DOI: 10.5061/dryad.rr4xgxdjz
title: Universal dynamics of a pair condensate

description: Pair condensates appear in multiple branches of physics, always
 introducing exotic phenomena. The pair condensate in quantum magnetism is
 the spin nematic, whose static (quadrupolar) order is difficult to access,
 favoring dynamical probes. Here, we perform high-resolution neutron
 spectroscopy to obtain 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. By combining
 precise experiments with quantitative theoretical and numerical analysis,
 we identify universal dynamics arising from the pair condensate. We show
 explicitly how the gapless Goldstone mode influences the dispersion and
 induces Cherenkov-like velocity-selective decay of the gapped
 single-quasiparticle band. These common spectral features shed new light
 on spin-nematic dynamics and underline the universal phenomenology shared
 by pair condensates across different physical systems.
description: # Universal dynamics of a pair condensate [Access this dataset on
 Dryad](10.5061/dryad.rr4xgxdjz) 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.