DOI: 10.5061/dryad.zcrjdfnp9
title: Data from: Possible observation of quadrupole waves in spin nematics

description: Discovery of new states of matter is a key objective in modern condensed
 matter physics, which often leads to revolutionary technological
 advancements such as superconductivity. Quantum spin nematic, a “hidden
 order” that evades conventional magnetic probes, is one such state.
 Na2BaNi(PO4)2 is a potential spin nematic material, suggested by the
 observation of a 2-magnon Bose-Einstein condensation from above the
 saturation field. However, direct confirmation of the spin nematicity
 remains elusive. This paper presents inelastic neutron scattering spectra
 from the putative spin nematic phases of Na2BaNi(PO4)2, revealing
 low-energy quadrupole waves that are absent in the neighboring
 conventional magnetic phases. A spin-1 model quantitatively captures the
 full details of the spin excitation spectra across all low-temperature
 phases, providing direct evidence of the spin nematic orders.
 Additionally, we show evidence of the 3-magnon continuum and 2-magnon
 bound states in the 1/3-magnetization plateau, revealing condensation of
 the 2-magnon bound state as the origin of the low-field spin nematic
 supersolid phase.
description: The computational and experimental methods used in this study —
 including neutron scattering measurements, the infinite projected
 entangled pair states (iPEPS) ansatz, and linear spin-wave (LSW) theory —
 are fully described in the Supplemental Material of the paper
 (<em>Phys. Rev. Lett. 135, 156704 (2025)</em>, DOI：https://doi.org/10.1103/pywx-vxfh).
description: # Data from: Possible observation of quadrupole waves in spin nematics
 *Phys. Rev. Lett. 135, 156704 (2025)*\ DOI:
 [https://doi.org/10.1103/pywx-vxfh](https://doi.org/10.1103/pywx-vxfh)
 **Overview** This dataset “NaBaNiPO” contains all data used in the main
 manuscript and Supplemental Material of the publication **“Possible
 Observation of Quadrupole Waves in Spin Nematics”** (*Phys. Rev. Lett.
 135, 156704 (2025)*). The dataset consists of main folder: NaBaNiPO.zip.
 Each data file corresponds directly to a specific figure or supplementary
 figure and is **named according to the figure label** (e.g.,
 Fig2a_dmrg.dat, Fig3a_exp_0T.dat for main text figures, and FigS1e.dat,
 FigS2.dat for Supplemental Material). This naming structure allows users
 to easily locate the dataset corresponding to each figure in the paper.
 **File Format & Access** * All files are in **plain text (.dat)**
 format and can be opened with any standard text editor (Notepad, TextEdit,
 VS Code, etc.). * Files use **standard delimiters** (comma, tab, or
 space). * All files **include a header line** specifying variable names. *
 Missing values, if any, are marked as **nan**. **Contents** * **Main
 Figures:** Data used to generate all plots in the main manuscript (named
 FigX_*.dat). * **Supplementary Figures:** Data used for all plots in the
 Supplemental Material (named FigSX_*.dat). **How to Use** 1. Identify the
 figure of interest in the published paper. 2. Open the data file with the
 corresponding figure name. 3. The data can be directly loaded into
 **Origin,** **Python, MATLAB, or any plotting software**. **Files and
 variables** **(1)   Fig2a_dmrg.dat** **Description:** Normalized
 magnetization from density matrix renormalization group (DMRG)
 calculations **Variables:** * Column 1: magnetic field * Column 2:
 calculated normalized magnetization **(2)   Fig2a_iPEPS4.dat**
 **Description:** Normalized magnetization using infinite projected
 entangled-pair states (iPEPS) calculations with bond dimension X=4
 **Variables:** * Column 1: magnetic field * Column 2: calculated
 normalized magnetization **(3)   Fig2a_iPEPS5.dat** **Description:**
 Normalized magnetization using infinite projected entangled-pair states
 (iPEPS) calculations with bond dimension X=5. **Variables:** * Column 1:
 magnetic field * Column 2: calculated normalized magnetization **(4)  
 Fig2a_EXP_M.dat** **Description:** Measured experimental normalized
 magnetization M/M~s~ at 50 mK **Variables:** * Column 1: magnetic field *
 Column 2: measured normalized magnetization **(5)   Fig2a_EXP_chi.dat**
 **Description:** Measured experimental magnetic susceptibility dM/dB~~ at
 50 mK **Variables:** * Column 1: magnetic field * Column 2: measured
 magnetic susceptibility dM/dB **(6)   Fig2b_Q4.dat** **Description:** Spin
 nematic order parameter using infinite projected entangled-pair states
 (iPEPS) calculations with bond dimension X=4.  **Variables:** * Column 1:
 magnetic field * Column 2: spin nematic order parameter **(7)  
 Fig2b_Q5.dat** **Description:** Spin nematic order parameter using
 infinite projected entangled-pair states (iPEPS) calculations with bond
 dimension X=5.  **Variables:** * Column 1: magnetic field * Column 2: spin
 nematic order parameter **(8)   Fig2b_Szz4.dat** **Description:** Solid
 order parameter using infinite projected entangled-pair states (iPEPS)
 calculations with bond dimension X=4.  **Variables:** * Column 1: magnetic
 field * Column 2: solid order parameter **(9)   Fig2b_Szz5.dat**
 **Description:** Solid order parameter using infinite projected
 entangled-pair states (iPEPS) calculations with bond dimension X=5 
 **Variables:** * Column 1: magnetic field * Column 2: solid order
 parameter **(10)  Fig3a_exp_0T.dat** **Description:** Spin excitation
 spectra of Na~2~BaNi(PO~4~)~2~ in the SN-supersolid state measured at 60mK
 and 0T. The inelastic neutron scattering data collected at 60 mK and 4 T
 were used as the background for subtraction. **Variables:** * Column 1:
 momentum along the high-symmetry directions * Column 2: excited energy *
 Column 3: spin excitation intensity **(11)  Fig3a_iPEPS_0T.dat**
 **Description:** Dynamic spin structure factor of Na~2~BaNi(PO~4~)~2~ in
 the SN-supersolid state using infinite projected entangled-pair states
 (iPEPS) calculations with bond dimension X=4. **Variables:** * Column 1:
 momentum along the high-symmetry directions * Column 2: excited energy *
 Column 3: spin excitation intensity **(12)  Fig3b_exp_0p8T.dat**
 **Description:** Spin excitation spectra of Na~2~BaNi(PO~4~)~2~ in the UUD
 state measured at 60mK and 0.8T. The inelastic neutron scattering data
 collected at 60 mK and 4 T were used as the background for subtraction.
 **Variables:** * Column 1: momentum along the high-symmetry directions *
 Column 2: excited energy * Column 3: spin excitation intensity **(13) 
 Fig3b_iPEPS_0p8T.dat** **Description:** Dynamic spin structure factor of
 Na~2~BaNi(PO~4~)~2~ in the UUD state using infinite projected
 entangled-pair states (iPEPS) calculations with bond dimension X=4.
 **Variables:** * Column 1: momentum along the high-symmetry directions *
 Column 2: excited energy * Column 3: spin excitation intensity **(14) 
 Fig3c_exp_1p72T.dat** **Description:** Spin excitation spectra of
 Na~2~BaNi(PO~4~)~2~ in the spin nematic state measured at 60mK and 1.72T.
 The inelastic neutron scattering data collected at 60 mK and 4 T were used
 as the background for subtraction. **Variables:** * Column 1: momentum
 along the high-symmetry directions * Column 2: excited energy * Column 3:
 spin excitation intensity **(15)  Fig3c_iPEPS_1p72T.dat** **Description:**
 Dynamic spin structure factor of Na~2~BaNi(PO~4~)~2~ in the spin nematic
 state using infinite projected entangled-pair states (iPEPS) calculations
 with bond dimension X=4. **Variables:** * Column 1: momentum along the
 high-symmetry directions * Column 2: excited energy * Column 3: spin
 excitation intensity **(16)  Fig3d_SN_supersolid_LSW_contour_data.dat**
 **Description:** Contour data of dynamic spin structure factor of
 Na~2~BaNi(PO~4~)~2~ in the SN-supersolid state using linear spin wave
 (LSW) calculations method.  **Variables:** * Column 1: momentum along the
 high-symmetry directions * Column 2: excited energy * Column 3: spin
 excitation intensity  **(17) 
 Fig3d_SN_supersolid_LSW_dispersion_data.dat** **Description:** Dispersion
 data of dynamic spin structure factor of Na~2~BaNi(PO~4~)~2~ in the
 SN-supersolid state using linear spin wave (LSW) calculations method. 
 **Variables:** * Column 1: momentum along the high-symmetry directions *
 Column 2: excited energy **(18)  Fig3d_SN_LSW_contour_data.dat**
 **Description:** Contour data of dynamic spin structure factor of
 Na~2~BaNi(PO~4~)~2~ in the spin nematic state using linear spin wave (LSW)
 calculations method.  **Variables:** * Column 1: momentum along the
 high-symmetry directions * Column 2: excited energy * Column 3: spin
 excitation intensity  **(19)  Fig3d_SN_LSW_dispersion_data.dat**
 **Description:** Dispersion data of dynamic spin structure factor of
 Na~2~BaNi(PO~4~)~2~ in the spin nematic state using linear spin wave (LSW)
 calculations method.  **Variables:** * Column 1: momentum along the
 high-symmetry directions * Column 2: excited energy **(20)  Fig3e_0T.dat**
 **Description:** Raw inelastic neutron scattering intensities at the K
 point measured at 60mK and 0T. **Variables:** * Column 1: excited energy *
 Column 2: spin excitation intensity * Column 3: error bar of spin
 excitation intensity **(21)  Fig3e_0p8T.dat** **Description:** Raw
 inelastic neutron scattering intensities at the K point measured at 60mK
 and 0.8T. **Variables:** * Column 1: excited energy * Column 2: spin
 excitation intensity * Column 3: error bar of spin excitation intensity
 **(22)  Fig3e_1p72T.dat** **Description:** Raw inelastic neutron
 scattering intensities at the K point measured at 60mK and 1.72T.
 **Variables:** * Column 1: excited energy * Column 2: spin excitation
 intensity * Column 3: error bar of spin excitation intensity **(23) 
 Fig3e_4T_and_fit.dat** **Description:** Raw inelastic neutron scattering
 intensities at the K point measured at 60mK, 4T and the Gaussian fits to
 the diffusive background. **Variables:** * Column 1: excited energy *
 Column 2: spin excitation intensity * Column 3: error bar of spin
 excitation intensity * Column 4: Gaussian fits to the diffusive background
 **(24)  Fig3f_K_point_0T.dat** **Description:** Raw inelastic neutron
 scattering intensities at the K point measured at 60mK and 0T.
 **Variables:** * Column 1: excited energy * Column 2: spin excitation
 intensity * Column 3: error bar of spin excitation intensity **(25) 
 Fig3f_HH0p05_0T.dat** **Description:** Raw inelastic neutron scattering
 intensities at the (H,H,0)=(0.05,0.05,0) point measured at 60mK and 0T.
 **Variables:** * Column 1: excited energy * Column 2: spin excitation
 intensity * Column 3: error bar of spin excitation intensity **(26) 
 Fig3f_K_point_1p72T.dat** **Description:** Raw inelastic neutron
 scattering intensities at the K point measured at 60mK and 1.72T.
 **Variables:** * Column 1: excited energy * Column 2: spin excitation
 intensity * Column 3: error bar of spin excitation intensity **(27) 
 Fig3f_HH0p05_1p72T.dat** **Description:** Raw inelastic neutron scattering
 intensities at the (H,H,0)=(0.05,0.05,0) point measured at 60mK and 1.72T.
 **Variables:** * Column 1: excited energy * Column 2: spin excitation
 intensity * Column 3: error bar of spin excitation intensity **(28) 
 Fig4a_magnons.dat** **Description:** One-magnon excitations (solid lines)
 and two magnon bound states (dashed lines) at the Γ point calculated by
 generalized linear spin wave (GLSW) theory at 0K. **Variables:** * Column
 1: magnetic field * Column 2: one of branches of one-magnon excitations *
 Column 3: one of branches of one-magnon excitations * Column 4: one of
 branches of one-magnon excitations * Column 5: one of branches of two
 magnon bound states * Column 6: one of branches of two magnon bound states
 **(29)  Fig4a_continuum.dat** **Description:** Two boson continuum
 contribution to the transverse fluctuations at the Γ point calculated by
 generalized linear spin wave (GLSW) theory at 0K. **Variables:** * Column
 1: magnetic field * Column 2: excited energy * Column 3: excitation
 intensity **(30)  Fig4b.dat** **Description:** One-magnon excitations
 (squares) and two magnon bound states (circles) at the Γ point calculated
 by infinite projected entangled-pair states (iPEPS) method with bond
 dimension
 ![](data:image/png;base64,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). **Variables:** * Column 1: magnetic field * Column 2: one of branches of one-magnon excitations * Column 3: one of branches of one-magnon excitations * Column 4: one of branches of one-magnon excitations * Column 5: one of branches of two magnon bound states * Column 6: one of branches of two magnon bound states **(31)  Fig4c.dat** **Description:** DMRG results of the lowest-energy two-magnon bound states calculated on a 9 × 6 TL torus with bond dimension 3500 at 0K. **Variables:** * Column 1: magnetic field * Column 2: one of branches of two magnon bound states * Column 3: one of branches of two magnon bound states **(32)  Fig4d.dat** **Description:** iPEPS results of the dynamic spin structure factor calculated at B=0.8T with bond dimension X=4. **Variables:** * Column 1: momentum along the high-symmetry directions * Column 2: excited energy * Column 3: spin excitation intensity  **(33)  FigS1e.dat** **Description:** Elastic scattering plane (H, K, L=0) collected at T= 60 mK and B= 0 T. **Variables:** * Column 1: momentum along the (H, H, 0) direction * Column 2: momentum along the (-K, K, 0) direction * Column 3: scattering intensity  **(34)  FigS1f.dat** **Description:** A line cut of the diffraction pattern along the (H,H,0) direction around (0,1,0) peak. The blue line represents a Gaussian fit to the experimental data **Variables:** * Column 1: momentum along the (H, H, 0) direction * Column 2: scattering intensity * Column 3: error bar of scattering intensity * Column 4: Gaussian fits of (0,1,0) bragg peak **(35)  FigS2.dat** **Description:** Dynamic spin structure factor of Na~2~BaNi(PO~4~)~2~ at 0K and 2T using infinite projected entangled-pair states (iPEPS) calculations with bond dimension X=4. **Variables:** * Column 1: momentum along the high-symmetry directions * Column 2: excited energy * Column 3: spin excitation intensity **(36)  FigS3a.dat** **Description:** iPEPS results of the total spin excitation spectra at 0 K and 0 T with bond dimension X = 4. **Variables:** * Column 1: momentum along the high-symmetry directions * Column 2: excited energy * Column 3: spin excitation intensity **(37)  FigS3b.dat** **Description:** iPEPS results of the transverse component of spin excitation spectra at 0 K and 0 T with bond dimension X = 4. **Variables:** * Column 1: momentum along the high-symmetry directions * Column 2: excited energy * Column 3: spin excitation intensity **(38)  FigS3c.dat** **Description:** iPEPS results of the longitudinal component of spin excitation spectra at 0 K and 0 T with bond dimension X = 4. **Variables:** * Column 1: momentum along the high-symmetry directions * Column 2: excited energy * Column 3: spin excitation intensity **(39)  FigS3d.dat** **Description:** iPEPS results of the total spin excitation spectra at 0 K and 1.72 T with bond dimension X = 4. **Variables:** * Column 1: momentum along the high-symmetry directions * Column 2: excited energy * Column 3: spin excitation intensity **(40)  FigS3e.dat** **Description:** iPEPS results of the transverse component of spin excitation spectra at 0 K and 1.72 T with bond dimension X = 4. **Variables:** * Column 1: momentum along the high-symmetry directions * Column 2: excited energy * Column 3: spin excitation intensity **(41)  FigS3f.dat** **Description:** iPEPS results of the longitudinal component of spin excitation spectra at 0 K and 1.72 T with bond dimension X = 4. **Variables:** * Column 1: momentum along the high-symmetry directions * Column 2: excited energy * Column 3: spin excitation intensity **(42)  FigS4a.dat** **Description:** iPEPS results of the dynamic quadrupolar structure factors S ^--++ (K, E)^ at 0 K and 0.2 T with bond dimension X = 4. **Variables:** * Column 1: momentum along the high-symmetry directions * Column 2: excited energy * Column 3: spin excitation intensity **(43)  FigS4b.dat** **Description:** iPEPS results of the dynamic quadrupolar structure factors S ^--++ (K, E)^ at 0 K and 0.2 T with bond dimension 𝜒 = 4. **Variables:** * Column 1: momentum along the high-symmetry directions * Column 2: excited energy * Column 3: spin excitation intensity **(44)  FigS5b_exp_1p4T.dat** **Description:** Spin excitation spectra of Na~2~BaNi(PO~4~)~2~ in the UUD state measured at 60mK and 1.4T. The inelastic neutron scattering data collected at 60 mK and 4 T were used as the background for subtraction. **Variables:** * Column 1: momentum along the high-symmetry directions * Column 2: excited energy * Column 3: spin excitation intensity **(45)  FigS5b_iPEPS_1p4T.dat** **Description:** Dynamic spin structure factor of Na~2~BaNi(PO~4~)~2~ at 0 K and 1.4 T using infinite projected entangled-pair states (iPEPS) calculations with bond dimension X=4. **Variables:** * Column 1: momentum along the high-symmetry directions * Column 2: excited energy * Column 3: spin excitation intensity **(46)  FigS6a_exp_0p8T.dat** **Description:** Spin excitation spectra of Na~2~BaNi(PO~4~)~2~ measured at 60mK and 0.8T. The inelastic neutron scattering data collected at 60 mK and 4 T were used as the background for subtraction. **Variables:** * Column 1: momentum along the high-symmetry directions * Column 2: excited energy * Column 3: spin excitation intensity **(47)  FigS6a_exp_1p4T.dat** **Description:** Spin excitation spectra of Na~2~BaNi(PO~4~)~2~ measured at 60mK and 1.4T. The inelastic neutron scattering data collected at 60 mK and 4 T were used as the background for subtraction. **Variables:** * Column 1: momentum along the high-symmetry directions * Column 2: excited energy * Column 3: spin excitation intensity **(48)  FigS6a_exp_1p72T.dat** **Description:** Spin excitation spectra of Na~2~BaNi(PO~4~)~2~ measured at 60mK and 1.72T. The inelastic neutron scattering data collected at 60 mK and 4 T were used as the background for subtraction. **Variables:** * Column 1: momentum along the high-symmetry directions * Column 2: excited energy * Column 3: spin excitation intensity **(49)  FigS6b_contour_data.dat** **Description:** Background-subtracted spin excitation spectra of Na~2~BaNi(PO~4~)~2~ measured at momentum (0.1, 0.1,0) point with magnetic fields 0.8T, 1.4T, 1.72T. **Variables:** * Column 1: magnetic field * Column 2: excited energy * Column 3: spin excitation intensity **(50)  FigS6b_data_points.dat** **Description:** One magnon excitations measured at (0.1, 0.1,0) momentum point with magnetic fields 0.8T, 1.4T, 1.72T. * Column 1: magnetic field * Column 2: excited energy * Column 3: error bar of the fitted excited energy of one magnon **(51)  FigS7_HH_0p05.dat** **Description:** Background-subtracted inelastic neutron scattering intensities measured at (0.05, 0.05,0) momentum point with 60 mK and 0 T. The blue solid lines are the Gaussian fits to the residual incoherent scattering near the elastic line **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity * Column 4: Gaussian fits to the residual incoherent scattering near the elastic line **(52)  FigS7_HH_0p075.dat** **Description:** Background-subtracted inelastic neutron scattering intensities measured at (0.075, 0.075,0) momentum point with 60 mK and 0 T. The blue solid lines are the Gaussian fits to the residual incoherent scattering near the elastic line **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity * Column 4: Gaussian fits to the residual incoherent scattering near the elastic line **(53)  FigS7_HH_0p1.dat** **Description:** Background-subtracted inelastic neutron scattering intensities measured at (0.1, 0.1,0) momentum point with 60 mK and 0 T. The blue solid lines are the Gaussian fits to the residual incoherent scattering near the elastic line **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity * Column 4: Gaussian fits to the residual incoherent scattering near the elastic line **(54)  FigS7_HH_0p125.dat** **Description:** Background-subtracted inelastic neutron scattering intensities measured at (0.125, 0.125,0) momentum point with 60 mK and 0 T. The blue solid lines are the Gaussian fits to the residual incoherent scattering near the elastic line **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity * Column 4: Gaussian fits to the residual incoherent scattering near the elastic line **(55)  FigS7_HH_0p15.dat** **Description:** Background-subtracted inelastic neutron scattering intensities measured at (0.15, 0.15,0) momentum point with 60 mK and 0 T. The blue solid lines are the Gaussian fits to the residual incoherent scattering near the elastic line **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity * Column 4: Gaussian fits to the residual incoherent scattering near the elastic line **(56)  FigS7_HH_0p175.dat** **Description:** Background-subtracted inelastic neutron scattering intensities measured at (0.175, 0.175,0) momentum point with 60 mK and 0 T. The blue solid lines are the Gaussian fits to the residual incoherent scattering near the elastic line **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity * Column 4: Gaussian fits to the residual incoherent scattering near the elastic line **(57)  FigS7_HH_0p2.dat** **Description:** Background-subtracted inelastic neutron scattering intensities measured at (0.2, 0.2,0) momentum point with 60 mK and 0 T. The blue solid lines are the Gaussian fits to the residual incoherent scattering near the elastic line **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity * Column 4: Gaussian fits to the residual incoherent scattering near the elastic line **(58)  FigS7_HH_0p25.dat** **Description:** Background-subtracted inelastic neutron scattering intensities measured at (0.25, 0.25,0) momentum point with 60 mK and 0 T. The blue solid lines are the Gaussian fits to the residual incoherent scattering near the elastic line **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity * Column 4: Gaussian fits to the residual incoherent scattering near the elastic line **(59)  FigS7_HH_0p3.dat** **Description:** Background-subtracted inelastic neutron scattering intensities measured at (0.3, 0.3,0) momentum point with 60 mK and 0 T. The blue solid lines are the Gaussian fits to the residual incoherent scattering near the elastic line **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity * Column 4: Gaussian fits to the residual incoherent scattering near the elastic line **(60)  FigS7_HH_0p35.dat** **Description:** Background-subtracted inelastic neutron scattering intensities measured at (0.35, 0.35,0) momentum point with 60 mK and 0 T. The blue solid lines are the Gaussian fits to the residual incoherent scattering near the elastic line **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity * Column 4: Gaussian fits to the residual incoherent scattering near the elastic line **(61)  FigS7_HH_0p4.dat** **Description:** Background-subtracted inelastic neutron scattering intensities measured at (0.4, 0.4,0) momentum point with 60 mK and 0 T. The blue solid lines are the Gaussian fits to the residual incoherent scattering near the elastic line **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity * Column 4: Gaussian fits to the residual incoherent scattering near the elastic line **(62)  FigS7_HH_0p45.dat** **Description:** Background-subtracted inelastic neutron scattering intensities measured at (0.45, 0.45,0) momentum point with 60 mK and 0 T. The blue solid lines are the Gaussian fits to the residual incoherent scattering near the elastic line **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity * Column 4: Gaussian fits to the residual incoherent scattering near the elastic line **(63)  FigS8_HH_0p05.dat** **Description:** Inelastic neutron scattering intensities at (0.05, 0.05,0) point derived from Fig. S7 after subtracting the fitted residual incoherent scattering (dark blue shaded areas in Fig. S7). The light blue area in the 0.01–0*.*05 meV range indicates spectral weight that mainly arises from the quadrupole wave contribution. **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity **(64)  FigS8_HH_0p075.dat** **Description:** Inelastic neutron scattering intensities at (0.075, 0.075,0) point derived from Fig. S7 after subtracting the fitted residual incoherent scattering (dark blue shaded areas in Fig. S7). The light blue area in the 0.01–0*.*05 meV range indicates spectral weight that mainly arises from the quadrupole wave contribution. **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity **(65)  FigS8_HH_0p1.dat** **Description:** Inelastic neutron scattering intensities at (0.1, 0.1,0) point derived from Fig. S7 after subtracting the fitted residual incoherent scattering (dark blue shaded areas in Fig. S7). The light blue area in the 0.01–0*.*05 meV range indicates spectral weight that mainly arises from the quadrupole wave contribution. **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity **(66)  FigS8_HH_0p125.dat** **Description:** Inelastic neutron scattering intensities at (0.125, 0.125,0) point derived from Fig. S7 after subtracting the fitted residual incoherent scattering (dark blue shaded areas in Fig. S7). The light blue area in the 0.01–0*.*05 meV range indicates spectral weight that mainly arises from the quadrupole wave contribution. **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity **(67)  FigS8_HH_0p15.dat** **Description:** Inelastic neutron scattering intensities at (0.15, 0.15,0) point derived from Fig. S7 after subtracting the fitted residual incoherent scattering (dark blue shaded areas in Fig. S7). The light blue area in the 0.01–0*.*05 meV range indicates spectral weight that mainly arises from the quadrupole wave contribution. **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity **(68)  FigS8_HH_0p175.dat** **Description:** Inelastic neutron scattering intensities at (0.175, 0.175,0) point derived from Fig. S7 after subtracting the fitted residual incoherent scattering (dark blue shaded areas in Fig. S7). The light blue area in the 0.01–0*.*05 meV range indicates spectral weight that mainly arises from the quadrupole wave contribution. **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity **(69)  FigS8_HH_0p2.dat** **Description:** Inelastic neutron scattering intensities at (0.2, 0.2,0) point derived from Fig. S7 after subtracting the fitted residual incoherent scattering (dark blue shaded areas in Fig. S7). The light blue area in the 0.01–0*.*05 meV range indicates spectral weight that mainly arises from the quadrupole wave contribution. **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity **(70)  FigS8_HH_0p25.dat** **Description:** Inelastic neutron scattering intensities at (0.25, 0.25,0) point derived from Fig. S7 after subtracting the fitted residual incoherent scattering (dark blue shaded areas in Fig. S7). The light blue area in the 0.01–0*.*05 meV range indicates spectral weight that mainly arises from the quadrupole wave contribution. **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity **(71)  FigS8_HH_0p3.dat** **Description:** Inelastic neutron scattering intensities at (0.3, 0.3,0) point derived from Fig. S7 after subtracting the fitted residual incoherent scattering (dark blue shaded areas in Fig. S7). The light blue area in the 0.01–0*.*05 meV range indicates spectral weight that mainly arises from the quadrupole wave contribution. **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity **(72)  FigS8_HH_0p35.dat** **Description:** Inelastic neutron scattering intensities at (0.35, 0.35,0) point derived from Fig. S7 after subtracting the fitted residual incoherent scattering (dark blue shaded areas in Fig. S7). The light blue area in the 0.01–0*.*05 meV range indicates spectral weight that mainly arises from the quadrupole wave contribution. **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity **(73)  FigS8_HH_0p4.dat** **Description:** Inelastic neutron scattering intensities at (0.4, 0.4,0) point derived from Fig. S7 after subtracting the fitted residual incoherent scattering (dark blue shaded areas in Fig. S7). The light blue area in the 0.01–0*.*05 meV range indicates spectral weight that mainly arises from the quadrupole wave contribution. **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity **(74)  FigS8_HH_0p45.dat** **Description:** Inelastic neutron scattering intensities at (0.45, 0.45,0) point derived from Fig. S7 after subtracting the fitted residual incoherent scattering (dark blue shaded areas in Fig. S7). The light blue area in the 0.01–0*.*05 meV range indicates spectral weight that mainly arises from the quadrupole wave contribution. **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity **(75)  FigS9_HH_0p05.dat** **Description:** Background-subtracted inelastic neutron scattering intensities measured at (0.05, 0.05,0) momentum point with 60 mK and 1.72 T. The blue solid lines are the Gaussian fits to the residual incoherent scattering near the elastic line **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity * Column 4: Gaussian fits to the residual incoherent scattering near the elastic line **(76)  FigS9_HH_0p075.dat** **Description:** Background-subtracted inelastic neutron scattering intensities measured at (0.075, 0.075,0) momentum point with 60 mK and 1.72 T. The blue solid lines are the Gaussian fits to the residual incoherent scattering near the elastic line **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity * Column 4: Gaussian fits to the residual incoherent scattering near the elastic line **(77)  FigS9_HH_0p1.dat** **Description:** Background-subtracted inelastic neutron scattering intensities measured at (0.1, 0.1,0) momentum point with 60 mK and 1.72 T. The blue solid lines are the Gaussian fits to the residual incoherent scattering near the elastic line **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity * Column 4: Gaussian fits to the residual incoherent scattering near the elastic line **(78)  FigS9_HH_0p125.dat** **Description:** Background-subtracted inelastic neutron scattering intensities measured at (0.125, 0.125,0) momentum point with 60 mK and 1.72 T. The blue solid lines are the Gaussian fits to the residual incoherent scattering near the elastic line **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity * Column 4: Gaussian fits to the residual incoherent scattering near the elastic line **(79)  FigS9_HH_0p15.dat** **Description:** Background-subtracted inelastic neutron scattering intensities measured at (0.15, 0.15,0) momentum point with 60 mK and 1.72 T. The blue solid lines are the Gaussian fits to the residual incoherent scattering near the elastic line **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity * Column 4: Gaussian fits to the residual incoherent scattering near the elastic line **(80)  FigS9_HH_0p175.dat** **Description:** Background-subtracted inelastic neutron scattering intensities measured at (0.175, 0.175,0) momentum point with 60 mK and 1.72 T. The blue solid lines are the Gaussian fits to the residual incoherent scattering near the elastic line **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity * Column 4: Gaussian fits to the residual incoherent scattering near the elastic line **(81)  FigS9_HH_0p2.dat** **Description:** Background-subtracted inelastic neutron scattering intensities measured at (0.2, 0.2,0) momentum point with 60 mK and 1.72 T. The blue solid lines are the Gaussian fits to the residual incoherent scattering near the elastic line **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity * Column 4: Gaussian fits to the residual incoherent scattering near the elastic line **(82)  FigS9_HH_0p25.dat** **Description:** Background-subtracted inelastic neutron scattering intensities measured at (0.25, 0.25,0) momentum point with 60 mK and 1.72 T. The blue solid lines are the Gaussian fits to the residual incoherent scattering near the elastic line **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity * Column 4: Gaussian fits to the residual incoherent scattering near the elastic line **(83)  FigS9_HH_0p3.dat** **Description:** Background-subtracted inelastic neutron scattering intensities measured at (0.3, 0.3,0) momentum point with 60 mK and 1.72 T. The blue solid lines are the Gaussian fits to the residual incoherent scattering near the elastic line **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity * Column 4: Gaussian fits to the residual incoherent scattering near the elastic line **(84)  FigS9_HH_0p35.dat** **Description:** Background-subtracted inelastic neutron scattering intensities measured at (0.35, 0.35,0) momentum point with 60 mK and 1.72 T. The blue solid lines are the Gaussian fits to the residual incoherent scattering near the elastic line **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity * Column 4: Gaussian fits to the residual incoherent scattering near the elastic line **(85)  FigS9_HH_0p4.dat** **Description:** Background-subtracted inelastic neutron scattering intensities measured at (0.4, 0.4,0) momentum point with 60 mK and 1.72 T. The blue solid lines are the Gaussian fits to the residual incoherent scattering near the elastic line **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity * Column 4: Gaussian fits to the residual incoherent scattering near the elastic line **(86)  FigS9_HH_0p45.dat** **Description:** Background-subtracted inelastic neutron scattering intensities measured at (0.45, 0.45,0) momentum point with 60 mK and 1.72 T. The blue solid lines are the Gaussian fits to the residual incoherent scattering near the elastic line **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity * Column 4: Gaussian fits to the residual incoherent scattering near the elastic line **(87)  FigS10_HH_0p05.dat** **Description:** Inelastic neutron scattering intensities at (0.05, 0.05,0) point derived from Fig. S9 after subtracting the fitted residual incoherent scattering (dark blue shaded areas in Fig. S9). The light blue area in the 0.01–0*.*05 meV range indicates spectral weight that mainly arises from the quadrupole wave contribution. **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity **(88)  FigS10_HH_0p075.dat** **Description:** Inelastic neutron scattering intensities at (0.075, 0.075,0) point derived from Fig. S9 after subtracting the fitted residual incoherent scattering (dark blue shaded areas in Fig. S9). The light blue area in the 0.01–0.05 meV range indicates spectral weight that mainly arises from the quadrupole wave contribution. **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity **(89)  FigS10_HH_0p1.dat** **Description:** Inelastic neutron scattering intensities at (0.1, 0.1,0) point derived from Fig. S9 after subtracting the fitted residual incoherent scattering (dark blue shaded areas in Fig. S9). The light blue area in the 0.01–0.05 meV range indicates spectral weight that mainly arises from the quadrupole wave contribution. **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity **(90)  FigS10_HH_0p125.dat** **Description:** Inelastic neutron scattering intensities at (0.125, 0.125,0) point derived from Fig. S9 after subtracting the fitted residual incoherent scattering (dark blue shaded areas in Fig. S9). The light blue area in the 0.01–0.05 meV range indicates spectral weight that mainly arises from the quadrupole wave contribution. **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity **(91)  FigS10_HH_0p15.dat** **Description:** Inelastic neutron scattering intensities at (0.15, 0.15,0) point derived from Fig. S9 after subtracting the fitted residual incoherent scattering (dark blue shaded areas in Fig. S9). The light blue area in the 0.01–0.05 meV range indicates spectral weight that mainly arises from the quadrupole wave contribution. **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity **(92)  FigS10_HH_0p175.dat** **Description:** Inelastic neutron scattering intensities at (0.175, 0.175,0) point derived from Fig. S9 after subtracting the fitted residual incoherent scattering (dark blue shaded areas in Fig. S9). The light blue area in the 0.01–0.05 meV range indicates spectral weight that mainly arises from the quadrupole wave contribution. **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity **(93)  FigS10_HH_0p2.dat** **Description:** Inelastic neutron scattering intensities at (0.2, 0.2,0) point derived from Fig. S9 after subtracting the fitted residual incoherent scattering (dark blue shaded areas in Fig. S9). The light blue area in the 0.01–0.05 meV range indicates spectral weight that mainly arises from the quadrupole wave contribution. **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity **(94)  FigS10_HH_0p25.dat** **Description:** Inelastic neutron scattering intensities at (0.25, 0.25,0) point derived from Fig. S9 after subtracting the fitted residual incoherent scattering (dark blue shaded areas in Fig. S9). The light blue area in the 0.01–0.05 meV range indicates spectral weight that mainly arises from the quadrupole wave contribution. **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity **(95)  FigS10_HH_0p3.dat** **Description:** Inelastic neutron scattering intensities at (0.3, 0.3,0) point derived from Fig. S9 after subtracting the fitted residual incoherent scattering (dark blue shaded areas in Fig. S9). The light blue area in the 0.01–0.05 meV range indicates spectral weight that mainly arises from the quadrupole wave contribution. **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity **(96)  FigS10_HH_0p35.dat** **Description:** Inelastic neutron scattering intensities at (0.35, 0.35,0) point derived from Fig. S9 after subtracting the fitted residual incoherent scattering (dark blue shaded areas in Fig. S9). The light blue area in the 0.01–0.05 meV range indicates spectral weight that mainly arises from the quadrupole wave contribution. **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity **(97)  FigS10_HH_0p4.dat** **Description:** Inelastic neutron scattering intensities at (0.4, 0.4,0) point derived from Fig. S9 after subtracting the fitted residual incoherent scattering (dark blue shaded areas in Fig. S9). The light blue area in the 0.01–0.05 meV range indicates spectral weight that mainly arises from the quadrupole wave contribution. **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity **(98)  FigS10_HH_0p45.dat** **Description:** Inelastic neutron scattering intensities at (0.45, 0.45,0) point derived from Fig. S9 after subtracting the fitted residual incoherent scattering (dark blue shaded areas in Fig. S9). The light blue area in the 0.01–0.05 meV range indicates spectral weight that mainly arises from the quadrupole wave contribution. **Variables:** * Column 1: excited energy * Column 2: spin excitation intensity * Column 3: error bar of spin excitation intensity **(99) FigS11a_blue_points.dat** **Description:** Momentum dependence of the quadrupole wave intensity at 60 mK and 0 T. Blue symbols: integrating the intensity of Figs. S7 in the energy range [0.01, 0.05] meV. **Variables:** * Column 1: momentum along the (H, H, 0) direction * Column 2: quadrupole wave intensity * Column 3: error bar of quadrupole wave intensity **(100) FigS11a_red_points.dat** **Description:** Momentum dependence of the quadrupole wave intensity at 60 mK and 0 T. red symbols: integrating the intensity of Figs. S8 in the energy range [0.01, 0.05] meV. **Variables:** * Column 1: momentum along the (H, H, 0) direction * Column 2: quadrupole wave intensity * Column 3: error bar of quadrupole wave intensity **(101)  FigS11b_blue_points.dat** **Description:** Momentum dependence of the quadrupole wave intensity at 60 mK and 1.72 T. Blue symbols: integrating the intensity of Figs. S9 in the energy range [0.01, 0.05] meV. **Variables:** * Column 1: momentum along the (H, H, 0) direction * Column 2: quadrupole wave intensity * Column 3: error bar of quadrupole wave intensity **(102) FigS11b_red_points.dat** **Description:** Momentum dependence of the quadrupole wave intensity at 60 mK and 1.72 T. red symbols: integrating the intensity of Figs. S10 in the energy range [0.01, 0.05] meV. **Variables:** * Column 1: momentum along the (H, H, 0) direction * Column 2: quadrupole wave intensity * Column 3: error bar of quadrupole wave intensity **Contact** For questions regarding data interpretation or reuse, please contact:\ **Jieming Sheng**\ Great Bay University\ **Email:** [shengjm@gbu.edu.cn](mailto:shengjm@gbu.edu.cn)