Preempting fermion sign problem: Unveiling quantum criticality through nonequilibrium dynamics in imaginary time

Dryad DOI: https://doi.org/10.5061/dryad.dncjsxmch

This repository contains the raw numerical data underlying all figures in the main text and supplementary materials of the paper "Preempting Fermion Sign Problem: Unveiling Quantum Criticality through Nonequilibrium Dynamics in Imaginary Time" (Science Advances; see also arXiv:2410.18854). The dataset is organized to mirror the figure numbering and panel structure used in the manuscript for straightforward cross-referencing and reuse.

Overview

• Scope: Raw numerical data for every plotted figure in the main text and Supplementary Materials (SM).
• Organization: main-text.zip holds main-text figure data; supplementary-materials.zip holds SM data organized by section.
  After unzipping these archives, you obtain the main-text/ and supplementary-materials/ directories described below.
• Formats:
  • .xlsx: Excel tables (most main-text and SM QMC data).
  • .csv: plain text, whitespace-separated columns (selected SM data for the SU(3) Hubbard model).
  • .json: structured records (TEBD/quantum Potts data in SM Sec. 1).
• Conventions:
  • For .xlsx files, most tables follow "one row = one curve" at fixed model parameters, with columns giving the grid of the
    x-axis variable (e.g., imaginary time tau or interaction strength U, V).
  • Files with the _errorbar suffix provide pointwise 1-sigma statistical uncertainties aligned with the companion _data files.
  • .csv files with names of the form (...Y-X).csv contain two columns: x-variable X (first column) and y-variable Y
    (second column).
  • .json files contain explicit field names for all variables; these are described below.

At the top level (after unzipping):

• main-text/: directory with all main-text figure data.
• supplementary-materials/: directory with all Supplementary Materials figure data.

Directory and file contents

main-text/

Fig. 1 - Single-Dirac-fermion Hubbard model (fig1_single-Dirac-fermion-Hubbard-model/)

All files in this folder contain projector QMC data for the single-Dirac-fermion Hubbard model at half filling.

• fig1a(inset)_data.xlsx, fig1a(inset)_errorbar.xlsx
  • Quantity: average Monte Carlo sign sign
  • Axes: sign vs imaginary time tau.
  • Rows correspond to different system sizes L; columns correspond to a grid of tau values.
  • The _errorbar file gives the 1-sigma uncertainty for each entry in the _data file.
• fig1b_data.xlsx, fig1b_errorbar.xlsx
  • Quantity: correlation-length ratio R.
  • Axes: R vs Hubbard interaction strength U.
  • Rows correspond to different system sizes L; columns correspond to a grid of U values.
  • The _errorbar file gives the 1-sigma uncertainty for each R value.
• fig1d_data(S-tau).xlsx, fig1d_errorbar(S-tau).xlsx
  • Quantity: antiferromagnetic structure factor S.
  • Axes: S vs imaginary time tau.
  • Rows correspond to different L; columns correspond to tau grid points.
• fig1e_data(G-tau).xlsx, fig1e_errorbar(G-tau).xlsx
  • Quantity: fermion correlation G.
  • Axes: G vs imaginary time tau.
  • Rows correspond to different L; columns correspond to tau grid points.

Fig. 2 - Spinless t-V model (fig2_spinless-t-V-model/)

All files in this folder contain projector QMC data for the spinless t-V model.

• fig2a_data.xlsx, fig2a_errorbar.xlsx
  • Quantity: correlation-length ratio R.
  • Axes: R vs nearest-neighbor interaction strength V.
  • Rows correspond to different system sizes L; columns correspond to V grid points.
• fig2a(inset)_data.xlsx, fig2a(inset)_errorbar.xlsx
  • Quantity: average Monte Carlo sign sign
  • Axes: sign vs imaginary time tau.
  • Rows correspond to different L; columns correspond to tau grid points.
• fig2c_data(S-tau).xlsx, fig2c_errorbar(S-tau).xlsx
  • Quantity: charge-density-wave structure factor S.
  • Axes: S vs imaginary time tau.
  • Rows correspond to different L; columns correspond to tau grid points.
• fig2d_data(G-tau).xlsx, fig2d_errorbar(G-tau).xlsx
  • Quantity: fermion correlation G.
  • Axes: G vs imaginary time tau.
  • Rows correspond to different L; columns correspond to tau grid points.

Fig. 3 - SU(3) Hubbard model (fig3_SU(3)-Hubbard-model/)

All files in this folder contain projector QMC data for the SU(3) Hubbard model.

• fig3b_data.xlsx, fig3b_errorbar.xlsx
  • Quantity: correlation-length ratio R.
  • Axes: R vs Hubbard interaction strength U.
  • Rows correspond to different system sizes L; columns correspond to U grid points.
  • For large U or large L, some cells are intentionally left empty when the sign problem prevents obtaining reliable
    statistics (see the "Missing values and empty cells" section below).
• fig3b(inset)_data.xlsx, fig3b(inset)_errorbar.xlsx
  • Quantity: average Monte Carlo sign sign
  • Axes: sign vs imaginary time tau.
  • Rows correspond to different L; columns correspond to tau grid points.
• fig3d_data(S-tau).xlsx, fig3d_errorbar(S-tau).xlsx
  • Quantity: lambda_8 antiferromagnetic structure factor S.
  • Axes: S vs imaginary time tau.
  • Rows correspond to different L; columns correspond to tau grid points.
• fig3e_data(G-tau).xlsx, fig3e_errorbar(G-tau).xlsx
  • Quantity: fermion correlation G.
  • Axes: G vs imaginary time tau.
  • Rows correspond to different L; columns correspond to tau grid points.

supplementary-materials/

Sec. 1 - PQMC / quantum Potts model (sec1_PQMC/)

These files contain TEBD simulation data for the quantum Potts model used to support the PQMC analysis.

• figS2a_data.json
  • Top-level structure:
    • parameters: global simulation parameters.
      • L_values: list of system sizes L.
      • g_values: list of transverse-field / coupling values g.
      • tau_over_L: value of tau/L used in this scan.
      • dtau: imaginary-time step size used in the evolution.
    • results: list of data records. Each record contains
      • L: system size.
      • g: coupling value.
      • tau: imaginary time.
      • tau_over_L: ratio tau/L.
      • energy_density: energy per site.
      • M2, M4: second and fourth moments of the order parameter.
      • binder_ratio: Binder cumulant constructed from M2 and M4.
      • chi_max: maximum bond dimension used in the TEBD simulation.
• figS2b_data.json
  • Top-level structure:
    • metadata: describes the grid and input files.
      • L_list: list of system sizes L.
      • g: fixed coupling value used in this data set.
      • tau_over_L_grid: grid of tau/L values.
      • tau_over_L_range: minimum and maximum tau/L.
      • n_points: number of points in the grid.
    • grid_data: list of data for each system size. Each entry contains
      • L: system size.
      • tau_over_L: grid of tau/L values.
      • tau: corresponding imaginary times tau.
      • M: order parameter as a function of tau/L.
      • energy_density: energy per site.
      • chi_max: maximum bond dimension used.

Sec. 2 - Single-Dirac-fermion Hubbard model (sec2_single-Dirac-fermion-Hubbard-model/)

These files contain supplementary projector QMC data for the single-Dirac-fermion Hubbard model.

• figS3a_data(sign-tau).xlsx, figS3a_errorbar(sign-tau).xlsx, figS3b_data(sign-tau).xlsx, figS3b_errorbar(sign-tau).xlsx
  • Quantity: average Monte Carlo sign sign.
  • Axes: sign vs tau.
  • Rows correspond to different system sizes L; columns correspond to tau grid points.
• figS4a_data(R-U).xlsx, figS4a_errorbar(R-U).xlsx, figS4c_data(R-U).xlsx, figS4c_errorbar(R-U).xlsx
  • Quantity: correlation-length ratio R.
  • Axes: R vs U.
  • Rows correspond to different L; columns correspond to U grid points.
• figS5a_data(S-tau).xlsx, figS5a_errorbar(S-tau).xlsx, figS5c_data(G-tau).xlsx, figS5c_errorbar(G-tau).xlsx
  • Quantities: spin structure factor S and fermion correlation G.
  • Axes: S or G vs tau.
  • Rows correspond to different L; columns correspond to tau grid points.
• figS7* files (e.g., figS7a_data(R-U).xlsx, figS7b_data(R-tau).xlsx, etc.)
  • Quantities: dimensionless ratios R as functions of U or tau, for various choices of imaginary time and other
    parameters.
  • Files ending with (R-U) contain R vs U; files ending with (R-tau) contain R vs tau.
  • Rows correspond to different L; columns correspond to either U or tau, as indicated in the filename.

Sec. 3 - Spinless t-V model (sec3_spinless-t-V-model/)

These files contain supplementary projector QMC data for the spinless t-V model.

• figS8a_data(sign-tau).xlsx, figS8a_errorbar(sign-tau).xlsx, figS8b_data(sign-tau).xlsx, figS8b_errorbar(sign-tau).xlsx
  • Quantity: average sign sign.
  • Axes: sign vs tau.
  • Rows correspond to different L; columns correspond to tau grid points.
• figS9a_data(R-V).xlsx, figS9a_errorbar(R-V).xlsx
  • Quantity: correlation-length ratio R.
  • Axes: R vs nearest-neighbor interaction V.
  • Rows correspond to different L; columns correspond to V grid points.
• figS10a_data(S-tau).xlsx, figS10a_errorbar(S-tau).xlsx, figS10c_data(G-tau).xlsx, figS10c_errorbar(G-tau).xlsx
  • Quantities: structure factor S and fermion correlation G.
  • Axes: S or G vs tau.
  • Rows correspond to different L; columns correspond to tau grid points.

Sec. 4 - SU(3) Hubbard model (sec4_SU(3)-Hubbard-model/)

These files contain supplementary data for the SU(3) Hubbard model. Most of the .xlsx files are projector QMC data, while the .csv files in this section come from self-consistent mean-field calculations.

• figS11a_data(m-U).csv and figS11b_data(phi*_m-U).csv
  • Origin: self-consistent mean-field calculations.
  • Quantity: mean-field order parameter m as a function of Hubbard interaction U for different system sizes and flux values phi.
  • Format: two whitespace-separated columns without a header row.
    • Column 1: interaction strength U.
    • Column 2: order parameter m.
  • Filenames of the form phiX_m-U indicate the value of the flux parameter phi used in that data set.
• figS12_data(E1-U).csv, figS12_data(E8-U).csv
  • Origin: self-consistent mean-field calculations.
  • Quantities: ground-state energy densities in different SU(3) channels as functions of U.
    • E1: ground-state energy density in the lambda_1 channel.
    • E8: ground-state energy density in the lambda_8 channel.
  • Format: two-column whitespace-separated text, U (first column) and energy density E1 or E8 (second column).
• figS13*, figS14*, figS15*, figS16*, figS19* (all .xlsx files)
  • figS13* and figS14*: correlation-length ratios R, R1, ..., R7 vs interaction U.
    • Axes: R, R1, ..., R7 vs U, as indicated in the filename (e.g., (R-U), (R1-U), ...).
    • Rows correspond to different L; columns correspond to U grid points.
  • figS15* and figS16*: structure factor S and fermion correlation G vs imaginary time tau, similar to the main-text and
    Sec. 2/3 files with (S-tau) and (G-tau) in the name.
  • figS19a* and figS19b*: average sign as a function of imaginary time or system size.
    • Files with (sign-tau) contain sign vs tau at fixed L.
    • Files with (sign-L) contain sign vs L at fixed tau or other parameters.
    • Rows/columns follow the same "one row = one curve" convention as in the main-text sign files.

Figure mapping and filename convention

Filenames map 1:1 to the paper's figure numbers and subpanels, including insets. Examples:

• main-text/fig1_single-Dirac-fermion-Hubbard-model/fig1b_data.xlsx -> Main-text Fig. 1b.
• main-text/fig1_single-Dirac-fermion-Hubbard-model/fig1a(inset)_data.xlsx -> Main-text Fig. 1a (inset).
• supplementary-materials/sec4_SU(3)-Hubbard-model/figS11a_data(m-U).csv -> SM Fig. S11a.
• supplementary-materials/sec4_SU(3)-Hubbard-model/figS19b_errorbar(sign-L).xlsx -> SM Fig. S19b (error bars).

General naming pattern:

fig[no][panel][(inset)]_[data|errorbar][(Y-X)].ext

• fig[no][panel] for main text (e.g., fig1b), figS[no][panel] for SM (e.g., figS11a).
• (Y-X) encodes the plotted variables (Y vs X), e.g., S-tau, G-tau, R-U, sign-tau, sign-L, m-U, E1-U, E8-U.

Variable names and units

Below we summarize the variables that appear as column names, JSON fields, or in filename suffixes. Units follow the conventions used in the main text and Supplementary Materials; for details and precise definitions, please refer to the published article.

• L: linear system size of the lattice (number of sites along one spatial direction).
• U: on-site interaction strength of the Hubbard model (in units of the hopping amplitude, as in the main text).
• V: nearest-neighbor interaction strength in the spinless t-V model (also in units of the hopping amplitude).
• g: transverse field strength in the quantum Potts model (dimensionless in the coupling J units).
• tau: imaginary time used in the projector / TEBD dynamics (units of inverse energy; same convention as in the paper).
• tau_over_L (tau/L): ratio of imaginary time to system size, used to set the imaginary time in finite-size scaling analyses.
• R, R1, ..., R7: correlation-length ratios constructed from the structure factors; all are dimensionless.
• S: structure factor (spin, charge-density-wave, or lambda_8-AFM depending on the model and context; dimensionless).
• G: fermion correlation; dimensionless in the chosen units.
• m: order parameter (e.g., staggered magnetization or charge-density-wave order parameter); dimensionless.
• sign: average Monte Carlo sign ; dimensionless and bounded between -1 and 1.
• M: order parameter of the quantum Potts model (dimensionless).
• M2, M4: second and fourth moments of M; used to construct the Binder cumulant.
• binder_ratio: Binder cumulant (dimensionless) defined from M2 and M4 (see SM for the precise formula).
• energy_density: energy per site (in units of the model's energy scale, e.g., hopping amplitude).
• chi_max: maximum bond dimension used in the TEBD simulations (dimensionless integer).
• E1, E8: ground-state energy densities obtained from self-consistent mean-field calculations in different SU(3) channels
  (lambda_1 and lambda_8, respectively), given as functions of U.
• phi: flux parameter appearing in filenames such as phi0.1_m-U; it specifies the value of the staggered flux used in the
  SU(3) simulations (in units of pi).

Notes

1. In all .xlsx files, the first row (starting from the second column) stores the x-axis values, and the first column (starting
   from the second row) stores the labels of each curve (e.g., system size L). The top-left cell is therefore empty by design.
   When reading these tables with scripts, please treat the first row and first column as axis metadata rather than data points.

2. In a few files, some table entries are intentionally left empty:

   • main-text/fig3_SU(3)-Hubbard-model/fig3b_data.xlsx
   • main-text/fig3_SU(3)-Hubbard-model/fig3b_errorbar.xlsx
   • supplementary-materials/sec4_SU(3)-Hubbard-model/figS13c_data(R-U).xlsx
   • supplementary-materials/sec4_SU(3)-Hubbard-model/figS13a_errorbar(R-U).xlsx

   These empty cells correspond to parameter points (typically at large U and/or large L) where the fermion sign problem prevented us from obtaining reliable statistics, so the data are simply not available. These gaps are also visible in the corresponding curves in the figures of the paper and occur at parameter values far from the critical points of interest, so they do not affect the physical conclusions. When analyzing these files, please treat empty cells as missing values and ignore them in fits or averages.

How to use the data

• To reproduce a given figure panel, locate the corresponding folder and file using the naming rule described above, then read
  the .xlsx, .csv, or .json file into your preferred analysis environment (e.g., Excel, Python, MATLAB, R).
• For _data/_errorbar pairs, plot the values from the _data file and use the _errorbar entries as 1-sigma error bars.
• For JSON files in sec1_PQMC, use the parameters / metadata blocks to reconstruct the parameter grids, and the results
  or grid_data lists to access the observables.
• When reading .xlsx tables programmatically, remember that the first row and first column encode axis values and curve labels
  rather than data points; see the "Notes" section above.

For more detailed definitions of the observables, model Hamiltonians, and simulation protocols, please refer to the main text and Supplementary Materials of the associated paper.