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This dataset contains the raw data files and Julia scripts used to generate all figures in the manuscript “Universal dynamics of a pair condensate” submitted to Science. The data includes high-resolution neutron spectroscopy measurements that provide direct evidence for the presence of two spin-nematic phases induced in the triangular-lattice antiferromagnet Na₂BaNi(PO₄)₂ by controlling the applied magnetic field. These measurements, along with theoretical and numerical analyses, identify universal dynamics arising from the pair condensate, showing how the gapless Goldstone mode influences the dispersion and induces Cherenkov-like velocity-selective decay of the gapped single-quasiparticle band.

Description of the data and file structure

The dataset is organized by figure, with separate folders for each figure in the main manuscript (Figure 2 through Figure 5) and a separate folder for all supplementary information figures (SI). Figure 1 contains only illustrations and has no associated data files.

The general structure of the dataset is as follows:

/
├── Figure2/
│   ├── data/               # Raw data files for Figure 2
│   ├── panel_A.jl          # Julia script for panel A
│   ├── panel_B.jl          # Julia script for panel B
│   └── ...                 # Additional panel scripts
├── Figure3/
│   ├── data/
│   ├── panel_A.jl
│   └── ...
├── Figure4/
│   ├── data/
│   ├── panel_A.jl
│   └── ...
├── Figure5/
│   ├── data/
│   ├── panel_A.jl
│   └── ...
└── SI/
    ├── data/               # Raw data for all SI figures
    ├── panel_A.jl          # Script for first SI figure panel
    └── ...

Each figure folder contains:

1. A data subfolder with all raw experimental data needed to generate that figure
2. Individual Julia script files (.jl) for each panel of the figure

The raw data files include neutron spectroscopy measurements of Na₂BaNi(PO₄)₂ under various magnetic field conditions, along with calculated values from theoretical and numerical analyses. Each script is self-contained and can be run independently to reproduce the corresponding figure panel from the manuscript.

File Types and Naming Conventions

.nxs files

All files with the .nxs extension are raw data from Inelastic Neutron Scattering (INS) measurements. These files can be read and analyzed using Dave or Mantid software, which are standard tools for neutron scattering data analysis. There are several types of .nxs files with different naming conventions:

1. Brillouin Zone 2D cuts: BZ_[Field]T_[Route]_sym.nxs
   • These files contain 2D (Q-E) cut data in the Brillouin Zone
   • [Field]: Applied magnetic field with ‘p’ replacing decimal point (e.g., “0p00T” means 0.00 Tesla)
   • [Route]: Indicates the cutting route (1 = Γ to M, 2 = M to K, 3 = K to Γ)
   • sym: Indicates that the data has been symmetrized to improve statistics
2. Direction-specific 2D cuts: 2H-HE_[Field]T_masked_bg_sub.nxs
   • These files contain 2D (Q-E) cut data along specific crystallographic directions
   • 2H-HE: Indicates the cut is along the (2H,-H,0) direction
   • [Field]: Applied magnetic field with ‘p’ replacing decimal point
   • masked: Indicates that spurious signals from the instrument have been masked
   • bg_sub: Indicates that the elastic background has been subtracted (background obtained from 5T data)
3. 1D energy cuts: 1D_cut_[Field]T_[QPosition]_masked_bg_sub.nxs
   • These files contain 1D energy cut data at specific Q points
   • [Field]: Applied magnetic field with ‘p’ replacing decimal point
   • [QPosition]: Position in reciprocal space (e.g., “K” for K-point)
   • masked: Indicates that spurious signals from the instrument have been masked
   • bg_sub: Indicates that the elastic background has been subtracted
4. Combined HH-L directions cut: H-HLcut_[Field]T_[Energy]meV.nxs
   • These files contain 2D cut data along combined directions (H,-H,0) and (0,0,L) on the elastic line (ΔE = 0)
   • [Field]: Applied magnetic field with ‘p’ replacing decimal point
   • [Energy]: Incident energy in meV
5. 00L direction cuts: 00Lcut_[Field]T_[Energy]meV.nxs
   • These files contain 1D cut data along (0,0,L) on the elastic line (ΔE = 0)
   • [Field]: Applied magnetic field with ‘p’ replacing decimal point
   • [Energy]: Incident energy in meV
6. 00L direction cuts: 00Lcut_[Field]T_[Energy]meV.nxs
   • These files contain 1D cut data along (0,0,L) on the elastic line (ΔE = 0)
   • [Field]: Applied magnetic field with ‘p’ replacing decimal point
   • [Energy]: Incident energy in meV
7. Symmetrized 1D cuts: 1D_cut_sym_[Field]T_[QPosition].nxs
   • These files contain 1D energy cut data at specific Q points that has been symmetrized to improve statistics
   • sym: Indicates that the data has been symmetrized
   • [Field]: Applied magnetic field with ‘p’ replacing decimal point
   • [QPosition]: Position in reciprocal space (e.g., “K” for K-point)
8. L-direction cuts at specific Q-points: LEcut_[Field]T_[QPosition]_[Energy]meV.nxs
   • These files contain 2D (Q-E) cut data along the (0,0,L) direction at a specific Q-point
   • [Field]: Applied magnetic field with ‘p’ replacing decimal point
   • [QPosition]: Position in reciprocal space (e.g., “K” for K-point)
   • [Energy]: Incident energy in meV

.txt files

The dataset includes the following text files:

1. AC Magnetization measurements: dMdH.txt
   • This file contains the raw AC magnetization measurements
   • The file layout is as follows:
     • Column 1: Magnetic field values (T)
     • Column 2: AC Magnetization values (arbitrary units)
2. Magnetization measurement: MH2.txt
   • This file contains integrated data from AC magnetization measurements
   • The file layout is as follows:
     • Column 1: Magnetic field values (T)
     • Column 2: Magnetization values (with the saturated moment normalized to 1)
3. Peak fitting results: [Field]T_X'.txt (e.g., 0p00T_X'.txt)
   • These files record the results of peak fitting of the 1D (energy) cuts at Q=X’ (K, X) point under various magnetic fields
   • The 1D curves are fitted by several peaks and linear background
   • File layout:
     • For fields ranging from 0T to 0.2T:
       • Last row: Background constant value
       • All preceding rows: Information for each peak (in sets of three rows per peak)
         • Row 1 of set: Intensity
         • Row 2 of set: Position (meV)
         • Row 3 of set: Width (meV)
     • For fields ranging from 1.76T to 2T:
       • Last row: Background slope
       • Second-to-last row: Background intercept
       • All preceding rows: Information for each peak (in sets of three rows per peak)
         • Row 1 of set: Intensity
         • Row 2 of set: Position (meV)
         • Row 3 of set: Width (meV)
4. Kinematic decay conditions: decay_kinematic_2.txt
   • This file records the kinematic conditions for single-magnon decay in the Ferro Quadrupolar phase
   • The file layout is as follows:
     • Column 1: Normalized position on the route Γ-M-K-Γ in the Brillouin Zone (from 0 to 1)
     • Column 2: Simulated energy level for the higher energy branch of spectra (meV)
     • Column 3: Simulated energy level for the lower energy branch of spectra (meV)
     • Column 4: High boundary of the two-magnon decay region (meV)
     • Column 5: Low boundary of the two-magnon decay region (meV)
     • Column 6: Decay rate

.dat files

The dataset includes the following .dat files:

1. GNLSW simulation result: Sqw_GNLSW_B=1.800T_new.dat
   • This file contains S(Q,ω) simulation results based on the Generalized Non-Linear Spin Wave (GNLSW) theory
   • The file follows the route Γ-M-K-Γ in the Brillouin Zone
   • The file layout is as follows:
     • Column 1: q - normalized position on the route (from 0 to 1)
     • Column 2: ω - energy (meV)
     • Column 3: S(Q,ω) - simulated intensity of signal (arbitrary units)
2. Quadrupole-quadrupole correlations: Qxx_vs_B.dat
   • This file records the quadrupole-quadrupole correlations calculated as a function of the applied field by Exact Diagonalization (ED) simulation
   • The file layout is as follows:
     • Column 1: Magnetic field (T)
     • The magnetic field rises from 1.76T to 1.9T four times, corresponding to calculations with different system sizes
     • From top to bottom, the system sizes are N=21, 24, 27, and 36

.csv files

Files with the .csv extension contain simulation results from the Exact Diagonalization (ED) method. There are two types of CSV files:

1. Broadened spectral data: [Component]_N_[CellSize]_B_[MagneticField]_q_[QPosition]_[BraggStatus].csv
   • [Component]: Indicates the component of exchange interaction (e.g., “Szz” or “Sxx”)
   • [CellSize]: Number of unit cells in the super cell (e.g., “21”)
   • [MagneticField]: Value of the applied magnetic field (e.g., “0.00”)
   • [QPosition]: Position in reciprocal space (e.g., “Gamma”)
   • [BraggStatus]: Indicates whether Bragg peak intensity has been removed (e.g., “noBragg”)
   • Example: Szz_N_21_B_0.00_q_Gamma_noBragg.csv contains Szz component data for a 21-unit cell system with 0.00 magnetic field at the Gamma point in reciprocal space, with Bragg peak intensity removed.
   • The layout of these CSV files is as follows:
     • Row 1: Energy values (in meV)
     • Row 2: Corresponding intensity values (Gaussian broadened)
2. Raw spectral pole data: poles_[Component]_N_[CellSize]_B_[MagneticField]_q_[QPosition].csv
   • These files contain the full spectral raw data calculated by ED
   • The naming convention follows the same pattern as the broadened data files, with “poles_” prefix
   • Example: poles_Sxx_N_21_B_0.00_q_Gamma.csv contains the raw spectral pole data for Sxx component
   • The layout of these CSV files is as follows:
     • Column 1: Energy of each pole (meV)
     • Column 2: Intensity of each pole
   • Note: The broadened data files are Gaussian broadened versions of these raw pole data files

The SI folder follows the same organization pattern but contains data and scripts for all supplementary figures.

Code/Software

All figure generation scripts are written in Julia. To run these scripts, the following software environment is required:

• Julia v1.11
• Required Julia packages:
  • Sunny v0.5.10
  • LinearAlgebra
  • Revise
  • CairoMakie
  • CSV
  • DataFrames
  • FileIO
  • Statistics
  • LsqFit
  • DelimitedFiles

Each script (.jl file) corresponds to a specific panel in a figure. The scripts are well-documented with comments explaining the data processing steps and plotting parameters.

The scripts read data from the corresponding data folder and generate publication-quality figures identical to those in the manuscript. No additional data processing is needed outside of what is included in the scripts.