DOI: 10.5061/dryad.z8w9ghxs8
title: Exact kinetic propagators for coherent state complex Langevin simulations

description: We introduce and benchmark an improved algorithm for complex Langevin
 simulations of bosonic coherent state path integrals in the preprint. Our
 approach utilizes a Strang splitting of the imaginary-time propagator
 rather than the conventional linear-order Taylor expansion, allowing us to
 construct an action that incorporates higher-order terms at negligible
 computational cost. The resulting algorithm enjoys guaranteed linear
 stability independent of the imaginary-time discretization, enabling more
 resource-efficient simulations. We demonstrate this improved performance
 for single-species bosons and for two-component bosons with Rashba
 spin-orbit coupling. The enclosed data set shows a method comparison
 between the standard "primitive" approach and the quadratic
 propagator technique used in this manuscript for many thermodynamic
 observables of interest.
description: Both methods involve complex Langevin sampling of a coherent
 state path integral, built from a second-quantized description of
 interacting bosons at finite temperature. Details are provided in the
 enclosed manuscript as well as the following publications: https://doi.org/10.1103/PhysRevLett.131.173403 and https://doi.org/10.1103/PhysRevLett.124.070601
description: # Exact kinetic propagators for coherent state complex Langevin
 simulations This README.txt file was generated on 2025-08-18 by Ethan
 McGarrigle. GENERAL INFORMATION Date of data collection: 2025-07-01
 through 2025-08-01 Geographic location of data collection: University of
 California Santa Barbara, Santa Barbara, California, USA Information about
 funding sources that supported the collection of the data: This work was
 enabled by field-theoretic simulation tools developed under support from
 the National Science Foundation (CMMT Program, DMR-2104255). Use was made
 of computational facilities purchased with funds from the NSF
 (CNS-1725797) and administered by the Center for Scientific Computing
 (CSC). This work made use of the BioPACIFIC Materials Innovation Platform
 computing resources of the National Science Foundation Award No.
 DMR-1933487. The CSC is supported by the California NanoSystems Institute
 and the Materials Research Science and Engineering Center (MRSEC; NSF DMR
 2308708) at UC Santa Barbara. Ethan McGarrigle acknowledges support from a
 Mitsubishi Chemical Fellowship. Thomas Kiely acknowledges support from the
 National Science Foundation under grant PHY-2309135 to the Kavli Institute
 for Theoretical Physics (KITP), and from the Gordon and Betty Moore
 Foundation through Grant GBMF8690 to the University of California, Santa
 Barbara. SHARING/ACCESS INFORMATION Links to publications that cite or use
 the data:
 [https://arxiv.org/abs/2508.11057](https://arxiv.org/abs/2508.11057)
 Recommended citation for this dataset:
 [https://doi.org/10.48550/arXiv.2508.11057](https://doi.org/10.48550/arXiv.2508.11057) DATA & FILE OVERVIEW File list: Files are organized based on figure number from the corresponding manuscript and simulation method used to generate the data. There are two figures. Each figure has the files "FigureX_Primitive_data.csv" and "FigureX_ExactPropagator_data.csv" are found with the same formatting and data. Files labelled "Primitive" contain data using the previous coherent state complex Langevin simulation method, which leveraged a primitive approximation within the path integral derivation. Files labelled "ExactPropagator" leverage the new technique detailed in the corresponding manuscript, where the quadratic portion of the propagator is applied in an exact manner during the path integral derivation. The files are listed here: Figure1_Primitive_data.csv Figure1_ExactPropagator_data.csv Figure2_Primitive_data.csv Figure2_ExactPropagator_data.csv ``` A. Figure1: Imaginary time convergence for a Bose fluid 1. Columns: Imaginary time points (N_{\tau}), average particle number, standard error for particle number, intensive canonical internal energy, standard error for intensive canonical internal energy, intensive free energy, standard error for intensive free energy 2. Filename denotes which path-integral technique was used to generate the data. B. Figure2: Imaginary time convergence for a Rashba spin-orbit coupled Bose fluid 1. Columns: Imaginary time points (N_{\tau}), average particle number for upspin particles, standard error for upspin particle number, intensive canonical internal energy, standard error for intensive canonical internal energy, intensive free energy, standard error for intensive free energy 2. Filename denotes which path-integral technique was used to generate the data. ``` METHODOLOGICAL INFORMATION Description of methods used for collection/generation of data: Some data were generated using an improved coherent state complex Langevin method, detailed in our preprint: [https://arxiv.org/abs/2508.11057](https://arxiv.org/abs/2508.11057) . Some data were generated using previous coherent state complex Langevin simulation methods, developed in the following publications: [https://doi.org/10.1103/PhysRevLett.131.173403](https://doi.org/10.1103/PhysRevLett.131.173403) [https://doi.org/10.1103/PhysRevLett.124.070601](https://doi.org/10.1103/PhysRevLett.124.070601) Methods for processing the data: Thermodynamic quantities were computed using field-operator functionals during the simulation, producing samples at some printing frequency. Statistics were taken for each thermodynamic quantity after an initial method equilibration period, producing a mean and a standard error of the mean, which are plotted in Figures 1 and 2. Describe any quality-assurance procedures performed on the data: After the simulation code was tested thoroughly to ensure that it agreed with known thermodynamic results, we generated each data set twice to ensure the results were consistent and statistically significant. People involved with sample collection, processing, analysis and/or submission: Ethan McGarrigle DATA-SPECIFIC INFORMATION FOR: Figure1_ExactPropagator_data.csv 1. Number of variables: 7 2. Number of cases/rows: 33 3. Variable List: Column 1: Ntau, Number of imaginary time points used Column 2: Average particle number, units = number of particles Column 3: Standard error of the mean for the particle number, units = number of particles Column 4: Dimensionless intensive canonical internal energy, units = dimensionless intensive energy per system area Column 5: Standard error of the mean for the intensive canonical internal energy, units = dimensionless energy per system area Column 6: Dimensionless intensive free energy, units = dimensionless intensive free energy per system area Column 7: Standard error of the mean for the dimensionless intensive free energy, units = dimensionless intensive free energy per system area 4. Missing data codes: nan: "Not a number", referring to simulations that diverged (field values blow up to infinite values, which cannot be represented meaningfully with a floating point number. The occurrence of "nan" represents conditions where the simulation technique fails. 5. Specialized formats or other abbreviations used: N/A 6. Notes: Data here utilized the exact quadratic propagator method that was introduced in this manuscript for coherent state, complex Langevin simulations of a one-component Bose fluid in two dimensions. DATA-SPECIFIC INFORMATION FOR: Figure1_Primitive_data.csv 1. Number of variables: 7 2. Number of cases/rows: 33 3. Variable List: Column 1: Ntau, Number of imaginary time points used Column 2: Average particle number, units = number of particles Column 3: Standard error of the mean for the particle number, units = number of particles Column 4: Dimensionless intensive canonical internal energy, units = dimensionless intensive energy per system area Column 5: Standard error of the mean for the intensive canonical internal energy, units = dimensionless energy per system area Column 6: Dimensionless intensive free energy, units = dimensionless intensive free energy per system area Column 7: Standard error of the mean for the dimensionless intensive free energy, units = dimensionless intensive free energy per system area 4. Missing data codes: nan: "Not a number", referring to simulations that diverged (field values blow up to infinite values, which cannot be represented meaningfully with a floating point number. Occurrence of "nan" represent conditions where the simulation technique fails. 5. Specialized formats or other abbreviations used: N/A 6. Notes: Data here utilized the primitive propagator method for coherent state, complex Langevin simulations of a one-component Bose fluid in two dimensions. DATA-SPECIFIC INFORMATION FOR: Figure2_ExactPropagator_data.csv 1. Number of variables: 7 2. Number of cases/rows: 40 3. Variable List: Column 1: Ntau, Number of imaginary time points used Column 2: Average up-spin particle number, units = number of particles Column 3: Standard error of the mean for the up-spin particle number, units = number of particles Column 4: Dimensionless intensive canonical internal energy, units = dimensionless intensive energy per system area Column 5: Standard error of the mean for the intensive canonical internal energy, units = dimensionless energy per system area Column 6: Dimensionless intensive free energy, units = dimensionless intensive free energy per system area Column 7: Standard error of the mean for the dimensionless intensive free energy, units = dimensionless intensive free energy per system area 4. Missing data codes: nan: "Not a number", referring to simulations that diverged (field values blow up to infinite values, which cannot be represented meaningfully with a floating point number. Occurrence of "nan" represent conditions where the simulation technique fails. 5. Specialized formats or other abbreviations used: N/A 6. Notes: Data here utilized the exact quadratic propagator method that was introduced in this manuscript for coherent state, complex Langevin simulations of a two-component Bose fluid with Rashba spin-orbit coupling in two dimensions. DATA-SPECIFIC INFORMATION FOR: Figure2_Primitive_data.csv 1. Number of variables: 7 2. Number of cases/rows: 40 3. Variable List: Column 1: Ntau, Number of imaginary time points used Column 2: Average up-spin particle number, units = number of particles Column 3: Standard error of the mean for the up-spin particle number, units = number of particles Column 4: Dimensionless intensive canonical internal energy, units = dimensionless intensive energy per system area Column 5: Standard error of the mean for the intensive canonical internal energy, units = dimensionless energy per system area Column 6: Dimensionless intensive free energy, units = dimensionless intensive free energy per system area Column 7: Standard error of the mean for the dimensionless intensive free energy, units = dimensionless intensive free energy per system area 4. Missing data codes: nan: "Not a number", referring to simulations that diverged (field values blow up to infinite values, which cannot be represented meaningfully with a floating point number. Occurrence of "nan" represent conditions where the simulation technique fails. 5. Specialized formats or other abbreviations used: N/A 6. Notes: Data here utilized the primitive propagator method for coherent state, complex Langevin simulations of a two-component Bose fluid with Rashba spin-orbit coupling in two dimensions.