for Fig. S7.

FigS7.txt contains the multi-peak fitting results for dI/dV spectra measured near the vortex-core edge. The horizontal coordinate for all curves is sample bias in millivolts (mV).

Columns 1–8 correspond to the four fitted peak components, with each pair of columns giving the horizontal and vertical coordinates of one peak component. Columns 9–10 give the horizontal and vertical coordinates of the background curve. Columns 11–12 give the horizontal and vertical coordinates of the total fitted curve. Columns 13–14 give the horizontal and vertical coordinates of the original dI/dV spectrum.

Fig. S8. Quantitative analysis of superconducting gaps and additional spectral features.

This folder contains source data for Fig. S8A and Fig. S8B.

FigS8A.txt contains the spatially averaged high-resolution dI/dV spectrum acquired at 20 mK and the corresponding second-derivative spectrum, d²I/dV², calculated from the same data. The first column gives the sample bias in millivolts (mV). The second column gives the dI/dV spectrum, and the third column gives the corresponding d²I/dV² spectrum.

FigS8B.txt contains the extracted superconducting gaps, denoted as Δ, and the associated additional spectral features, denoted as E. The offset energy Ω is obtained from the difference between the additional feature energy and the superconducting gap energy.

Fig. S9. Quantum vortex splitting.

This folder contains source data for Fig. S9A and Fig. S9B.

FigS9A_F1_1.txt to FigS9A_F1_5.txt, FigS9A_F2_1.txt to FigS9A_F2_5.txt, FigS9A_F3_1.txt to FigS9A_F3_5.txt, and FigS9A_F5_1.txt to FigS9A_F5_5.txt contain temperature-dependent zero-bias conductance (ZBC) maps recorded in the vicinity of vortex cores #F1, #F2, #F3, and #F5, respectively. For each vortex core, the five files correspond to measurements acquired at 1.8 K, 2.5 K, 3.0 K, 3.5 K, and 4.2 K, respectively. The spatial field of view is x = 0–35.019 nm and y = 0–25.566 nm. The matrix values represent the measured ZBC intensity at each spatial pixel. For the plotted figures, the scalar color-bar limits were multiplied by 1000 relative to the uploaded source data to make the labels in the figures more concise.

FigS9B_F1.txt, FigS9B_F2.txt, FigS9B_F3.txt, and FigS9B_F5.txt contain the spatial dependence of ZBC across vortex cores #F1, #F2, #F3, and #F5, respectively. The horizontal axis is distance in nanometers (nm).

Fig. S10. Quantum vortex splitting in different samples.

This folder contains source data for Fig. S10A and Fig. S10B.

FigS10A.txt and FigS10B.txt contain zero-bias conductance (ZBC) maps recorded on the “1 × 1” K-terminated surface at 20 mK and 4.3 K, respectively. The spatial field of view is x = 0–312.8 nm and y = 0–303.2 nm. The matrix values represent the measured ZBC intensity at each spatial pixel. For the plotted figures, the scalar color-bar limits were multiplied by 1000 relative to the uploaded source data to make the labels in the figures more concise.

Fig. S11. Statistical analysis of distinct vortices.

This folder contains source data for Fig. S11A–I.

FigS11.xlsx contains multiple sheets with zero-bias conductance (ZBC) maps recorded on the “1 × 1” K-terminated surface under different temperature and magnetic-field conditions. The sheets FigS11A, FigS11B, FigS11C, FigS11D, FigS11E, FigS11F, FigS11G, and FigS11H correspond to ZBC maps measured under the following conditions, respectively: 2 T at 0.3 K, 0 T at 4.2 K, 1 T at 4.2 K, 2 T at 4.2 K, 3 T at 4.2 K, 1.4 T at 4.2 K, 2 T with the magnetic field tilted by 45° in the x–z plane at 4.2 K, and 2 T with the magnetic field tilted by 60° in the x–z plane at 4.2 K. The spatial field of view for these ZBC maps is 365 nm × 375 nm. The matrix values represent the measured ZBC intensity at each spatial pixel. For the plotted figures, the scalar color-bar limits were multiplied by 1000 relative to the uploaded source data to make the labels in the figures more concise.

The sheet FigS11I contains a ZBC map measured on LiFeAs at 4.2 K under an applied magnetic field of 2 T. The spatial field of view is x = 0–410 nm and y = 0–410 nm. The matrix values represent the measured ZBC intensity at each spatial pixel.

Fig. S12. Rapid-scan measurements.

This folder contains source data for Fig. S12A and Fig. S12B.

FigS12A.xlsx contains zero-bias conductance (ZBC) maps measured at 4.2 K under a perpendicular magnetic field of 2 T. The spatial field of view is x = 0–50 nm and y = 0–50 nm. The matrix values represent the measured ZBC intensity at each spatial pixel. For the plotted figures, the scalar color-bar limits were multiplied by 1000 relative to the uploaded source data to make the labels in the figures more concise.

FigS12B.xlsx contains the spatial ZBC distributions of the splitting vortex obtained from consecutive scans. Columns 2–11 correspond to the ZBC line profiles obtained during forward scans. Columns 13–22 correspond to the ZBC line profiles obtained during backward scans. Columns 1 and 12 give the distance coordinates in nanometers (nm) for the forward-scan and backward-scan line profiles, respectively.

Fig. S13. Time-dependence of vortex-core splitting.

This folder contains source data for Fig. S13.

FigS13.xlsx contains two sheets corresponding to zero-bias conductance (ZBC) maps measured at 4.2 K under the same experimental conditions after a time interval of 24 days. The spatial field of view for each ZBC map is x = 0–400 nm and y = 0–400 nm. The matrix values represent the measured ZBC intensity at each spatial pixel. For the plotted figures, the scalar color-bar limits were multiplied by 1000 relative to the uploaded source data to make the labels in the figures more concise.

Fig. S14. Quantum vortex core fractionalization and gap mapping.

This folder contains source data for Fig. S14A and Fig. S14B.

FigS14A.txt and FigS14B.txt contain a zero-bias conductance (ZBC) map acquired under a perpendicular magnetic field of 2 T and a superconducting-gap map recorded at 300 mK in the same field of view, respectively. The spatial field of view is x = 0–100 nm and y = 0–100 nm. For FigS14A.txt, the matrix values represent the measured ZBC intensity at each spatial pixel. For the plotted figures, the scalar color-bar limits were multiplied by 1000 relative to the uploaded source data to make the labels in the figures more concise. For FigS14B.txt, the matrix values represent the extracted superconducting-gap value at each spatial pixel.

Fig. S15. Magnetic-field dependence of vortex-core splitting.

This folder contains source data for Fig. S15.

FigS15.xlsx contains multiple sheets with zero-bias conductance (ZBC) maps measured at 4.2 K under different magnetic-field configurations. The sheets 1T@4.2K, 2T_deg60@4.2K, 1.4T@4.2K, and 2T_deg45@4.2K correspond to ZBC maps measured under the indicated field conditions. The spatial field of view is x = 0–159.78 nm and y = 0–159.98 nm. The matrix values represent the measured ZBC intensity at each spatial pixel. For the plotted figures, the scalar color-bar limits were multiplied by 1000 relative to the uploaded source data to make the labels in the figures more concise.

Fig. S16. Time-dependent Ginzburg–Landau simulation.

This folder contains source data for Fig. S16.

FigS16.xlsx contains multiple sheets corresponding to a set of time-dependent Ginzburg–Landau (TDGL) simulations performed with different thermal-noise amplitudes. Each sheet contains the numerical simulation data for one thermal-noise condition. The matrix values represent the simulated field or order-parameter-related quantity shown in the corresponding sheet.

Fig. S17. VBS across the splitting vortices.

This folder contains source data for Fig. S17A and Fig. S17B.

FigS17.xlsx contains multiple sheets with source data for high-resolution zero-bias conductance (ZBC) maps and line-cut dI/dV intensity plots across splitting vortices.

The sheets FigS17A1 to FigS17A3 contain high-resolution ZBC maps around splitting vortices. For FigS17A1 and FigS17A2, the spatial field of view is x = 0–30 nm and y = 0–30 nm. For FigS17A3, the spatial field of view is x = 0–45 nm and y = 0–45 nm. The matrix values represent the measured ZBC intensity at each spatial pixel.

The sheets FigS17B1 to FigS17B3 contain line-cut dI/dV intensity plots across the splitting vortices. The horizontal axis is sample bias in millielectronvolts (meV), ranging from -10 to 10 meV. The vertical axis is distance in nanometers (nm), ranging from 0 to 30.8664 nm (FigS17B1), 0 to 35.6357 nm (FigS17B2), 0 to 50.3449 nm (FigS17B3).

Fig. S18. Spectroscopy contrast between normal vortices and fractional vortices.

This folder contains source data for Fig. S18A and Fig. S18B.

FigS18.xlsx contains multiple sheets with source data for spectroscopic maps recorded at the same surface location on the “1 × 1” K-terminated surface at 4.2 K under an applied magnetic field of 2 T.

The sheet FigS18A contains the zero-bias conductance (ZBC) map. The sheet FigS18B contains the dI/dV(r, V) map slice measured at V = -4 mV. For both maps, the spatial field of view is x = 0–100 nm and y = 0–100 nm. For FigS18A, the matrix values represent the measured ZBC intensity at each spatial pixel. For FigS18B, the matrix values represent the measured dI/dV intensity at -4 mV at each spatial pixel. For the plotted figures, the scalar color-bar limits were multiplied by 1000 relative to the uploaded source data to make the labels in the figures more concise.

Fig. S19. Simulation of the vortex splitting.

This folder contains source data for Fig. S19A–F.

FigS19.xlsx contains multiple sheets with theoretically calculated density of states (DOS), dI/dV spectra, and line-cut intensity plots for normal and split vortices.

The sheets FigS19A and FigS19B contain the theoretically calculated DOS at zero energy for the normal vortex and the split vortices, respectively. The matrix values represent the calculated DOS at zero energy at each spatial pixel.

The sheet FigS19C contains calculated dI/dV spectra at the center of the one-quanta vortex and the fractional vortex. Columns 1–2 correspond to the one-quanta vortex, with the first column giving the horizontal coordinate and the second column giving the calculated dI/dV value. Columns 3–4 correspond to the fractional vortex, with the third column giving the horizontal coordinate and the fourth column giving the calculated dI/dV value.  The horizontal axis is bias in normalized units, ranging from -1.2 to 1.2.

The sheets FigS19D to FigS19F contain calculated line-cut intensity plots across the normal vortex and fractional vortices. The horizontal axis is bias in normalized units, ranging from -1.2 to 1.2, and the vertical axis is distance in normalized units, ranging from -3.01331 to 3.01348.

Fig. S20. Magnetic-field dependence of domain walls.

This folder contains source data for Fig. S20.

FigS20.xlsx contains multiple sheets with zero-bias conductance (ZBC) maps measured at 4.2 K under different magnetic-field conditions. The sheets 1T@4.2K, 1.4T @ 4.2K, 2T @ 4.2K, 2Tdeg60 @ 4.2K, 2Tdeg45 @ 4.2K, and 3T @ 4.2K correspond to ZBC maps measured under the indicated magnetic-field configurations. The spatial field of view is x = 0–365 nm and y = 0–375 nm. The matrix values represent the measured ZBC intensity at each spatial pixel. For the plotted figures, the scalar color-bar limits were multiplied by 1000 relative to the uploaded source data to make the labels in the figures more concise.

Fig. S21. Coexistence of fractional vortices on a domain wall with regular vortices.

This folder contains source data for Fig. S21.

FigS21.txt contains numerical data for the coexistence of fractional vortices on a domain wall with regular vortices. The columns x and y give the spatial coordinates.

The column Bz corresponds to the magnetic-field component plotted as B. The columns Phasediff12 and Phasediff13 correspond to the phase-difference quantities used to plot sin𝜑₁₂ and sin𝜑₁₃, respectively. The columns N1, N2, and N3 correspond to |ψ₁|², |ψ₂|², and |ψ₃|², respectively, where ψ₁, ψ₂, and ψ₃ are the three superconducting components.

The current magnitude for component a, denoted as Ja, is obtained from the corresponding in-plane current components as |Ja| = sqrt(Jax² + Jay²), where Jax and Jay are the current components along the x and y directions, respectively.

Fig. S22. Magnetic-field evolution of domains with different chiralities.

This folder contains source data for Fig. S22A and Fig. S22B.

FigS22A1.txt and FigS22A2.txt contain calculated total-density data for the superconducting components. The columns N1, N2, and N3 give the densities of the three superconducting components. The total density of superconducting components plotted in Fig. S22A is Ψ†Ψ = N1 + N2 + N3.

FigS22B1.txt and FigS22B2.txt contain zero-bias conductance (ZBC) maps acquired at 4.2 K under applied magnetic fields of 2 T and 3 T, respectively. The spatial field of view is x = 0–365 nm and y = 0–375 nm. The matrix values represent the measured ZBC intensity at each spatial pixel. For the plotted figures, the scalar color-bar limits were multiplied by 1000 relative to the uploaded source data to make the labels in the figures more concise.

Fig. S23. Self-consistent BdG simulation of skyrmion topological charge.

This folder contains source data for Fig. S23A–D.

FigS23A.txt contains the calculated density of states (DOS) at zero energy for two composite vortices distributed on a domain wall with two pinning centers.

FigS23B.txt contains the calculated distribution of the skyrmionic charge density.

FigS23C.txt and FigS23D.txt contain the calculated phase differences between superconducting gaps in different bands.

Fig. S24. Self-consistent BdG simulation of the domain wall decoration.

This folder contains source data for Fig. S24A–L.

FigS24A-D.xlsx contains multiple sheets with experimental source data for Fig. S24A–D. The sheets correspond to the STM topographic image (FigS24A), the zero-bias conductance (ZBC) map measured at 0.3 K under an applied magnetic field of 2 T (FigS24B), the ZBC map measured at 4.2 K under zero magnetic field (FigS24C), and the ZBC map measured at 4.2 K under an applied magnetic field of 1 T (FigS24D). The spatial field of view is x = 0–365 nm and y = 0–375 nm. For the ZBC maps, the matrix values represent the measured ZBC intensity at each spatial pixel.

FigS24E.txt to FigS24H.txt contain numerical simulation results for a sample with a domain wall under an external magnetic field. These files correspond to the simulated density of states (DOS) at zero bias energy, magnetic-field distribution, and phase differences between superconducting gaps.

FigS24I.txt to FigS24L.txt contain the corresponding numerical simulation results for the same type of system, but with two pinning centers located on the domain wall. These files include the simulated DOS at zero bias energy, magnetic-field distribution, and phase differences between superconducting gaps.

Fig. S25. Domain wall stabilized by randomly located pinning centers.

This folder contains source data for Fig. S25.

FigS25.txt contains calculated data for the total density of all superconducting components, Ψ†Ψ, and the skyrmionic charge-density distribution in a system with randomly located pinning centers. The columns N1, N2, and N3 give the densities of the three superconducting components. The total density of superconducting components plotted in Fig. S25A is Ψ†Ψ = N1 + N2 + N3. The column dQ gives the skyrmionic charge density plotted in Fig. S25B.

Fig. S26. Pseudospin texture associated with domain walls and skyrmions.

This folder contains source data for Fig. S26.

FigS26.txt contains calculated data for the pseudospin texture associated with domain walls and skyrmions. The pseudospin vector n is defined as

n = (√2 Re(ψ₁* ψ₂ + ψ₂* ψ₃) / (|ψ₁|² + |ψ₂|² + |ψ₃|²),
√2 Im(ψ₁* ψ₂ + ψ₂* ψ₃) / (|ψ₁|² + |ψ₂|² + |ψ₃|²),
(|ψ₁|² − |ψ₃|²) / (|ψ₁|² + |ψ₂|² + |ψ₃|²)),

where ψ₁, ψ₂, and ψ₃ are the three superconducting components. The arrows in the plotted figure indicate the direction of the pseudospin vector n.

Software for opening data files

The tabulated experimental and numerical data files can be opened with standard spreadsheet or text-processing software, including Microsoft Excel, LibreOffice, or Python packages such as pandas and openpyxl.

Related software/code record

The analysis and simulation code associated with this dataset is deposited separately in a linked Zenodo software record under the MIT License. The Zenodo record is linked to this Dryad dataset through Dryad's simultaneous software-publication workflow. A summary README for the linked software/code record is provided at the end of this document.

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Linked Zenodo software/code README

Code/software information

This Zenodo software record contains the analysis and simulation code associated with the main-text and supplementary figures of the above Science article. The code was used to generate computational results, process simulation outputs, and/or reproduce the corresponding theoretical plots.

The code is organized into five subfolders according to the corresponding figures or figure panels:

• Fig4BC,S23,S24E-L: code used for Fig. 4B, Fig. 4C, Fig. S23, and Fig. S24E--L.
• Fig4DE,S21,S22A,S25,S26: code used for Fig. 4D, Fig. 4E, Fig. S21, Fig. S22A, Fig. S25, and Fig. S26.
• FigS4: code used for Fig. S4.
• FigS16: code used for Fig. S16.
• FigS19: code used for Fig. S19.

Each subfolder contains the scripts, input files, output files, or processed data required to generate the corresponding calculation results. Additional details are provided in the file names and comments within the code files where applicable.

License

The code in this Zenodo software record is released under the MIT License.

Some template code is adapted from the open-source CMakeTemplate repository:

https://github.com/DavidAce/CMakeTemplate/

The CMakeTemplate repository is distributed under the MIT License. The MIT License text and any required notices should be retained with the relevant code files.

Some code depends on external open-source software or libraries with their own licenses, including FreeFem++ and related dependencies. Users should consult the corresponding software packages for their license terms.

Code for Fig4BC,S23,S24E-L

This folder contains the code and raw simulation files used for Figs. 4B, 4C, S23, and S24E--L.

The notebook gen_data.ipynb generates .txt files from the raw .h5 files. It uses the Python scripts import_data.py and skyrmion_charge.py. The generated .txt files are the processed numerical data used for plotting the corresponding figure panels.

The raw .h5 files are:

• CV.h5: solution for a system with a conventional or composite vortex, used for Figs. 4B and 4C.
• FV.h5: solution for a system with a fractional vortex, used for Figs. 4B and 4C.
• DW_pin.h5: solution for a system under an external magnetic field with pinning centers, used for Fig. S24I--L.
• DW_nopin.h5: solution for a system under an external magnetic field without pinning centers, used for Fig. S24E--H.
• DW.h5: solution for a system with six fractional vortices, used for Fig. S23.

The Python scripts are:

• import_data.py: reads the .h5 files.
• skyrmion_charge.py: calculates the skyrmionic charge density, used for Fig. S23B.
• plot_file.py: generates plots from the .txt files obtained from gen_data.ipynb.

The scripts were run using Python 3.12.11.

Code for Fig4DE,S21,S22A,S25,S26

This folder contains two FreeFem++ scripts used to produce numerical data and prepare the data for visualization processing. These scripts were used for the computational results shown in Figs. 4D, 4E, S21, S22A, S25, and S26.

The scripts use the FreeFem++ library, which is distributed under the LGPL 3.0 license and can be obtained from:

https://freefem.org/

The scripts can be run after the parallel MPI version of FreeFem++ has been properly installed.

Scripts

• 3CGL-produce-mpi.edp: produces the numerical data.
• 3CGL-snapshot.edp: loads previously produced data, computes derived quantities, and exports tabulated .txt files for visualization.

The exported .txt files are space-separated and can be used in standard visualization software, such as gnuplot, Python, or other plotting tools.

Options

The following command-line options can be used:

• -index and -indexOld: restart from previous data. For example, -index 1 -indexOld 0 loads the iter=0 file and outputs the iter=1 file.
• -dirname: directory where the data are stored.
• -plots: activates the plot interface. Use 0 or 1.
• -lx and -ly: domain size.
• -npts: number of points on the mesh boundary; this controls the spatial resolution.
• -loops: number of minimization loops.
• -iMAX: number of iterations of the Ginzburg--Landau solver.
• -a11, -a22, -a33, -a12, -a13, -a23, -b11, -b22, -b33: parameters of the Ginzburg--Landau potential.
• -e: gauge-field coupling.
• -B0: applied magnetic field.

Example A: Produce data

In the following example, the initial state is random, and the parameters are chosen to be in a time-reversal-symmetry-breaking phase. The data can be produced using the following MPI command:

if [ ! -d tmp-here ]; then mkdir tmp-here; fi

mpirun --oversubscribe -np 4 FreeFem++-mpi -wg 3CGL-produce-mpi.edp \
       -dirname tmp-here -plots 1 \
       -index 0 -indexOld 0 \
       -lx 20 -ly 20 -npts 200 \
       -loops 10 -iMAX 100 \
       -a11 -2.0 -a22 -1.0 -a33 -1.0 \
       -a12 1.0 -a13 1.0 -a23 1.0 \
       -b11 1.0 -b22 1.0 -b33 1.0 \
       -e 0.250 -B0 3.250

Example B: Convert output for visualization

In the following example, the previously produced data are loaded and processed to compute derived quantities. These quantities are then exported as a space-separated .txt file, which can be used in standard visualization software such as gnuplot.

mpirun --oversubscribe -np 1 FreeFem++-mpi -wg 3CGL-snapshot.edp \
       -dirname tmp-here -plots 1 \
       -index 0 -indexOld 0 \
       -a11 -2.0 -a22 -1.0 -a33 -1.0 \
       -a12 1.0 -a13 1.0 -a23 1.0 \
       -b11 1.0 -b22 1.0 -b33 1.0 \
       -e 0.250 -B0 3.250

Notes

The parameters, iteration numbers, and number of mesh points can be adjusted to improve the accuracy of the solution. High-quality solutions comparable to those used in the paper may require computational resources beyond a personal computer. Such calculations should be ported to a computing cluster and run on a larger number of nodes.

Code for FigS4

This folder contains FPLO input and/or output files used for the first-principles calculations shown in Fig. S4. These files are provided to document the first-principles calculations used to generate the band structures. Re-running the calculations requires a properly licensed and installed FPLO environment.

Code for FigS16

This folder contains a time-dependent Ginzburg--Landau (TDGL) solver based on a finite-difference scheme. The code was used for the computational results shown in Fig. S16.

The gauge field is implemented using link variables. The boundary condition imposes no supercurrent through the boundary. The Euler--Lagrange equations are implemented using the Crank--Nicolson method, including a linearized version for the order-parameter evolution.

The code is based on the open-source CMake template:

https://github.com/DavidAce/CMakeTemplate/

Requirements

• NVIDIA GPU and NVCC compiler.
• C++17 compiler.
• CMake version 3.21 or above.
• Conan version 2, recommended.

Dependencies

By default, the direct dependencies are:

• h5pp: fast and simple binary storage. This package includes HDF5, Eigen, spdlog, and fmt.
• fmt: string formatting.
• spdlog: logging.

These dependencies are installed automatically when using the bundled CMake presets release-conan or debug-conan.

Note that h5pp itself has the same dependencies, so they should already be present if h5pp was installed previously.

Compatibility

This code has been tested only in Linux environments.

Usage with CMake presets

Build the code as a regular out-of-source CMake project after installing the dependencies above. The bundled CMake presets can be used to enable automatic dependency installation with Conan.

Step 1: Install Conan

pip install conan
conan profile detect

Step 2: Set CUDA architecture and list CMake presets

Set the CUDA architecture of your device in line 30 of the file CMakePresets.json.

Then open a terminal and run:

cmake --list-presets

The available configure presets are:

"release-conan" - Release | Conan package manager
"debug-conan" - Debug   | Conan package manager

Step 3: Configure and build

Select a preset listed in the previous step, and run:

cmake --preset=release-conan
cmake --build --preset=release-conan

These commands configure the project and build the executable at:

./build/release-conan/CMT

Step 4: Run calculations

Open a terminal and run:

python3 run.py

Code for FigS19

This folder contains MATLAB code used for the calculations shown in Fig. S19.

The MATLAB scripts generate the numerical results used to plot the corresponding panels in Fig. S19.

The scripts were run using MATLAB. No special toolboxes are required unless otherwise specified in the scripts.

Software environments

The code files require the software environments described above, including Python, FreeFem++, FPLO, C++/CUDA/CMake, or MATLAB, depending on the corresponding folder.

Data files

This Zenodo software record is intended for code/software. The corresponding source data for the manuscript figures are deposited in the linked Dryad dataset. Small input or output files required for the code to run or to reproduce the code-generated plots may be included here where necessary.

Code citation

When using the associated analysis or simulation code, please cite the associated Science article, the Dryad dataset, and the linked Zenodo software record.

Access information

There are no access restrictions on the dataset after publication. The dataset is intended to be publicly available through Dryad at the DOI listed above.

Contact

For questions about this dataset, please contact:

Quanxin Hu
Email: sjtu18810999766@sjtu.edu.cn

Citation

Please cite both the associated Science article and this Dryad dataset when using these data. If using the associated analysis or simulation code, please also cite the linked Zenodo software record.