DOI: 10.5061/dryad.dncjsxmch
title: Preempting fermion sign problem: Unveiling quantum criticality through nonequilibrium dynamics in imaginary time

description: The notorious fermion sign problem, arising from fermion statistics,
 presents a fundamental obstacle to the numerical simulation of quantum
 many-body systems. Here, we introduce an innovative framework that
 circumvents the sign problem in the studies of quantum criticality and the
 associated phases by leveraging imaginary-time nonequilibrium critical
 dynamics. We demonstrate that the critical properties can be accurately
 determined from the system’s short-time relaxation, a regime where the
 sign problem remains manageable for quantum Monte-Carlo (QMC) simulations.
 After validating this approach on two benchmark fermionic models, we apply
 it to the sign-problematic Hubbard model hosting SU(3)-symmetric Dirac
 fermions. We present the first numerically exact characterization of its
 quantum phase diagram, revealing a continuous transition between a Dirac
 semi-metal and a novel antiferromagnetic phase, which belongs to a new
 universality class, distinct from the previously known Gross-Neveu
 transitions. Our work provides a powerful tool for investigating
 sign-problematic systems and quantum criticality.
description: ## 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` -&gt;
 Main-text Fig. 1b. *
 `main-text/fig1_single-Dirac-fermion-Hubbard-model/fig1a(inset)_data.xlsx`
 -&gt; Main-text Fig. 1a (inset). *
 `supplementary-materials/sec4_SU(3)-Hubbard-model/figS11a_data(m-U).csv`
 -&gt; SM Fig. S11a. *
 `supplementary-materials/sec4_SU(3)-Hubbard-model/figS19b_errorbar(sign-L).xlsx` -&gt; 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.