Conventions in the data for the LLR Note some of the conventions in the data differ from those used in the paper: : Bennett, E., Lucini, B., Mason, D., Piai, M., Rinaldi, E., & Vadacchino, D. (2024). The density of states method for symplectic gauge theories at finite temperature. arXiv preprint arXiv:2409.19426. k is used to denote the energy interval n is the RM iteration a_k is defined to be negative a_k=-|a_k|, as opposed to positive in the paper a_k=|a_k| V = (\tilde{V}/a^4) Lattice volume, i.e. the number of lattice sites Nt*Ns*Ns*Ns, Nt number of temporal sites, Ns number of spatial sites The 'energy' in the data is defined as 6*V* average plaquette, whereas in the paper it is defined as 6*V*(1-average plaquette) ---------------------------------------------------------------------------------- ---------------------------------------------------------------------------------- IS CSV files Results from HiRep PureGauge when the input is input_file (within the same folder), giving the output output_file (within the same folder). Denoted data/IS_{Nt}x{Ns}/ where Nt and Ns are the size in the temporal and spatial directions The file full.csv is for each beta value in the folders IS_{Nt}x{Ns}/{beta}/ where beta is the beta value for that ensemble Within these folders are also the input and output files of the from HiRep called input_file and output_file respectively The files hist.csv and std.csv are combined results for all beta values and are located in IS_{Nt}x{Ns}/CSV/ ---------------------------------------------------------------------------------- full.csv Contains measurements of the average plaquette and Polyakov loop for each configuration from importance sampling One entry for each configuration measured Beta - the coupling value for this simulation Plaq - the measured value of the average plaquette for a given configuration Poly - the measured value of the absolute value of the Polyakov loop for a given configuration V - lattice volume, Nt*Ns^3 Lt - lattice size in the temporal direction (Nt) ---------------------------------------------------------------------------------- hist.csv Contains the histogram of the measured average plaquette values for all beta values One entry for each bin of each histogram at each beta value Beta - the coupling value for this simulation Hist - the height of the histogram for a given average plaquette bin of the histogram Bins - Centre of the average plaquette bin for the histogram ---------------------------------------------------------------------------------- std.csv Contains measurements and errors of the plaquette, specific heat, Polyakov loop and Polyakov loop susceptibility for different beta values from importance sampling results One entry for each beta value Beta - the coupling value for this simulation Plaq - the VEV of the average plaquette for a given beta value Plaq_err - the error on the VEV of the average plaquette for a given beta value Plaq_SH - the specific heat for a given beta value Plaq_SH_err - the error on the specific heat for a given beta value Poly - the VEV of the absolute value of the Polyakov loop for a given beta value Poly_err - the error on the VEV of the absolute value of the Polyakov loop for a given beta value Poly_sus - the Polyakov loop susceptibility for a given beta value Poly_sus_err - the error on the Polyakov loop susceptibility for a given beta value Plaq_binder - binder cumulant of the average plaquette for a given beta value Plaq_binder_err - the error on the binder cumulant of the average plaquette for a given beta value V - lattice volume, Nt*Ns^3 Lt - lattice size in the temporal direction (Nt) ---------------------------------------------------------------------------------- ---------------------------------------------------------------------------------- LLR input file the folder contains the initial an and Ek values for that specific ensemble in the file data/LLR_{Nt}x{NS}_{NRep}/pre.dat where Nt and Ns are the size in the temporal and spatial directions and NRep is the number of replicas with the format " E0 a0 dE E1 a1 dE ... ENrep aNrep dE " where En is the centre of the nth interval, an is the initial ak value for the centre of the interval and dE is the initial interval size ---------------------------------------------------------------------------------- ---------------------------------------------------------------------------------- LLR CSV files at finite DE for each repeat Within Folders Denoted data/LLR_{Nt}x{NS}_{NRep}/{n}/CSV/ where Nt and Ns are the size in the temporal and spatial directions and NRep is the number of replicas, and n = 0,...,NRepeats, where NRepeats is the number of repeats in some cases critical is used instead of NRep, this happens when there are two ensembles with the same number of replicas but different energy ranges, the ensemble with the smaller energy range is referred to as critical there is results for each repeat for that lattice and interval size containing the files RM.csv, final.csv, fa.csv, obs.csv, obs_critical.csv, comparison.csv, DG.csv, F.csv ---------------------------------------------------------------------------------- RM.csv Contains information from the RM iterations Has an entry for each interval at each RM iteration n - RM iteration a - the a_k^n value, note the notation in the data is a_k^n=-|a_k^n|, this differs from the paper which has a_k^n=|a_k^n|, n is the RM iteration and k is the energy intervals index Ek - Centre of the energy interval, note in the data E = 6V*up, rather than E = 6V*(1-up) as defined in the paper dE - E_k - E_{k-1} V - lattice volume, Nt*Ns^3 S - average plaquette at the end of the RM iteration Rep - Replica index Lt - lattice size in the temporal direction (Nt) ------------------------------------------------------------------------------------- final.csv Contains final values of a_k and E_k for each replica after finishing the RM iterations Has an entry for each interval a - the final a_k value Ek - Centre of the energy interval, note in the data E = 6V*up, rather than E = 6V*(1-up) as defined in the paper dE - E_k - E_{k-1} V - lattice volume, Nt*Ns^3 Lt - lattice size in the temporal direction (Nt) -------------------------------------------------------------------------------------- fa.csv (fixed a) Contains information from the fixed a iterations after the final a_k are found Has an entry for each interval for each fixed a iteration a - the final a_k value Ek - Centre of the energy interval, note in the data E = 6V*up, rather than E = 6V*(1-up) as defined in the paper S - measured average plaquette for this fixed a update V - lattice volume, Nt*Ns^3 Rep - Replica index Poly - measure value of the absolute value of the Polyakov loop |l_p| for a fixed a_k iteration PolySqr - measure value of the absolute value of the Polyakov loop squared |l_p|^2 for a fixed a_k iteration n - index of the configuration measured ------------------------------------------------------------------------------------------ obs.csv Contains reconstructed observables for a range of beta values b - \beta value u - reconstructed average plaquette value Cu - specific heat, variance of the average plaquette Bv - binder cumulant of the average plaquette lp - Polyakov loop Xlp - Polyakov loop susceptibility Blp - Unused V - lattice volume, Nt*Ns^3 Lt - lattice size in the temporal direction (Nt) ------------------------------------------------------------------------------------------ obs_critical.csv Contains reconstructed observables for a range of beta values focused on the critical region b - \beta value u - reconstructed average plaquette value Cu - specific heat, variance of the average plaquette Bv - binder cumulant of the average plaquette lp - Polyakov loop, not calculated 0 Xlp - Polyakov loop susceptibility, not calculated 0 Blp - Unused, not calculated 0 V - lattice volume, Nt*Ns^3 Lt - lattice size in the temporal direction (Nt) ------------------------------------------------------------------------------------------ comparison.csv Contains reconstructed observables at\beta values with direct comparison to standard importance sampling method b - \beta value u - reconstructed average plaquette value Cu - specific heat, variance of the average plaquette Bv - binder cumulant of the average plaquette lp - Polyakov loop Xlp - Polyakov loop susceptibility Blp - Unused V - lattice volume, Nt*Ns^3 Lt - lattice size in the temporal direction (Nt) ------------------------------------------------------------------------------------------ DG.csv Contains results for critical observables found by fitting double Gaussians to the plaquette distributions Bc- critical \beta value lh - 6*V*plaquette jump at the critical point dE - E_k - E_{k-1} V - lattice volume, Nt*Ns^3 Lt - lattice size in the temporal direction (Nt) dP - difference between the heights of the two maxima of the probability distribution N - 1 The remaining parameters are the coefficients for the fitted double Gaussian, A1,M1,S1,A2,M2,S2 y = A1 * exp(-(x- M1)^2 / (2*(S1^2))) + A2 * exp(-(x- M2)^2 / (2*(S2^2))) Where y is the probability distribution and x is the energy ------------------------------------------------------------------------------------------ F.csv shows results for critical observables found using the free energy Tc - the reciprocal of critical coupling found using the crossing point of the free energy tc = 1/ b_c Fcmin - free energy at the critical point on the meta-stable branch Fcmax - free energy at the critical point on the unstable-stable branch F - free energy at the centre of each interval T - micro-canonical temperature at the centre of each interval S - entropy at the centre of each interval U - internal energy at the centre of each interval F_int - free energy interpolated in critical region T_int - micro-canonical temperature interpolated in the critical region S_int - entropy interpolated in the critical region U_int - internal energy interpolated in the critical region up- - (lower) plaquette value corresponding to the maxima of the plaquette distribution at the critical point up+ - (upper) plaquette value corresponding to the maxima of the plaquette distribution at the critical point ind_c - interpolated indices corresponding to locations in the critical region E_Pb - plaquette values for the plaquette distribution given in Pb at the critical point Pb - plaquette distribution values for plaquette values given by E_Pb at the critical point ---------------------------------------------------------------------------------- ---------------------------------------------------------------------------------- LLR CSV files at finite DE combined for all repeats Within Folders Denoted data/LLR_{Nt}x{NS}_{NRep}/CSV/ where Nt and Ns are the size in the temporal and spatial directions and NRep is the number of replicas, in some cases critical is used instead of NRep, this happens when there are two ensembles with the same number of replicas but different energy ranges, the ensemble with the smaller energy range is referred to as critical there is results for a given lattice and interval size combined for all repeats containing the files final.csv, obs.csv, obs_critical.csv, comparison.csv, DG.csv, F.csv, critical.csv, info.csv ------------------------------------------------------------------------------------------ info.csv Information about results in this folder n_repeats - number of repeats n_replicas - number of replicas N_nr - Number of Newton-Raphson steps V - - lattice volume, Nt*Ns^3 Lt - lattice size in the temporal direction (Nt) Ls - lattice size in the spatial direction (Ns) umin - sets location of the centre of the first energy interval umax - sets location of the centre of the final energy interval IS_b - beta values for IS results to be used when finding initial a_n values betas - beta values for obs.csv betas_critical - beta values for obs_critical.csv dg_db - tuneable parameter for finding the critical point using the double gaussian dg_tol - tolerance for finding the critical point using the double gaussian num_samples - number of bootstrap samples error_type - type of error used for analysis std_files - location of the corresponding importance sampling results std_folder - location of folder containing the corresponding importance sampling results seed - random seed for analysis dE - interval size E_k - E_{k-1} ----------------------------------------------------------------------------------------- final.csv Contains final values of a_k and E_k for each replica for all repeats Has an entry for each interval a - the final a_k value a_err - the error on the final a_k value Ek - Centre of the energy interval, note in the data E = 6V*up, rather than E = 6V*(1-up) as defined in the paper dE - E_k - E_{k-1} V - lattice volume, Nt*Ns^3 Lt - lattice size in the temporal direction (Nt) ------------------------------------------------------------------------------------------ obs.csv Contains reconstructed observables for a range of beta values b - \beta value u - reconstructed average plaquette value u_err - the error on reconstructed average plaquette value Cu - specific heat, variance of the average plaquette Cu_err - the error on specific heat, variance of the average plaquette Bv - binder cumulant of the average plaquette Bv_err - the error on binder cumulant of the average plaquette lp - Polyakov loop lp_err - the error on Polyakov loop Xlp - Polyakov loop susceptibility Xlp_err - the error on Polyakov loop susceptibility ------------------------------------------------------------------------------------------ obs_critical.csv Contains reconstructed observables for a range of beta values focused on the critical region b - \beta value u - reconstructed average plaquette value u_err - the error on reconstructed average plaquette value Cu - specific heat, variance of the average plaquette Cu_err - the error on specific heat, variance of the average plaquette Bv - binder cumulant of the average plaquette Bv_err - the error on binder cumulant of the average plaquette lp - Polyakov loop lp_err - the error on Polyakov loop Xlp - Polyakov loop susceptibility Xlp_err - the error on Polyakov loop susceptibility ------------------------------------------------------------------------------------------ comparison.csv Contains reconstructed observables at\beta values with direct comparison to standard importance sampling method b - \beta value u - reconstructed average plaquette value u_err - the error on reconstructed average plaquette value Cu - specific heat, variance of the average plaquette Cu_err - the error on specific heat, variance of the average plaquette Bv - binder cumulant of the average plaquette Bv_err - the error on binder cumulant of the average plaquette lp - Polyakov loop lp_err - the error on Polyakov loop Xlp - Polyakov loop susceptibility Xlp_err - the error on Polyakov loop susceptibility ------------------------------------------------------------------------------------------ DG.csv Contains results for critical observables found by fitting double Gaussians to the plaquette distributions bc- critical \beta value bc_err- the error on the critical \beta value du - plaquette jump at the critical point du_err - the error on plaquette jump at the critical point dP - difference between the heights of the two maxima of the probability distribution dP_err - the error on difference between the heights of the two maxima of the probability distribution ------------------------------------------------------------------------------------------ F.csv shows results for critical observables found using the free energy Tc - the reciprocal of critical coupling found using the crossing point of the free energy tc = 1/ b_c Tc - the error on the reciprocal of critical coupling found using the crossing point of the free energy tc = 1/ b_c dF - the free energy jump at the critical point dF_err - the error on the free energy jump at the critical point Fcmax - free energy at the critical point on the unstable-stable branch F - free energy at the centre of each interval T - micro-canonical temperature at the centre of each interval S - entropy at the centre of each interval U - internal energy at the centre of each interval F_err - the error on the free energy at the centre of each interval T_err - the error on the micro-canonical temperature at the centre of each interval S_err - the error on the entropy at the centre of each interval U_err - the error on the internal energy at the centre of each interval F_int - free energy interpolated in critical region T_int - micro-canonical temperature interpolated in the critical region S_int - entropy interpolated in the critical region U_int - internal energy interpolated in the critical region up- - (lower) plaquette value corresponding to the maxima of the plaquette distribution at the critical point up+ - (upper) plaquette value corresponding to the maxima of the plaquette distribution at the critical point up-_err - the error on the (lower) plaquette value corresponding to the maxima of the plaquette distribution at the critical point up+_err - the error on the (upper) plaquette value corresponding to the maxima of the plaquette distribution at the critical point du - the plaquette jump at the critical point du_err - the error on the plaquette jump at the critical point ind_c - interpolated indices corresponding to locations in the critical region E_Pb - plaquette values for the plaquette distribution given in Pb at the critical point Pb - plaquette distribution values for plaquette values given by E_Pb at the critical point Pb_err - the error on the plaquette distribution values for plaquette values given by E_Pb at the critical point Sigma - the interface tension term Sigma_err - the error on the interface tension term ------------------------------------------------------------------------------------------ critical.csv contains the results for the critical betas based on the peaks of observables and the corresponding value of the observables b_Xlp - beta corresponding to the peak of the Polyakov loop susceptibility b_Xlp_err - the error on the beta corresponding to the peak of the Polyakov loop susceptibility Xlp - the value of the peak of the Polyakov loop susceptibiltiy Xlp_err - the error on the value of the peak of the Polyakov loop susceptibility b_Cu - beta corresponding to the peak of the specific heat b_Cu_err - the error on the beta corresponding to the peak of the specific heat Cu - the value of the peak of the specific heat Cu_err - the error on the value of the peak of the specific heat b_Bv - beta corresponding to the peak of the binder cumulant b_Bv_err - the error on the beta corresponding to the peak of the binder cumulant Bv - the value of the peak of the binder cumulant Bv_err - the error on the value of the peak of the binder cumulant ---------------------------------------------------------------------------------- ---------------------------------------------------------------------------------- LLR CSV files for results extrapolated to the vanishing interval size limit Within Folders Denoted data/LLR_{Nt}x{NS}/CSV/ where Nt and Ns are the size in the temporal and spatial directions and NRep is the number of replicas, there is results for that lattice and interval size combined for all repeats containing the files final.csv, obs.csv, obs_critical.csv, comparison.csv, DG.csv, F.csv, critical.csv, info.csv ------------------------------------------------------------------------------------------ F.csv shows results for critical observables found using the free energy Tc - the reciprocal of critical coupling found using the crossing point of the free energy tc = 1/ b_c Tc - the error on the reciprocal of critical coupling found using the crossing point of the free energy tc = 1/ b_c du - the plaquette jump at the critical point du_err - the error on the plaquette jump at the critical point Sigma - the interface tension term Sigma_err - the error on the interface tension term ------------------------------------------------------------------------------------------ critical.csv contains the results for the critical betas based on the peaks of observables and the corresponding value of the observables b_Xlp - beta corresponding to the peak of the Polyakov loop susceptibility b_Xlp_err - the error on the beta corresponding to the peak of the Polyakov loop susceptibility Xlp - the value of the peak of the Polyakov loop susceptibiltiy Xlp_err - the error on the value of the peak of the Polyakov loop susceptibiltiy b_Cu - beta corresponding to the peak of the specific heat b_Cu_err - the error on the beta corresponding to the peak of the specific heat Cu - the value of the peak of the specific heat Cu_err - the error on the value of the peak of the specific heat b_Bv - beta corresponding to the peak of the binder cumulant b_Bv_err - the error on the beta corresponding to the peak of the binder cumulant Bv - the value of the peak of the binder cumulant Bv_err - the error on the value of the peak of the binder cumulant ----------------------------------------------------------------------------------------- info.csv Information about results in this folder V - - lattice volume, Nt*Ns^3 Lt - lattice size in the temporal direction (Nt) Ls - lattice size in the spatial direction (Ns) dE - this just equal 0, its used to avoid key errors ----------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------- Summary data Generated in the data_products directory, these files contain combined data from a selection of the above CSV files for all ensembles. ---------------------------------------------------------------------------------- info_combined.csv contains all input metadata for all LLR ensembles. n_repeats - number of repeats n_replicas - number of replicas N_nr - Number of Newton-Raphson steps V - - lattice volume, Nt*Ns^3 Lt - lattice size in the temporal direction (Nt) Ls - lattice size in the spatial direction (Ns) umin - sets location of the centre of the first energy interval umax - sets location of the centre of the final energy interval IS_b - beta values for IS results to be used when finding initial a_n values betas - beta values for obs.csv betas_critical - beta values for obs_critical.csv dg_db - tuneable parameter for finding the critical point using the double gaussian dg_tol - tolerance for finding the critical point using the double gaussian num_samples - number of bootstrap samples error_type - type of error used for analysis std_files - location of the corresponding importance sampling results std_folder - location of folder containing the corresponding importance sampling results seed - random seed for analysis dE - interval size E_k - E_{k-1} ---------------------------------------------------------------------------------- critical_combined.csv contains results for critical observables found using the free energy, the results for the critical betas based on the peaks of observables and the corresponding value of the observables, and associated metadata, for Nt=4 ensembles and their extrapolations. n_repeats - number of repeats n_replicas - number of replicas N_nr - Number of Newton-Raphson steps V - - lattice volume, Nt*Ns^3 Lt - lattice size in the temporal direction (Nt) Ls - lattice size in the spatial direction (Ns) umin - sets location of the centre of the first energy interval umax - sets location of the centre of the final energy interval IS_b - beta values for IS results to be used when finding initial a_n values betas - beta values for obs.csv betas_critical - beta values for obs_critical.csv dg_db - tuneable parameter for finding the critical point using the double gaussian dg_tol - tolerance for finding the critical point using the double gaussian num_samples - number of bootstrap samples error_type - type of error used for analysis std_files - location of the corresponding importance sampling results std_folder - location of folder containing the corresponding importance sampling results seed - random seed for analysis dE - interval size E_k - E_{k-1} Tc - the reciprocal of critical coupling found using the crossing point of the free energy tc = 1/ b_c Fcmin - free energy at the critical point on the meta-stable branch Fcmax - free energy at the critical point on the unstable-stable branch F - free energy at the centre of each interval T - micro-canonical temperature at the centre of each interval S - entropy at the centre of each interval U - internal energy at the centre of each interval F_int - free energy interpolated in critical region T_int - micro-canonical temperature interpolated in the critical region S_int - entropy interpolated in the critical region U_int - internal energy interpolated in the critical region up- - (lower) plaquette value corresponding to the maxima of the plaquette distribution at the critical point up+ - (upper) plaquette value corresponding to the maxima of the plaquette distribution at the critical point ind_c - interpolated indices corresponding to locations in the critical region E_Pb - plaquette values for the plaquette distribution given in Pb at the critical point Pb - plaquette distribution values for plaquette values given by E_Pb at the critical point b_Xlp - beta corresponding to the peak of the Polyakov loop susceptibility b_Xlp_err - the error on the beta corresponding to the peak of the Polyakov loop susceptibility Xlp - the value of the peak of the Polyakov loop susceptibiltiy Xlp_err - the error on the value of the peak of the Polyakov loop susceptibiltiy b_Cu - beta corresponding to the peak of the specific heat b_Cu_err - the error on the beta corresponding to the peak of the specific heat Cu - the value of the peak of the specific heat Cu_err - the error on the value of the peak of the specific heat b_Bv - beta corresponding to the peak of the binder cumulant b_Bv_err - the error on the beta corresponding to the peak of the binder cumulant Bv - the value of the peak of the binder cumulant Bv_err - the error on the value of the peak of the binder cumulant b_F - beta corresponding to equality of the free energy between the metastable branches b_F_err - the error on the beta corresponding to equality of the free energy between the metastable branches ---------------------------------------------------------------------------------- infvol.csv contains the results for the critical observables and couplings in the thermodynamic limit b_Xlp - beta corresponding to the peak of the Polyakov loop susceptibility b_Xlp_err - the stochastic error on the beta corresponding to the peak of the Polyakov loop susceptibility b_Xlp_comb_err - an estimate of the total error including systematic error on the beta corresponding to the peak of the Polyakov loop susceptibility Xlp - the value of the peak of the Polyakov loop susceptibiltiy Xlp_err - the stochastic error on the value of the peak of the Polyakov loop susceptibiltiy Xlp_comb_err - an estimate of the total error including systematic error on the value of the peak of the Polyakov loop susceptibility b_Cu - beta corresponding to the peak of the specific heat b_Cu_err - the stochastic error on the beta corresponding to the peak of the specific heat b_Cu_comb_err - an estimate of the total error including systematic error on the beta corresponding to the peak of the specific heat Cu - the value of the peak of the specific heat Cu_err - the stochastic error on the value of the peak of the specific heat Cu_comb_err - an estimate of the total error including systematic error on the value of the peak of the specific heat b_Bv - beta corresponding to the peak of the binder cumulant b_Bv_err - the stochastic error on the beta corresponding to the minima of the binder cumulant b_Bv_comb_err - an estimate of the total error including systematic error on the beta corresponding to the minima of the binder cumulant Bv - the value of the peak of the Binder cumulant Bv_err - the stochastic error on the value of the peak of the Binder cumulant Bv_comb_err - an estimate of the total error including systematic error on the value of the peak of the Binder cumulant b_F - beta corresponding to equality of the free energy between the metastable branches b_F_err - the stochastic error on the beta corresponding to equality of the free energy between the metastable branches b_F_comb_err - an estimate of the total error including systematic error on the beta corresponding to equality of the free energy between the metastable branches du - the plaquette jump at the critical point du_err - the stochastic error on the plaquette jump at the critical point du_comb_err - an estimate of the total error including systematic error on the value of the plaquette jump at the critical point Sigma - the interface tension term Sigma_err - the stochastic error on the interface tension term Sigma_comb_err - an estimate of the total error including systematic error on the value of the interface tension term