DOI: 10.5061/dryad.q83bk3jts
title: Spectroscopy of elementary excitations from quench dynamics in a dipolar XY Rydberg simulator

description: We use a Rydberg quantum simulator to demonstrate a new form of
 spectroscopy, called quench spectroscopy, which probes the low-energy
 excitations of a many-body system. We illustrate the method on a
 two-dimensional simulation of the spin-1/2 dipolar XY model. Through
 micro- scopic measurements of the spatial spin correlation dynamics
 following a quench, we extract the dispersion relation of the elementary
 excitations for both ferro- and anti-ferromagnetic cou- plings. We observe
 qualitatively different behaviors between the two cases that result from
 the long-range nature of the interactions, and the frustration inherent in
 the antiferromagnet. In particular, the ferromagnet exhibits elementary
 excitations behaving as linear spin waves. In the anti-ferromagnet, spin
 waves appear to decay, suggesting the presence of strong nonlinear- ities.
 Our demonstration highlights the importance of power-law interactions on
 the excitation spectrum of a many-body system.
description: # Spectroscopy of elementary excitations from quench dynamics in a dipolar
 XY Rydberg simulator Dataset DOI:
 [https://doi.org/10.5061/dryad.q83bk3jts](&lt;
 https://doi.org/10.5061/dryad.q83bk3jts&gt;) ## Description of the data
 and file structure These files contain all the data for the figures in the
 article, including Figure 1.xlsx, Figure 2.xlsx, Figure 3.xlsx and Figure
 4.xlsx. #### Data Organization in Excel Files: For clarity and
 consistency, data in Excel files (e.g., Figure 1.xlsx) are structured to
 align with figures in the paper. Each file contains multiple labeled
 sheets (e.g., panel C, panel D), with distinct naming conventions to
 differentiate data types: * Experimental data: Sheets or variables include
 the label "data" (e.g., *panel E_data_(nx,ny)=(1,0)*). This
 denotes experimental results for panel E of Figure 1 under the condition
 (nx, ny) = (1, 0). - Theoretical methods: Sheets or variables use labels
 like "tVMC" (time-dependent variational Monte Carlo) or
 "LSW OBC" (linear spin-wave theory with open boundary
 conditions). Example: In the sheet *panel E_data_(nx,ny)=(1,0)*, the
 Variables include: * Time(hbar/J)_data: Evolution time for experimental
 data. * Structure factor: Measured structure factors. * lower_errors and
 upper_errors: Error bounds for plotting. #### Consistency Across Files:
 All Excel files follow this labeling logic: Sheet names map directly to
 figure panels (e.g., panel X corresponds to panel X in the paper).
 Variable names (e.g., Time, Structure factor, errors) correspond to axis
 or colorbar labels in the figures. Method-specific labels (e.g., tVMC, LSW
 OBC) distinguish theoretical predictions from experimental results. This
 structured approach ensures alignment between the data sheets and their
 graphical representations in the paper. ### Files and variables ####
 Figure 1.xlsx\panel C This data sheet provides the data of space-time
 dynamics of the spin correlations following the quench into the coherent
 spin state (FM case, Figure 1C). Variable list: * "data: d
 (sites)": distances between the sites of the experimental data in the
 upper left plot of Figure 1C; The data range spans from A2 to A51. *
 "data: Corr_zz": zz correlations of the experimental data in the
 upper left plot of Figure 1C; The data range spans from cell C2 to AQ51. *
 "Time (hbar/J)": Evolution time of the experimental data in the
 upper left plot of Figure 1C; The data range spans from C53 to AQ53. *
 "tVMC: d (sites)": distances between the sites of the tVMC data
 in the lower left plot of Figure 1C; The data range spans from A60 to
 A109. * "tVMC: Corr_zz": zz correlations of the tVMC data in the
 lower left plot of Figure 1C; The data range spans from cell C60 to BN109.
 * "Time (hbar/J)": Evolution time of the tVMC data in the lower
 left plot of Figure 1C; The data range spans from C111 to BN111. ####
 Figure 1.xlsx\panel D This data sheet provides the data of space-time
 dynamics of the spin correlations following the quench into the staggered
 coherent spin state (AFM case, Figure 1D). Variable list: * "data: d
 (sites)": distances between the sites of the experimental data in the
 upper plot of Figure 1D; The data range spans from A2 to A51. *
 "data: Corr_zz": zz correlations of the experimental data in the
 upper plot of Figure 1D; The data range spans from cell C2 to AI51. *
 "Time (hbar/J)": Evolution time of the experimental data in the
 upper plot of Figure 1D; The data range spans from C53 to AI53. *
 "tVMC: d (sites)": distances between the sites for the tVMC data
 in the lower right plot of Figure 1D; The data range spans from A60 to
 A77. * "tVMC: Corr_zz": zz correlations of the tVMC data in the
 lower right plot of Figure 1D; The data range spans from cell C60 to AQ77.
 * "Time (hbar/J)": Evolution time of the tVMC data in the lower
 right of Figure 1D; The data range spans from C79 to AQ79. #### Figure
 1.xlsx\panel E\_data\_(nx,ny)=(1,0) #### Figure 1.xlsx\panel
 E\_data\_(nx,ny)=(3,0) #### Figure 1.xlsx\panel E\_data\_(nx,ny)=(3,3)
 #### Figure 1.xlsx\panel E\_data\_(nx,ny)=(5,1) #### Figure 1.xlsx\panel
 E\_data\_(nx,ny)=(5,5) These sheets provide experimental data of structure
 factor dynamics in the ferromagnetic case with different (nx,
 ny). Variable list: * "Time(hbar/J)_data":  Evolution time of
 the experimental data in Figure 1E; The data range spans from A2 to A44. *
 "Structure factor": Structure factor of the experimental data in
 Figure 1E; The data range spans from B2 to B44. *
 "lower_errors": lower bound of the error bar for experimental
 structure factor in Figure 1E; The data range spans from C2 to C44. *
 "upper_errors": upper bound of the error bar for experimental
 structure factor in Figure 1E; The data range spans from D2 to D44. ####
 Figure 1.xlsx\panel E\_LSW\_OBC\_(nx,ny)=(1,0) #### Figure 1.xlsx\panel
 E\_LSW\_OBC\_(nx,ny)=(3,0) #### Figure 1.xlsx\panel
 E\_LSW\_OBC\_(nx,ny)=(3,3) #### Figure 1.xlsx\panel
 E\_LSW\_OBC\_(nx,ny)=(5,1) #### Figure 1.xlsx\panel
 E\_LSW\_OBC\_(nx,ny)=(5,5) These sheets provide the structure factor
 dynamics of linear spin-wave theory with open boundary conditions in the
 ferromagnetic case with different (nx, ny). Variable list: *
 "Time(hbar/J)_LSW_OBC":  Evolution time of the LSW OBC data in
 Figure 1E; The data range spans from A2 to A631. * "Structure
 factor": Structure factor of the LSW OBC data in Figure 1E; The data
 range spans from B2 to B631. #### Figure 1.xlsx\panel
 E\_tVMC\_(nx,ny)=(1,0) #### Figure 1.xlsx\panel E\_tVMC\_(nx,ny)=(3,0)
 #### Figure 1.xlsx\panel E\_tVMC\_(nx,ny)=(3,3) #### Figure 1.xlsx\panel
 E\_tVMC\_(nx,ny)=(5,1) #### Figure 1.xlsx\panel E\_tVMC\_(nx,ny)=(5,5)
 These sheets provide the structure factor dynamics of time-dependent
 variational Monte Carlo in the ferromagnetic case with different (nx,
 ny). Variable list: * "Time(hbar/J)_tVMC":  Evolution time of
 the tVMC data in Figure 1E; The data range spans from A2 to A63. *
 "Structure factor": Structure factor of the tVMC data in Figure
 1E; The data range spans from B2 to B63. #### Figure 1.xlsx\panel
 F\_data\_(nx,ny)=(1,0) #### Figure 1.xlsx\panel F\_data\_(nx,ny)=(3,0)
 #### Figure 1.xlsx\panel F\_data\_(nx,ny)=(3,3) #### Figure 1.xlsx\panel
 F\_data\_(nx,ny)=(5,1) #### Figure 1.xlsx\panel F\_data\_(nx,ny)=(5,5)
 These sheets provide experimental data of structure factor dynamics in the
 antiferromagnetic case with different (nx, ny). Variable list: *
 "Time(hbar/J)_data":  Evolution time of the experimental data in
 Figure 1F; The data range spans from A2 to A52. * "Structure
 factor": Structure factor of the experimental data in Figure 1F; The
 data range spans from B2 to B52. * "lower_errors": lower bound
 of the error bar for experimental structure factor in Figure 1F; The data
 range spans from C2 to C52. * "upper_errors": upper bound of the
 error bar for experimental structure factor in Figure 1F; The data range
 spans from D2 to D52. #### Figure 1.xlsx\panel F\_LSW\_OBC\_(nx,ny)=(1,0)
 #### Figure 1.xlsx\panel F\_LSW\_OBC\_(nx,ny)=(3,0) #### Figure
 1.xlsx\panel F\_LSW\_OBC\_(nx,ny)=(3,3) #### Figure 1.xlsx\panel
 F\_LSW\_OBC\_(nx,ny)=(5,1) #### Figure 1.xlsx\panel
 F\_LSW\_OBC\_(nx,ny)=(5,5) These sheets provide the structure factor
 dynamics of linear spin-wave theory with open boundary conditions in the
 antiferromagnetic case with different (nx, ny). Variable list: *
 "Time(hbar/J)_LSW_OBC":  Evolution time of the LSW OBC data in
 Figure 1F; The data range spans from A2 to A1601. * "Structure
 factor": Structure factor of the LSW OBC data in Figure 1F; The data
 range spans from B2 to B1601. #### Figure 1.xlsx\panel
 F\_tVMC\_(nx,ny)=(1,0) #### Figure 1.xlsx\panel F\_tVMC\_(nx,ny)=(3,0)
 #### Figure 1.xlsx\panel F\_tVMC\_(nx,ny)=(3,3) #### Figure 1.xlsx\panel
 F\_tVMC\_(nx,ny)=(5,1) #### Figure 1.xlsx\panel F\_tVMC\_(nx,ny)=(5,5)
 These sheets provide the structure factor dynamics of time-dependent
 variational Monte Carlo in the antiferromagnetic case with different (nx,
 ny). Variable list: * "Time(hbar/J)_tVMC":  Evolution time of
 the tVMC data in Figure 1F; The data range spans from A2 to A21. *
 "Structure factor": Structure factor of the tVMC data in Figure
 1F; The data range spans from B2 to B21. #### Figure 2.xlsx\panel A\_data
 This data sheet provide the data of dispersion relation from the
 experiment in the ferromagnetic case. Variable list: *
 "k": modulus of wave-vector. The data range spans from A2 to
 A18. * "hbar*omega/J": Frequency in the unit of interaction
 J. The data range spans from B2 to B18. * "error bars": Error
 bars. The data range spans from C2 to C18. #### Figure 2.xlsx\panel
 A\_tVMC This data sheet provide the data of dispersion relation from the
 tVMC in the ferromagnetic case. Variable list: * "k": modulus of
 wave-vector. The data range spans from A2 to A22. *
 "hbar*omega/J": Frequency in the unit of interaction J. The data
 range spans from B2 to B22. * "error bars": Error bars. The data
 range spans from C2 to C22. #### Figure 2.xlsx\panel A\_LSW\_OBC This data
 sheet provide the data of dispersion relation from the LSW OBC in the
 ferromagnetic case. Variable list: * "k": modulus of
 wave-vector. The data range spans from A2 to A21. *
 "hbar*omega/J": Frequency in the unit of interaction J. The data
 range spans from B2 to B21. * "error bars": Error bars. The data
 range spans from C2 to C21. #### Figure 2.xlsx\panel A\_LSW\_mPBC This
 data sheet provide the data of dispersion relation from the LSW mPBC in
 the ferromagnetic case. Variable list: * "k": modulus of
 wave-vector. The data range spans from A2 to A101. *
 "hbar*omega/J": Frequency in the unit of interaction J. The data
 range spans from B2 to B101. #### Figure 2.xlsx\panel B\_data This data
 sheet provide the data of dispersion relation from the experiment in the
 antiferromagnetic case. Variable list: * "k": modulus of
 wave-vector. The data range spans from A2 to A18. *
 "hbar*omega/J": Frequency in the unit of interaction J. The data
 range spans from B2 to B18. * "error bars": Error bars. The data
 range spans from C2 to C18. #### Figure 2.xlsx\panel B\_tVMC This data
 sheet provide the data of dispersion relation from the tVMC in the
 antiferromagnetic case. Variable list: * "k": modulus of
 wave-vector. The data range spans from A2 to A22. *
 "hbar*omega/J": Frequency in the unit of interaction J. The data
 range spans from B2 to B22. * "error bars": Error bars. The data
 range spans from C2 to C22. #### Figure 2.xlsx\panel B\_LSW\_OBC This data
 sheet provide the data of dispersion relation from the LSW OBC in the
 antiferromagnetic case. Variable list: * "k": modulus of
 wave-vector. The data range spans from A2 to A21. *
 "hbar*omega/J": Frequency in the unit of interaction J. The data
 range spans from B2 to B21. * "error bars": Error bars. The data
 range spans from C2 to C21. #### Figure 2.xlsx\panel B\_LSW\_mPBC This
 data sheet provide the data of dispersion relation from the LSW mPBC in
 the antiferromagnetic case. Variable list: * "k": modulus of
 wave-vector. The data range spans from A2 to A101. *
 "hbar*omega/J": Frequency in the unit of interaction J. The data
 range spans from B2 to B101. #### Figure 2.xlsx\panel C\_data This data
 sheet provide the data of dispersion relation from the experiment in the
 ferromagnetic case. Variable list: * "k": modulus of
 wave-vector. The data range spans from A2 to A18. *
 "gamma*sqrt(2C-1)": Frequency obtained from the offset Ck. The
 data range spans from B2 to B18. * "error bars": Error bars. The
 data range spans from C2 to C18. #### Figure 2.xlsx\panel C\_tVMC This
 data sheet provide the data of dispersion relation from the tVMC in the
 ferromagnetic case. Variable list: * "k": modulus of
 wave-vector. The data range spans from A2 to A22. *
 "gamma*sqrt(2C-1)": Frequency obtained from the offset Ck. The
 data range spans from B2 to B22. * "error bars": Error bars. The
 data range spans from C2 to C22. #### Figure 2.xlsx\panel C\_LSW\_OBC This
 data sheet provide the data of dispersion relation from the LSW OBC in the
 ferromagnetic case. Variable list: * "k": modulus of
 wave-vector. The data range spans from A2 to A21. *
 "gamma*sqrt(2C-1)": Frequency obtained from the offset Ck. The
 data range spans from B2 to B21. * "error bars": Error bars. The
 data range spans from C2 to C21. #### Figure 2.xlsx\panel C\_LSW\_mPBC
 This data sheet provide the data of dispersion relation from the LSW mPBC
 in the ferromagnetic case. Variable list: * "k": modulus of
 wave-vector. The data range spans from A2 to A101. *
 "gamma*sqrt(2C-1)": Frequency obtained from the offset Ck. The
 data range spans from B2 to B101. #### Figure 2.xlsx\panel D\_data This
 data sheet provide the data of dispersion relation from the experiment in
 the antiferromagnetic case. Variable list: * "k": modulus of
 wave-vector. The data range spans from A2 to A18. *
 "gamma*sqrt(2C-1)": Frequency obtained from the offset Ck. The
 data range spans from B2 to B18. * "error bars": Error bars. The
 data range spans from C2 to C18. #### Figure 2.xlsx\panel D\_tVMC This
 data sheet provide the data of dispersion relation from the tVMC in the
 antiferromagnetic case. Variable list: * "k": modulus of
 wave-vector. The data range spans from A2 to A22. *
 "gamma*sqrt(2C-1)": Frequency obtained from the offset Ck. The
 data range spans from B2 to B22. * "error bars": Error bars. The
 data range spans from C2 to C22. #### Figure 2.xlsx\panel D\_LSW\_OBC This
 data sheet provide the data of dispersion relation from the LSW OBC in the
 antiferromagnetic case. Variable list: * "k": modulus of
 wave-vector. The data range spans from A2 to A21. *
 "gamma*sqrt(2C-1)": Frequency obtained from the offset Ck. The
 data range spans from B2 to B21. * "error bars": Error bars. The
 data range spans from C2 to C21. #### Figure 2.xlsx\panel D\_LSW\_mPBC
 This data sheet provide the data of dispersion relation from the LSW mPBC
 in the antiferromagnetic case. Variable list: * "k": modulus of
 wave-vector. The data range spans from A2 to A101. *
 "gamma*sqrt(2C-1)": Frequency obtained from the offset Ck. The
 data range spans from B2 to B101. #### Figure 3.xlsx\panel
 A\_data\_(nx,ny)=(0,1) #### Figure 3.xlsx\panel A\_data\_(nx,ny)=(0,2)
 #### Figure 3.xlsx\panel A\_data\_(nx,ny)=(1,1) #### Figure 3.xlsx\panel
 A\_data\_(nx,ny)=(1,2) #### Figure 3.xlsx\panel A\_data\_(nx,ny)=(2,2)
 These sheets provide experimental data of structure factor dynamics in the
 ferromagnetic 4 by 4 case with different (nx, ny). Variable list: *
 "Time(hbar/J)_data":  Evolution time of the experimental data in
 Figure 3A; The data range spans from A2 to A102. * "Structure
 factor": Structure factor of the experimental data in Figure 3A; The
 data range spans from B2 to B102. * "lower_errors": lower bound
 of the error bar for experimental structure factor in Figure 3A; The data
 range spans from C2 to C102. * "upper_errors": upper bound of
 the error bar for experimental structure factor in Figure 3A; The data
 range spans from D2 to D102. #### Figure 3.xlsx\panel A\_ED\_(nx,ny)=(0,1)
 #### Figure 3.xlsx\panel A\_ED\_(nx,ny)=(0,2) #### Figure 3.xlsx\panel
 A\_ED\_(nx,ny)=(1,1) #### Figure 3.xlsx\panel A\_ED\_(nx,ny)=(1,2) ####
 Figure 3.xlsx\panel A\_ED\_(nx,ny)=(2,2) These sheets provide the results
 of structure factor dynamics from exact diagonalization simulations in the
 ferromagnetic 4 by 4 case with different (nx, ny). Variable list: *
 "Time(hbar/J)_ED":  Evolution time of the exact diagonalization
 data in Figure 3A; The data range spans from A2 to A201. * "Structure
 factor": Structure factor of the exact diagonalization data in Figure
 3A; The data range spans from B2 to B201. #### Figure 3.xlsx\panel
 A\_ED\_err\_(nx,ny)=(0,1) #### Figure 3.xlsx\panel
 A\_ED\_err\_(nx,ny)=(0,2) #### Figure 3.xlsx\panel
 A\_ED\_err\_(nx,ny)=(1,1) #### Figure 3.xlsx\panel
 A\_ED\_err\_(nx,ny)=(1,2) #### Figure 3.xlsx\panel
 A\_ED\_err\_(nx,ny)=(2,2) These sheets provide the results of structure
 factor dynamics from exact diagonalization simulations with imperfections
 taken into account in the ferromagnetic 4 by 4 case with different (nx,
 ny). Variable list: *
 "Time(hbar/J)_ED_imperfections":  Evolution time of the results
 from exact diagonalization simulations with imperfections taken into
 account in Figure 3A; The data range spans from A2 to A597. *
 "lower_bound": lower bound of the shadow area for the exact
 diagonalization simulations with imperfections taken into account in
 Figure 3A; The data range spans from B2 to B597. *
 "upper_bound": upper bound of the shadow area for the exact
 diagonalization simulations with imperfections taken into account in
 Figure 3A; The data range spans from C2 to C597. #### Figure 3.xlsx\panel
 B\_data\_(nx,ny)=(0,1) #### Figure 3.xlsx\panel B\_data\_(nx,ny)=(0,2)
 #### Figure 3.xlsx\panel B\_data\_(nx,ny)=(1,1) #### Figure 3.xlsx\panel
 B\_data\_(nx,ny)=(1,2) #### Figure 3.xlsx\panel B\_data\_(nx,ny)=(2,2)
 These sheets provide experimental data of structure factor dynamics in the
 antiferromagnetic 4 by 4 case with different (nx, ny). Variable list: *
 "Time(hbar/J)_data":  Evolution time of the experimental data in
 Figure 3B; The data range spans from A2 to A101. * "Structure
 factor": Structure factor of the experimental data in Figure 3B; The
 data range spans from B2 to B101. * "lower_errors": lower bound
 of the error bar for experimental structure factor in Figure 3B; The data
 range spans from C2 to C101. * "upper_errors": upper bound of
 the error bar for experimental structure factor in Figure 3B; The data
 range spans from D2 to D101. #### Figure 3.xlsx\panel B\_ED\_(nx,ny)=(0,1)
 #### Figure 3.xlsx\panel B\_ED\_(nx,ny)=(0,2) #### Figure 3.xlsx\panel
 B\_ED\_(nx,ny)=(1,1) #### Figure 3.xlsx\panel B\_ED\_(nx,ny)=(1,2) ####
 Figure 3.xlsx\panel B\_ED\_(nx,ny)=(2,2) These sheets provide the results
 of structure factor dynamics from exact diagonalization simulations in the
 antiferromagnetic 4 by 4 case with different (nx, ny). Variable list: *
 "Time(hbar/J)_ED":  Evolution time of the exact diagonalization
 data in Figure 3B; The data range spans from A2 to A201. * "Structure
 factor": Structure factor of the exact diagonalization data in Figure
 3B; The data range spans from B2 to B201. #### Figure 3.xlsx\panel
 B\_ED\_err\_(nx,ny)=(0,1) #### Figure 3.xlsx\panel
 B\_ED\_err\_(nx,ny)=(0,2) #### Figure 3.xlsx\panel
 B\_ED\_err\_(nx,ny)=(1,1) #### Figure 3.xlsx\panel
 B\_ED\_err\_(nx,ny)=(1,2) #### Figure 3.xlsx\panel
 B\_ED\_err\_(nx,ny)=(2,2) These sheets provide the results of structure
 factor dynamics from exact diagonalization simulations with imperfections
 taken into account in the antiferromagnetic 4 by 4 case with different
 (nx, ny). Variable list: *
 "Time(hbar/J)_ED_imperfections":  Evolution time of the results
 from exact diagonalization simulations with imperfections taken into
 account in Figure 3B; The data range spans from A2 to A567. *
 "lower_bound": lower bound of the shadow area for the exact
 diagonalization simulations with imperfections taken into account in
 Figure 3B; The data range spans from B2 to B567. *
 "upper_bound": upper bound of the shadow area for the exact
 diagonalization simulations with imperfections taken into account in
 Figure 3B; The data range spans from C2 to C567. #### Figure 4.xlsx\panel
 A This data sheet provides the data of time-averaged structure factor of
 the experimental data (FM case, Figure 4A). Variable list: * "kx
 ": The data range spans from C14 to M14. * "ky ": The data
 range spans from cell A2 to A12. * "Time-averaged structure factor
 ": Time-averaged structure factor of the experimental data in Figure
 4A; The data range spans from C2 to M12. #### Figure 4.xlsx\panel B This
 data sheet provides the structure factor of the Quantum Monte Carlo
 simulations (FM case, Figure 4B). Variable list: * "kx ": The
 data range spans from C14 to M14. * "ky ": The data range spans
 from cell A2 to A12. * "Time-averaged structure factor
 ": Time-averaged structure factor of the Quantum Monte Carlo (QMC)
 simulations in Figure 4B; The data range spans from C2 to M12. #### Figure
 4.xlsx\panel C This data sheet provides the time-averaged structure factor
 along the path connecting the high-symmetry points of the Brillouin zone
 (FM case, Figure 4C). Variable list: * "k ": The wave-vector.
 The data range spans from A3 to A16. * "data":
 "Time-averaged SF": Structure factor of the experimental data.
 The data range spans from cell C3 to C16. * "data": "error
 bars": error bars of the experimental data. The data range spans from
 cell D3 to D16. * "tVMC": "Time-averaged
 SF": Structure factor of the tVMC data. The data range spans from
 cell F3 to F16. * "QMC ": "Time-averaged
 SF": Structure factor of the QMC data. The data range spans from cell
 H3 to H16. * "QMC ": "error bars": error bar of the
 QMC data. The data range spans from cell I3 to I16. #### Figure
 4.xlsx\panel D This data sheet provides the data of time-averaged
 structure factor of the experimental data (AFM case, Figure 4D). Variable
 list: * "kx ": The data range spans from C14 to M14. * "ky
 ": The data range spans from cell A2 to A12. * "Time-averaged
 structure factor ": Time-averaged structure factor of the
 experimental data in Figure 4D; The data range spans from C2 to M12. ####
 Figure 4.xlsx\panel E This data sheet provides the structure factor of the
 Quantum Monte Carlo simulations (AFM NN case, Figure 4E). Variable list: *
 "kx ": The data range spans from C14 to M14. * "ky ":
 The data range spans from cell A2 to A12. * "Time-averaged structure
 factor ": Time-averaged structure factor of the Quantum Monte Carlo
 (QMC) simulations with only nearest-neighbor coupling in Figure 4E; The
 data range spans from C2 to M12. #### Figure 4.xlsx\panel F This data
 sheet provides the time-averaged structure factor along the path
 connecting the high-symmetry points of the Brillouin zone (AFM case,
 Figure 4F). Variable list: * "k ": The wave-vector. The data
 range spans from A3 to A16. * "data": "Time-averaged
 SF": Structure factor of the experimental data. The data range spans
 from cell C3 to C16. * "data": "error bars": error
 bars of the experimental data. The data range spans from cell D3 to D16. *
 "tVMC": "Time-averaged SF": Structure factor of the
 tVMC data . The data range spans from cell F3 to F16. * "LSW":
 "Time-averaged SF": Structure factor of the LSW data. The data
 range spans from cell H3 to H16. * "QMC  NN":
 "Time-averaged SF": Structure factor of the QMC data. The data
 range spans from cell J3 to J16. * "QMC NN ": "error
 bars": error bar of the QMC data. The data range spans from cell K3
 to K16. ## Code/software You can use the following software to view the
 data: (1) Python 3.12.3 + pandas 2.2.2 (2) LibreOffice (3) Microsoft
 Excel.