ΩMaxEnt user guide - Département de physique
Contents
1. 25 2 11 Spectrum at sample frequencies vs amp aooo a 26 212 Output ales o E A es re BY 27 2 13 When is the calculation over 1 22 e4 4 25 244 30 2 14 How to improve the results 4 24 4e6 fe0 254 ls 31 List of input parameters 32 3 1 OPTIONAL PREPROCESSING TIME PARAMETERS 32 3 1 1 DATA PARAMETERS o 32 3 1 2 INPUT FILES PARAMETERS 35 3 1 3 FREQUENCY GRID PARAMETERS 37 3 1 4 COMPUTATION OPTIONS 38 3 1 5 PREPROCESSING EXECUTION OPTIONS 39 3 2 OPTIONAL MINIMIZATION TIME PARAMETERS 40 3 2 1 OUTPUT FILES PARAMETERS 40 3 2 2 COMPUTATION PARAMETERS 41 3 2 3 MINIMIZATION EXECUTION OPTIONS 42 32 4 DISPLAY OPTIONS A jala 42 A How the program estimates the width of a low energy peak 43 B Other parameters 44 1 Introduction 1 1 Overview This is the user guide for the program 2MaxEnt a tool to perform the analytic contin uation of Matsubara data using a maximum entropy aproach It can treat fermionic or bosonic data given as a function of Matsubara frequency or imaginary time with either a diagonal or general covariance matrix For the fermionic case the output is a positive spectral function A w and for the bosonic case it is the function o w A w w also positive A specific feature of QMaxEnt is the method it uses to find the optimal value of the entropy weight pa2. small so that it barely affects the result As will be mentioned in section 2 12 the spectra are saved for a certain range of a around the optimal one Therefore you can also choose yourself the value of a that you find the most reasonable from the output figures and use the spectrum saved in the corre sponding file instead of the one saved in optimal_spectral_function_tem _alpha dat see section 2 12 If for some reason the value of a chosen by the program is not the one you think is optimal for example if there are more than one peak in the curvature log x vs log a in the region where the optimal alpha should be located you can also choose the region where the program will search for the optimal a with the parameters maximum optimal alpha and minimum optimal alpha in subsection COMPUTATION PARAMETERS When the error standard deviation or covariance is known precisely the crossover region is narrow and it is clear what is the optimal value This is because in that ideal case the ratio of AG and the error tends to stay comparable at different w or covariance eigenvector index 2 as decreases and thus the noise in Gin starts to be fitted around the same value of a at all frequencies wp te On the other hand if the error is not well estimated the crossover region where both noise and information are fitted at the same time becomes wider The choice of optimal a is then more arbitrary In that
3. spectral function sample frequencies Use this parameter to choose which sample frequencies of the spectrum will be plotted as a function of a See section 2 11 for a discussion about that figure 40 3 2 2 COMPUTATION PARAMETERS e initial value of alpha Use this parameter only if you do not see a plateau at high a in the figure of log x versus log a Most of the time the value determined by the program is correct If not set this parameter to a value a few orders of magnitudes larger than the largest value in the figure and increase it the same way until you see the plateau e minimum value of alpha Use this parameter to adjust the width of the noise fitting region namely the quasi plateau at low a in log x versus log q if necessary This region should cover at least a few decades to make sure that the calculation is really over but not too many since it is located below the optimal a and thus useless to the results themselves The program is designed to make sure that this region is wide enough so this parameter might be useful mostly to reduce the width of the noise fitting region e maximum optimal alpha Use this parameter if the program does not choose the correct peak in the curvature of log x versus log a namely the peak located in the crossover region between the information and noise fitting ranges of a See subsection 2 9 for a discussion about this topic e minimum optimal alpha Use this
4. sponding to those moments The moments are also used during the definition of the real frequency grid and the default model since they tell us about global properties of the spectrum among which the center and the width of the spectrum 13 2 4 1 Use QMaxEnt to extract moments If the Matsubara frequency cutoff w of your data is in the asymptotic part of G iw namely that the condition Wna gt W is satisfied the program can extract moments from your data by fitting a high frequency expansion to the asymptotic part and determine the onset of the asymptotic regime at the same time Unless you provide enough moments or real frequency grid parameters QMaxEnt will try to extract them and use them in the calculation instead of the high frequencies they are extracted from When it performs that procedure if you set the parameter display preprocessing figures to yes in sub section PREPROCESSING EXECUTION OPTIONS you can check for convergence in the computation of the moments from the figures of the norm Mp and the first moment M plotted as a function of the starting frequency of the fit If there is a plateau in those curves above a certain frequency then the values are converged In any case you can force the program to extract the moments by setting evaluate moments to yes in subsection COMPUTATION OPTIONS You can also control how many moments are used during the computation of the spectrum with the pa
5. with the first column being the Matsubara frequency wn when the input data is G iw By default the next two columns are assumed to be the real and imaginary parts of G respectively Wn Re G Im G 2 1 If G is not on the 2 and 3 columns write the correct column numbers on the lines Re G column in data file and Im G column in data file in the subsection INPUT FILES PARAMETERS 2 2 2 Temperature If the Matsubara frequencies do not have high precision in your data file or if the first col umn gives only the frequency indices set the parameter temperature in the subsection DATA PARAMETERS 2 2 3 G T as input For a Green function given in imaginary time G 7 first set the parameter imaginary time data to yes in the subsection DATA PARAMETERS By default the columns in the data file are assumed to be T G 2 2 If G is not on the 2 column put the correct column number on the line column of G tau in data file in subsection INPUT FILES PARAMETERS Note that the imaginary time grid must be uniform and go from 0 to 1 T where T is the temperature If the temperature is specified in the subsection DATA PARAMETERS the imaginary time grid will be redefined using the provided temperature Otherwise T is assumed to be 1 Tmax Where Tmax is the last value of 7 in the input file To get good results if your data is G T the step Ar must be small enough See subsection 2 4 4 for
6. AG seems to contain only noise that the autocorrelation does not have a noisy Kronecker delta shape for lt amp opt you should use a non diagonal covariance matrix instead of a diagonal one 2 11 Spectrum at sample frequencies vs a When the execution stops during minimization the program also plots the spectrum at a number of sample frequencies as a function of log a By default the program uses Nwsamp uniformly distributed frequencies in the main spectral region where Nwsamp is defined in file OmegaMaxEnt_other_params dat To choose the sample frequencies yourself use parameter spectral function sample frequencies in subsection OUTPUT FILES PARAMETERS in file OmegaMaxEnt_input_params dat In an ideal case where the error standard deviation or covariance is known pre cisely the curves corresponding to different frequencies have stable or quasi stable point at the same a which happens to be located in the crossover between the information and the noise fitting regions where the optimal a opt is chosen using the log x vs 26 log a curve see subsection 2 9 If the error is small enough the curves are actually stable around a and eventually have plateaus that overlap around a as the error gets even smaller which indicates quantitatively accurate results Otherwise the curves are quasi stable at a namely they have either extrema or inflexion points with a non vanishing derivative B
7. blank and OmegaMaxEnt_other_params dat will be created with the default values of its parameters When you define a parameter in OmegaMax Ent_input_params dat that parameter is printed on the terminal as the program reads the file As for the parameters in OmegaMaxEnt_other_params dat they are printed out only if you modify the default value It is important not to modify any parameter s name namely any caracter up to on a line If you define a parameter or modify one s default value in one of the two parameter files and this parameter is not printed on screen when the program reads the file it means you inadvertently modified its name in the parameter file and the program cannot find it anymore If that happens destroy the file and execute OMaxEnt to create the default version of the missing parameter file Unless mentioned otherwise all the input parameters referred below are defined in the file OmegaMaxEnt_input_params dat 2 2 Set the input data Put the input file name on the line data file the first line in OmegaMaxEnt_input_params dat If your data file is not in the same directory as the code you can set input directory in subsection INPUT FILES PARAMETERS which tells the program where to look first for your files If a file is not found in that directory the program then looks for it in the current directory 2 2 1 Gliw as input The data is always assumed to be given by columns in the data file
8. more details 10 2 2 4 Bosonic data If G is bosonic set the parameter bosonic data to yes in subsection DATA PARAM ETERS If G iw is also real or G T is symmetric with respect to 7 8 2 set Im G column in data file to a value lt 0 The code will then assume an odd spectral function a w even This is the case of a correlation function of observables having the same signature under time reversal 2 2 5 Self energy If your data is a self energy and the static part the Hartree Fock value has been sub tracted it can be treated the same way as a Green function Otherwise if you know the value of the static part set parameter static part of data in subsection DATA PARAM ETERS If you do not know it an estimate is the value of Re 2 iw at high frequency If you have enough frequencies the latter function should become constant If you have put the correct value when the moments are evaluated see section 2 4 1 there should be a plateau in the value of M that is plotted as a function of the fit starting frequency If it is not the case then the provided value is incorrect However if there are not many frequencies in the asymptotic part of the data it is possible that only the beginning of a saturation can be seen Note that display preprocessing figures must be set to yes in section PREPROCESSING EXECUTION OPTIONS to see the figures when the mo ments are evaluated 2 2 6 Mo
9. totic part of the data and the provided one if applicable A warning is given if that tolerance is exceeded tol_M2 Tolerance on the difference between the second moment extracted from the asymp totic part of the data and the provided one if applicable A warning is given if that tolerance is exceeded tol_M3 Tolerance on the difference between the third moment extracted from the asymp totic part of the data and the provided one if applicable A warning is given if that tolerance is exceeded default_error_G Default error on the input data When no error file is provided a constant error of default_error_G G maz 18 used where Gmax is the maximum absolute value of the imaginary part of the data for fermions or the real part for bosons 46 err_norm Relative error on norm default_error_M Default error on moments tol_mean_Cl Tolerance used to find the onset of the asymptotic region of G from the fitted norm as a function of the starting frequency of the fit tol_std_Cl Tolerance used to find the onset of the asymptotic region of G from the standard deviation of the fitted norm as a function of the starting frequency of the fit tol_rdw Tolerance used to decide if consecutive frequency intervals are equal Used to detect if a frequency grid has been generated by the program Rmin_Dw_dw Minimum number of steps accepted in a grid interval when using a parameter ized real frequency grid grid para
10. AG reduced by ferr but no effect will be seen on the spectrum at the new optimal a However if the ratio of the errors in different frequency ranges is not well estimated Gout will eventually contain some of the noise of Gin in a given frequency range namely AG will be smaller than the error in that range while not yet containing all the information in another range AG will then have both noisy regions and structured ones If that problem is too serious distorsions appear in some parts of the spectrum A w although some of the information in Gin does not yet appear in it That problem can be reduced by rescaling the errors in one frequency 25 range with respect to the other This can be done by multiplying the errors by a smooth function of frequency 2 There is no correlations between neighboring frequencies or te of AG Look at the autocorrelation of AG as a function of Matsubara index differences An or Aie to see if the noise in AG is correlated or not between neighboring frequencies At the optimal a the autocorrelation should look like a noisy Kronecker delta function The value at the origin is equal to y N 2 where N 2 is the number of terms in x which should be close to 1 at a if the magnitude of the error is correct In most cases the actual error is not very precise therefore only the structure of the autocorrelation really matters If you have assumed that the covariance is diagonal and observe even though
11. QMaxEnt user guide Dominic Bergeron August 25 2015 Contents 1 Introduction 4 Lil CONSENSO aoe paom a aag a E A ds a wie a 4 1 2 Type of data accepted a a a ls al da 5 1 3 What the program does 00002 ee eee 6 2 Step by step guide to 0MaxEnt 8 2 1 The parameter files gos Lo dc A tee Bok ke Sh oh Bel 8 2 2 Setthernp tdata ae dla dt dee ate Mag aE ae Ei dis 9 221 SAP AAA Ee A 9 2 2 2 Temperature aa o ad ao SA we ne ew o e EN 10 22 3 NGC AS IMU A E ee 10 2 24 SBOSOMe daa a Gay ela e Ea ating a Be oa a By ei 11 ASS CONE a ath wisest eek ad a uae ed ok Ie 11 2 2 6 Most basic execution ps Ue a gig ede Re A Soe oe 11 Zod WORPEIEOIS A e a Slee eet be eS i A 12 2 3 1 Uncorrelated errors 50 2 v2 ve o o 2d o a 12 2 32 Correlated errors sosa 4 soia Eo EK PR AE CA 12 23 3 Basic execution DS 13 2 4 Control how moments are used o 13 2 4 1 Use QMaxEnt to extract moments 14 3 2 4 2 Special case of the norm of the spectrum 15 2 4 3 Providing moments pe A HD EE eS 15 2 4 4 G r as input data Ed a Ep aks 16 DAD e a EA E EEE E used T Mule 17 2 5 Provide a user defined grid os re a 17 2 6 Provide a user defined default model 19 2T EXecuUton yo apo a eS a e aui REESE eS 20 2 8 Output figures es il a RO A SS 22 2 9 How the optimal is chosen aoaaa a 23 2 10 Properties of A in Gy at the optimala
12. SC M the M s are the moments of the spectrum The number of frequencies in that region is defined in two possible ways if there is a sharp and well isolated peak at low frequency the program can estimate its width directly from the Mat subara data see appendix A and use a step appropriate to resolve that peak otherwise the step is defined as SW Rmin_SW_dw where Rmin_SW_dw is defined in file Omega MaxEnt_other_params dat This grid is of course not optimal in many cases especially for spectra possessing very broad and very sharp features at the same time Fortunately QMaxEnt offers a few ways to define a better adapted grid with the parameters of sub section FREQUENCY GRID PARAMETERS 17 1 The simplest way is by setting one or more of the parameters spectral function width spectral function center and real frequency step The first two of those parameters are used for SW and SC instead of the default ones defined from the moments to define the main spectral range as SC SW 2 SC SW 2 with a uniform step given by real frequency step The default values given in the previous paragraph are used for those of these parameters that are left blank You can also make sure that a specific frequency is included in the grid with the parameter real frequency grid origin This is useful for example if the position of a peak is known precisely to make sure that there is a grid point at the maximum If rea
13. a function when localizing the asymp totic part of the function Nn_fit_fin During computation of the moments maximum number of frequencies used in the final fit of the asymptotic form of the Matsubara function Nn_as_min Minimum number of frequencies required in the asymptotic region for the com puted moments to be used in the calculation Nwn_test_metal Number of low frequencies used to look for the signature of a low energy peak If the test is conclusive the programs tries to determine the position width and weight of the peak Niter_dA_max Maximum number of iterations in the calculation of the spectrum at a given value of a using Newton s method to solve VQ 0 Nwsamp Default number of sample frequencies of the spectral function to be plotted and saved as a function of alpha f_w_range Ratio of the total real frequency range and the main spectral range Rmin_SW_dw Minimum ratio of spectral function width and frequency step 45 tol_tem Tolerance on the relative difference between the temperature extracted from the first finite Matsubara frequency and the input temperature A warning is given if that tolerance is exceeded tol_norm Tolerance on the difference between the norm extracted from the asymptotic part of the data and the provided norm if applicable A warning is given if that tolerance is exceeded tol MI Tolerance on the difference between the first moment extracted from the asymp
14. arameter to yes grid parameters Use this line to provide the parameters of a non uniform grid that will be generated by the program The left and right boundaries will define the main spectral region 37 3 1 4 COMPUTATION OPTIONS e evaluate moments If you set this parameter to yes the program will try to extract the spectral func tion s moments from your Matsubara data Note that depending on the other input parameters the moments may also be extracted even if this option is disabled e maximum moment Use this parameter to set the maximum moment to be imposed to the spectrum For example to impose only the norm and the first moment set it to 1 e default model center Use this parameter if you want the default model to be centered on a frequency different than the normalized first moment e default model half width Use this parameter if you want the half width of the default model to be different from the standard deviation e default model shape parameter Use this parameter if you want the default model to be different from a gaussian shape parameter 2 A shape parameter of 1 corresponds to a Laplace distri bution and as the shape parameter increases the default model becomes closer to a uniform distribution A shape parameter gt 1 is recommended so that the derivative has no singularity e default model file Use this parameter to provide a user defined default model e initial spectral funct
15. ariance or covari ance eigenvector index 7 general covariance at the optimal a if Qro lt Qopt 5 e Autocorrelation of AG vs An diagonal covariance or Ai general covariance at the lowest a 22 e Autocorrelation of AG opi vs An diagonal covariance or Ai general covariance at the optimal a if Qigw lt Qopt In the figure showing the spectrum at dopt a spectrum at a value of a slightly below Qopt and one at a value slightly above are also plotted Those two other values of a say Aop and Qopt delimit the range of the crossover region between the information and the noise fitting regimes Along with the optimal spectrum the spectra at amp opt and Qopt tell how stable the spectrum is around a and thus how accurate the results are see also section 2 11 2 9 How the optimal a is chosen If you look at log x as a function of log a for a large enough range of a you will notice that there are three regimes in that function At high a the entropy term aS dominates in the quantity Q x aS that is minimized with respect to the spectrum A see section 1 3 Thus x decreases very slowly with a and the spectrum stays close to default model We may call this the default model regime At intermediate values of a x decreases rapidly as decreases This is the regime where the information in input data Gin is gradually integrated in the spectrum A w or Gou KA as a decreases It is thus an inform
16. ation fitting regime Finally when most of the information in Gin is already present in A w the noise in G starts to be integrated into A w as AG Gin Gout becomes smaller than the error on Gout At that point the rate of change of x with a decreases rapidly and becomes very small in what we may call the noise fitting regime The optimal alpha should clearly be chosen where most of the information in Gin is contained in A w or Gout but not its noise It should therefore be somewhere in the crossover region between the information and the noise fitting regime The change of decreasing rate of x with a can be identified by a peak in the curvature of log x vs log a The optimal a say amp opt can simply be chosen at the highest peak 23 maximum However in some cases peaks other than the one marking the onset of the noise fitting regime may be present in the curvature at values of a larger than amp opt To avoid choosing those peaks by accident instead of the correct one one can choose the optimal a at the highest peak maximum in the curvature of log x vs ylog a where y is typically between 0 1 and 0 5 instead The factor y increases the ratio of the amplitudes of the correct peak and the other peaks located at higher a which reduces the probability of choosing the wrong peak It displaces the position of the crossover peak maximum toward the noise fitting region however but only very slightly if 7 is not too
17. be on the first column of the file Note however that unless it is a grid that was previously generated by MaxEnt it will be used only in the main spectral range which is defined either by the default values of SW and SC or by spectral function width and spectral function center On the other hand if the file you provide is a spectrum obtained by the program in a previous calculation the whole grid including the high frequency parts will be identical to the provided one As for the grid in the regions outside the central region it is always defined in the same way by the program Instead of having a uniform step Aw the grid in those regions is such that Au u 1 u is constant where u 1 w woz on the left side and u 1 w wor on the right side Here wo and wo are frequencies determined by the extrema of the grid which can be controlled by the parameter f_w_range defined in file OmegaMaxEnt_other_params dat More precisely f_w_range is the ratio of the total size of the grid Wmax Wmin to the width of the main spectral range SW 2 6 Provide a user defined default model By default the default model is a gaussian of width equal to the standard deviation AWsid VM M fermionic case if M and Ma are defined provided by you or extracted from the data or equal to parameter spectral function width subsec tion FREQUENCY GRID PARAMETERS and centered on M fermionic case or sp
18. case the best solution is of course to improve the error if that is possible If not you 24 can also use the other output figures to help you choose the a that corresponds to the best compromise according to your judgement In particular in addition to the spectrum itself the figures described in subsections 2 10 and 2 11 can help you diagnose more precisely the problems with the errors in order to improve it or help you find the best compromise in the choice of the optimal a 2 10 Properties of AG Gi Gag at the optimal a The analytic continuation operation consists essentially in fitting the function Cas KA to the input data G where G is a Green function vector expressed in the eigenbasis of the covariance matrix and normalized by the standard deviation in that basis The function AG Ge Ce as a function of wpn or covariance eigenvector index te is very useful to assess the quality of the fit To have a good fit that function must satisfy the two following conditions 1 AG is essentially noise at the optimal a When most of the information contained in Gin is also in Gout but not the noise AG is essentially the noise in Gin at the optimal a In addition because AG is normalized it has a standard deviation of 1 This is what we obtain when the covariance matrix of the data is accurate If the error is simply multiplied by an global factor ferr Qop Will simply be reduced by f2 and the standard deviation of
19. e of the step in log p a e chi2_alpha_smooth_range Length of the part of the curve logio x vs y log o a used to compute its local curvature y is the parameter f_scale_lalpha_Ichi2 e f_scale_lalpha_Ichi2 y in the calculation of the curvature of log p x vs y log p a e FNfitTauW Factor that determines the number of values of 7 used in the polynomial fit in the calculation of the moments from imaginary time data e std_norm_peak_max For fermions tolerance on the standard deviation of the low frequency peak weight as a function of fitting power in the estimation of a low frequency peak width Used to decide if there is a well defined peak or not e varM2_peak_max For bosons relative tolerance on the low frequency peak variance as a function of fitting power in the estimation of a low frequency peak width Used to decide if there is a well defined peak or not e R_d2G_chi_peak For bosons ratio used to decide if the program looks for a low frequency peak 49 e RMAX_dlchi2_lalpha Maximum ratio of dlog x dlog a at the lowest a and the maximum value Used to determine if a is deep enough in the noise fitting region e f_alpha_min Factor by which amin is reduced when estimated to be too high e save_alpha_range Range of a above and below the optimal value for which the results are saved in log scale If r save_alpha_range the range saved is 10 Qp4 100 ops 50
20. ectral function center subsection FREQUENCY GRID PARAMETERS if M is not defined If you want to use a different default model there are two options 1 You can use the parameters default model center DC default model half width DW and default model shape parameter DS in subsection COMPU TATION OPTIONS to use a default model that has a generalized normal distribu 19 tion form As special cases if DS 1 this corresponds to a Laplace distribution if DS 2 itis a gaussian and as DS oo the distribution becomes uniform in the range DC DW DC DW It is recommended to set default model shape parameter to a value larger than 1 since for DS lt 1 there is a discontinuity in the derivative at DC which is not physical in general except in the presence of a Van Hove singularity You can also use your own default model by setting the parameter default model file subsection COMPUTATION OPTIONS It is assumed that the first column is the grid and the second column is the function value The grid on which your function is defined can be arbitrary the program uses cubic splines to interpolate your function on the active grid namely the one set previously during preprocess ing If your grid does not extend over the whole active grid the program will extend your function with gaussians matching it as smoothly as possible at the extrema of your grid The default model is always
21. ectral representation 1 1 It is recommended to set evaluate moments to yes subsection COM PUTATION OPTIONS to verify that your value is close to the one extracted from your data and printed in the terminal 31 moment error If you provided the third moment use that parameter to provide its absolute error Otherwise the value default_error_M Aw will be used where default_error_M is defined in file OmegaMaxEnt_other_params dat and Awgsta is the standard de viation of the spectrum static part of data If your data is a self energy that contains a static part use that parameter to provide 1ts value If you do not know that value see section 2 2 5 for instructions 34 3 1 2 INPUT FILES PARAMETERS e input directory If your data is not in the same directory as the program use that parameter to set the input directory This way the paths are not required for parameters that are file names The directory can be relative to the program s directory Note that if the program does not find a file in the provided directory it will also look for it in the current directory This allows you to use both the current and another directory at the same time Just remember that if files with the same name exist in the current and the provided directory the ones in the latter will be used e Re G column in data file For Matsubara frequency data use that parameter if the real part is not on the second column of you
22. elow amp opt in the noise fitting region the results eventually become unstable and the curves can increase or decrease very quickly On the other hand if the error is not well known there is necessarily some imbalance between different Matsubara frequency ranges The result of that imbalance will be that the quasi stable points in different real frequency regions of the spectrum will not be aligned Those regions will have different optimal values of a and thus there will be no well defined single optimal a In that case if possible it is preferable to modify the error to improve balance between frequency ranges Otherwise if the value determined automatically by the program does not appear to be the best the spectrum sample frequencies curves combined with the figures of AG and its autocorrelation can help you choose a better compromise To summarize the spectrum sample frequencies vs a curves tell you 1 what the optimal a is in a given real frequency range 2 if the errors are well balanced between different Matsubara frequency regions 3 if your data is precise enough to have quantitatively accurate results 2 12 Output files You can control how and where the output files are saved with the parameters in subsec tion OUTPUT FILES PARAMETERS of section OPTIONAL MINIMIZATION TIME PARAMETERS of OmegaMaxEnt_input_params dat When the minimization stage starts the program creates two directories for the output file
23. eter to yes for imaginary time data temperature If only the Matsubara frequency indices are given in your data file or if the fre quencies are not very precise you can use this parameter to provide the tempera ture It must be given in energy units kg 1 norm of spectral function If the Green function is not normalized to 1 and you know the norm you can use this parameter to provide it The norm corresponds to the coefficient of 1 iw 32 in the high frequency expansion of Matsubara functions having the spectral repre sentation 1 1 It is recommended to set evaluate moments to yes subsection COMPUTATION OPTIONS to verify that the provided value is close to the norm extracted from your data and printed in the terminal If the Green function is not normalized to 1 and you do not know the norm see subsection 2 4 2 for instructions The program assumes that the norm is known with relative precision err_norm default value 1079 a parameter in file OmegaMaxEnt_other_params dat For real bosonic Matsubara data the norm is 0 However in that case set Im G column in data file subsection INPUT FILES PARAMETERS to a value lt 0 instead and do not put anything for the norm 1 moment If you know the first moment of the spectral function use that parameter to pro vide it The first moment corresponds to the coefficient of 1 iwn in the high frequency expansion of of Matsubara func
24. f the spectral function The next two subsection of this introduction present the basic theoretical aspects necessary to understand what you are doing when using this program Then section 2 describes how to use MaxEnt step by step You can execute the program already at the end of subsection 2 2 to get some preliminary results The following subsections give you the possibility to use more information you might possess to improve the result Some explanations on how the program works and advices on how to interpret and improve the results are given at the end of the section Lists of the output figures and the output files are given in subsections 2 8 and 2 12 respectively Finally section 3 presents a list of all the input parameters with a short description of each one For simplicity throughout this guide the Matsubara data is referred to as the Green function G but the data can be a correlation function a self energy or any Matsubara function satisfying the conditions given in the next section The result obtained with the program is also referred to as the spectral function A although for bosonic data it is actually o w A w w 1 2 Type of data accepted The program can treat data that can be expressed as Gliwn IS Ea 1 1 2k Wn wW or due TA w where A w gt 0 for fermions or o w A w w gt 0 for bosons and the and signs in the denominator of 1 2 are for fermions and bosons re
25. f the results 2 Step by step guide to MaxEnt 2 1 The parameter files The program uses two parameter files OmegaMaxEnt_input_params dat and Omega MaxEnt_other_params dat The main parameter file used to control the calculation is OmegaMaxEnt_input_params dat The other parameter file should be modified by expe rienced users only or according to instructions in this guide The only mandatory param eter in OmegaMaxEnt_input_params dat is the data file name on the first line Although not recommended you can obtain results even if all the other fields are left blank When a parameter can have the values yes or no leaving the field blank or putting anything else than yes corresponds to the default value no The parameters other than the data file name are divided in two sections The parameters defined in the section OPTIONAL PREPROCESSING TIME PARAMETERS are used to set all the variables in the opti mization problem The parameters defined in the section OPTIONAL MINIMIZATION TIME PARAMETERS control what happens during the search for the optimal solution to the problem They can be modified without triggering the preprocessing Some of them can be modified during the optimization itself as explained in subsection 2 7 Before doing your first analytic continuation with XMaxEnt create the two parameter files by executing the file OmegaMaxEnt The file OmegaMaxEnt_input_params dat will be created with all the fields
26. fications in section OPTIONAL PREPROCESSING TIME PARAMETERS the preprocessing will also restart 3 2 4 DISPLAY OPTIONS e display results at each value of alpha 42 If that option is enabled the values of Q x aS S and x will be printed in the terminal at each value of a computed show optimal alpha figures Use this parameter if you do not want the program to display the figures corre sponding to the optimal a show lowest alpha figures Use this parameter if you do not want the program to display the figures corre sponding to the lowest a show alpha dependant curves Use this parameter if you do not want the program to display the a dependant curves namely log x vs log a its curvature and the spectrum sample fre quencies vs log a reference spectral function file You can use that parameter to provide a reference spectrum that will be plotted on the same figures as the spectrum How the program estimates the width of a low energy peak The presence of a peak in the spectrum at low energy produces a specific signature in the Matsubara data For fermions the imaginary part will seem divergent at low frequency while for bosons it produces a jump in the real part at low frequency In addition if that peak is isolated enough from the rest of the spectrum one can obtain its weight position and width from the coefficients of a Laurent series fitted to the low frequency part of G iw When t
27. he signature of the presence of a well defined peak at low energy is 43 detected by the program it performs that fit and print the three values in the terminal if the fit is good The parameters Nwn_test_metal R_d2G_chi_peak std_norm_peak_max and varM2_peak_max all defined in file OmegaMaxEnt_other_params dat are used to determine if the fit is performed and if so if it is good enough to take the result into account The only parameter actually used by the program is the width which provides an order of magnitude for the step that should be used in the low frequency part of the grid B Other parameters The following is a complete list of the parameters defined in file OmegaMaxEnt_other_params dat e Nn_max Maximum number of Matsubara frequencies Once the asymptotic part has been removed if the number of frequencies exceeds that value a non uniform Matsub ara grid will be used e Nw_min Minimum number of real frequencies e Nw_max Maximum number of real frequencies The execution is paused and cannot con tinue if the number of frequencies exceeds that number If that happens you can either increase that value or use parameters of subsection FREQUENCY GRID PARAMETERS in file OmegaMaxEnt_input_params dat to modify the grid The latter option is more efficient and thus recommended e Nn_fit_max 44 During computation of the moments maximum number of frequencies used in the fit of the asymptotic form of the Matsubar
28. ion file 38 3 1 5 The program uses the default model as the initial spectral function This is actu ally the solution when a is very large However if for some reason the program is interupted not just paused before the optimal result was reached you can use this parameter and initial value of alpha in subsection COMPUTATION PARAME TERS to restart it at a value of a different from the initial one This is useful only if the frequency grid is very large however N or N gt 1000 and the calculation time is long For numbers of Matsubara and real frequencies in the hundreds the calculation is fast and you will not need that option If you do need it it may be useful to read section 2 12 to learn how and where the results are saved PREPROCESSING EXECUTION OPTIONS e preprocess only Set this parameter to yes to pause execution at the end of preprocessing As long as this parameter is enabled when resuming execution the preprocessing stage restarts if changes were made in section OPTIONAL PREPROCESSING TIME PARAMETERS during the pause or else nothing happens You can disable that option during the pause to start the actual calculation This option along with display preprocessing figures are very useful to verify the input parameters input files frequency grid moments default model etc and modify them if necessary before starting the actual computation display preprocessing figures Se
29. ists in minimizing Q xX aS 1 3 where x Y Gm KmA Cok Gn KnA 1 4 mn where A is the vector obtained after discretizing w K is a row vector such that K A is an approximation to 1 1 and is the covariance matrix S is a differential entropy defined as d A s Fawn 1 5 2T W where D w is called the default model and a is a parameter to be determined The solution that minimize 1 3 if x is negligible namely that maximizes the entropy S is e D w hence the name default model D w is defined in a way to include what is known a priori about the spectrum The maximum entropy approach is based on Bayesian inference which consists ba sically in using the information you know in advanced about the spectrum to deduce it from incomplete or noisy data which alone is not sufficient to obtain acceptable results In the present case the entropy term aS is used to include that information To find the optimal value for a the program first computes the solution to min x aS for a large range of a The location of the optimal a can then be found by a simple analysis of the shape of the function log x as a function of log a The procedure is described in subsection 2 9 Note that if we diagonalize the covariance matrix C defining UCU 1 6 x can be written x G KA G KA 1 7 where 1 8 This form is more useful for practical calculations and analysis o
30. l frequency grid origin is not defined but spectral function center is the two values will coincide 2 One can also define a non uniform grid in the central range by setting the param eter use parameterized real frequency grid to yes and by defining intervals and their respective steps on the line grid parameters The format of that line is wo Aw wy Aw Wy 1 Awy 1 wy where the w s are the fre quencies delimiting the intervals and the Aw s are the steps in those intervals The program then generates a grid with a step varying smoothly with a hyperbolic tan gent shape between intervals This grid is then used in the main spectral range Note that it is important not to change the step too quickly between contiguous intervals There should not be more than a factor of three from one interval to it neighbors otherwise spurious structures may appear in the spectrum at the bound aries between intervals Each interval should also contain at least 10 points to en sure that Aw vary smoothly with w a necessary condition to obtain high quality re sults Another parameter RW_grid defined in OmegaMaxEnt_other_params dat controls how that non uniform grid is generated It defines the ratio of an interval width to the transition region between its step and the neighboring intervals steps 3 Finally you can provide your own grid with the parameter real frequency grid 18 file The grid must
31. m G w Aopt w w Copt w Wn Re G Normalized output error at Aopt Re AG Im AG moments of the spectral function at Qopt M M autocorrelation of AG at Qopts An ACre An ACrm An diagonal covariance Ai ACI Ai AC2 Ai general co variance and the following files are saved in directory OmegaMaxEnt_output_files 29 spectral_function_tem _alpha dat G_out_tem _alpha dat error_G_out_tem _alpha dat moments_G_out_tem _alpha dat chi2_vs_alpha_tem dat Asamp_vs_alpha_tem dat spectral function at a given a w A w fermions w A w w bosons output Green function Wn Re G Im G Normalized output error Re AG Im AG moments of the spectral function MP Meta x versus a a 32 A w at sample frequencies versus a a AO AGS a Alas The opt and opt subscripts for the data in file optimal_spectral_functions_tem _alpha _ _ dat refer to values of a slightly above and below a Those values delimit the crossover region between information and noise fitting regimes see also section 2 8 2 13 When is the calculation over The program assumes that the calculation is over when 1 the optimal a has been found and 2 the following condition is satisfied A min dlog x2 PEN lt A diene Tara LEON 2 5 dy log a dy log a where RMAX_dlchi2_lalpha and y f_scale_lalpha_lchi2 are defined in OmegaMax Ent_other_params dat You can also
32. meters in subsection FREQUENCY GRID PARAMETERS of file OmegaMaxEnt_input_params dat Rdw_max Maximum ratio of steps recommended in consecutive grid interval when using a parameterized real frequency grid grid parameters RW_grid Ratio of grid interval length and the transition region between intervals in the pa rameterized real frequency grid 47 RWD_grid Width of the transition region of the hyperbolic tangent used to define the param eterized real frequency grid minDefM Minimum value of default model f_alpha_init Initial ratio of the entropy and the x contributions to the spectrum R_width_ASmin Width of the minimum entropy spectrum peaks relative to the width of the default model f_Smin Ratio of the minimum entropy and the optimal x Used to estimate amin during the preprocessing diff_chi2_max Maximum relative difference between the x of consecutive values of a If this ratio is exceeded the step in log a is decreased tol_int_dA Tolerance on consecutive values of the integral of 5 A where A is the variation in the spectrum A between iterations in the Newton s method used to compute the spectrum rc2H Maximum ratio of the penalization parameter and the maximum eigenvalue of the hessian of x The penalization parameter is used to avoid negative values of the spectrum 48 e pow_alpha_step_init Initial value of the step in log q e pow_alpha_step_min Minimal valu
33. normalized to the same value as the spectrum namely Mo in the fermionic case or G iw 0 in the bosonic case 2 7 Execution OMaxEnt runs in an interactive loop There are a few ways given below to pause the execution to see the result at a given stage and with a given set of input parameters At the pause the program always asks if you want to continue execution To do so simply press ENTER or enter anything else than y to stop During a pause you can make changes to the parameter files Depending on the type of changes you made execution can resume either from the point it halted or at an earlier stage If you modify a param eter in section OPTIONAL PREPROCESSING TIME PARAMETERS during a pause the preprocessing starts over On the other hand the parameters in section OPTIONAL 20 MINIMIZATION TIME PARAMETERS with the exception of initialize preprocess ing can be modified during a pause without triggering the preprocessing when execu tion continues Some parameters in that subsection will however cause the minimization stage to restart at the initial highest a if they are modified namely output directory 39 Ges spectral function sample frequencies initial value of alpha and initialize maxent To pause execution you can use one of the following parameters e preprocess only subsection PREPROCESSING EXECUTION OPTIONS e minimum value of alpha Amin s
34. nts you can either increase the step everywhere in the main spectral region or use a non uniform grid to improve the resolution of those peaks It may also happen that the main spectral region is too narrow and not optimally aligned with the spectrum You should adjust it so that it covers all the frequency region where the spectral weight 1s located The rest of the grid should cover only the tails of the spectrum See section 2 5 for more details on how to define the grid The default model may also be too wide too narrow or misaligned with the spec trum If so you can modify its parameters in subsection COMPUTATION OPTIONS Note however that the default model should stay mostly featureless The structure in the 31 spectrum should come from the data and not from the default model Finally if AG and its autocorrelation do not have the properties given in section 2 10 or the spectrum sample frequencies curves as a function of log a do not have the same quasi stable points see section 2 11 then you should improve your errors if it is possible Advices on how to improve the error are given in section section 2 10 3 List of input parameters The following is a complete list of the parameters defined in file OmegaMaxEnt_input_params dat 3 1 3 1 1 OPTIONAL PREPROCESSING TIME PARAMETERS DATA PARAMETERS bosonic data Set this parameter to yes if your data is bosonic imaginary time data Set this param
35. of W where Ar is the imaginary time step This condition is equivalent to w gt W Ideally A7 should be at least a few times smaller than 1 W Thus if you provide G 7 as the input data and do not know M and M make sure the imaginary time step is small enough so that this condition is satisfied If you do know M and M with good accuracy note however that even though the Fourier transform of G r will be accurate 16 the Fourier transform of the error will not be if Ar is not small enough which may degrade the quality of the result It is thus preferable that the condition Ar lt 1 W be well satisfied in any case 2 4 5 Basic execution You can execute QMaxEnt here See section 2 7 for more details on the execution Oth erwise follow the next steps or see section 3 to learn about other functionalities 2 5 Provide a user defined grid The real frequency grid used by 2MaxEnt is separated in two different types of grid a central dense part let us call it the main spectral range where most of the spectral weight should be found and two non uniform high frequency parts where the step size increases approximately quadratically with the module of the frequency Let us address how to control the grid in the main spectral range first By default in that range the program uses a uniform grid of width SW 2Awsta where Aw ig VM M fermionic case is the standard deviation of the spectrum and centered on
36. of G and you have the covariance matrix a l N 1 gt Gi G G Gy 2 3 l 1 Cij 12 where N is the number of sample values for each element G G is the sample of G and y Y Gi Gi 2 4 F Le 24 then in subsection INPUT FILES PARAMETERS provide that matrix using the param eters re re covariance file im im covariance file and re im covariance file if the input is G iwn or imaginary time covariance file if the input is G T Those files must contain an equal number of lines and columns 2 3 3 Basic execution If you do not have more information you can execute 2MaxEnt now See section 2 7 for more details on the execution If you have more information or want to learn about other functionalities follow the next steps or see section 3 2 4 Control how moments are used The part of the Green function at frequencies w gt W where A w gt W 0 can be expressed as a high frequency expansion of which the coefficients are the moments of the spectral function A w However because of finite precision and noise the informa tion on high order moments is not present in the data Therefore the high frequencies of G iw contain the information about a few moments only It is thus numerically equivalent but computationally more efficient to replace the large number of constraints corresponding to those high frequencies in the calculation by the few contraints corre
37. parameter to ignore the peaks below a certain a in the curvature of log x versus log a in case there are more than one peak in the region where the optimal a should be chosen 4 3 2 3 MINIMIZATION EXECUTION OPTIONS e number of values of alpha computed in one execution Use this parameter to force a pause once the spectrum has been computed for a certain number of values of a The program will display figures showing the results at the current stage of the calculation You can make some changes in section OPTIONAL MINIMIZATION TIME PARAMETERS without triggering the preprocessing except for option initialize preprocessing Once you have closed the figures the program will ask you if you want to continue the execution If you do computation will resume at the point it halted e initialize maxent Use this option to restart the minimization stage at the initial value of a after a pause Do not forget to disable that option if the execution is paused again before the calculation is over and you modify the file OmegaMaxEnt_input_params dat during the pause unless you want the minimization stage to start over again 113 Note that 1f you modify either output directory spectral function sample fre quencies or initial value of alpha during a pause the minimization stage will also start over e initialize preprocessing Use this option if you want to restart the preprocessing Note that if you make modi
38. r data file e Im G column in data file For Matsubara frequency data use that parameter if the imaginary part is not on the third column of your data file For the bosonic case if the imaginary part vanishes exactly set this parameter to a value lt 0 The spectrum o w is then assumed to be even and only the positive part of the grid is used in the calculation e error file Use this parameter if you have an error file where the error is given by columns This is for the case of diagonal covariance Otherwise use the covariance file parameters e Re G column in error file For Matsubara frequency data use that parameter if an error file is provided and the error on the real part is not on the second column of the file 35 Im G column in error file For Matsubara frequency data use that parameter if an error file is provided and the error on the imaginary part is not on the third column of the file re re covariance file For Matsubara frequency data if you have a covariance matrix use that parameter to provide the Re G Re G part of the matrix There must be an equal number of rows and columns in the file im im covariance file For Matsubara frequency data if you have a covariance matrix use that parameter to provide the Im G Im G part of the matrix There must be an equal number of rows and columns in the file re im covariance file For Matsubara frequency data if you have a covariance matrix use
39. rameter maximum moment in the same subsection By default QMaxEnt imposes the first 4 moments including the norm Mo to the spectrum for fermions and the first 3 moments for bosons For example if you want the program to impose only the norm and the first moment set maximum moment to 1 You can also control the maximum frequency to be used in the calculation with the parameter Matsubara frequency cutoff in subsection FREQUENCY GRID PARAME TERS For example 1f you do not have enough frequencies in the asymptotic frequency range it is probably better to use all the frequencies instead of the moments by setting maximum moment to 0 if the norm is known exactly and Matsubara frequency cut off to a value larger than the last frequency in your data 14 2 4 2 Special case of the norm of the spectrum If your spectrum is normalized to 1 namely if the leading high frequency term of the Green function is 1 iw you do not have to provide the norm Also if you have real bosonic Matsubara data and have set Im G column in data file to O or less the norm is automatically assumed to be 0 Otherwise if the norm is not 1 use norm of spectral function in subsection DATA PARAMETERS to provide it If you do not know it you can let the program extract it first and use the value printed in the terminal To do so set evaluate moments to yes in subsection COMPUTATION OPTIONS and preproces
40. rameter a It does not use either the classic approach or Bryan s but instead locates the range of a where only information is fitted and the range where noise is also fitted The optimal value is then chosen in the crossover between the two regimes The program also offers useful graphical diagnostic tools for assessing the quality of the results In addition the program uses a cubic spline to model the spectral function and uses the Matsubara frequency spectral representation of the Green function instead of the imaginary time version Those two choices allow to compute the Green function cor responding to a given spectral function with high accuracy by a piecewise analytical integral The use of a non uniform grid for the spectral function and contraints on mo ments instead of a high Matsubara frequency cutoff also allows to improve calculation efficiency without losing accuracy Basically the code is optimized to ensure that the errors in the results comes from the errors in the data and not from numerical approxi mations You can obtain results by providing only a data file name to the program It is however highly recommended to provide also errors standard deviation or covariance matrix if reliable estimates are available You can also further improve the results by providing more information like some known moments of the spectral function a de fault model or by providing a frequency grid adapted to the expected structures o
41. res the pro gram asks you if you want to continue execution You can also choose which groups of 21 99 66 figures are displayed with parameters show optimal alpha figures show lowest alpha figures and show alpha dependant curves in subsection DISPLAY OPTIONS If you have set minimum value of alpha and a is reached before the condition for the calculation to be over is satisfied see section 2 13 the program will tell you that Amin is not small enough and will reduce it by the factor f_alpha_min defined in file OmegaMaxEnt_other_params dat before asking if you want to continue execution If you have set number of values of alpha computed in one execution the spectrum will be computed for N values of a each time the calculation is resumed unless that max parameter is emptied during a pause 2 8 Output figures The following functions are plotted when the execution stops during the minimization stage e log x versus log a e Curvature of log x vs log a e Spectral function at the lowest a Qlow A VS Ww e Spectral function at the optimal a opt Aopt VS W Cif Qlow lt Qopt gt e Spectral function at sample frequencies A Wsamp Versus a e Normalized error AG Ge Goa versus frequency index n diagonal covari ance or covariance eigenvector index 7 general covariance at the lowest a e Normalized error AG versus frequency index n diagonal cov
42. s OmegaMaxEnt_output_files 27 and OmegaMaxEnt_final_result which are located in output directory if this param eter is defined else in input directory subsection DATA PARAMETERS if defined or in the current directory otherwise Typically the solutions to min y aS are computed for hundreds of values of a be fore the optimal spectral function is obtained The results in a certain range above and be low the optimal a may be useful but the rest of the results are not By default in directory OmegaMaxEnt_output_files the program saves the results in a range 107 pz 10 Qopt where r is equal to parameter save_alpha_range defined in file OmegaMaxEnt_other_params dat To define differently the range of a within which the results are saved you can use pa rameters maximum alpha for which results are saved and minimum alpha for which results are saved in subsection OUTPUT FILES PARAMETERS The following files with their respective formats are saved in directory OmegaMax Ent_final_result 28 optimal_spectral_function_tem _alpha dat optimal_spectral_functions_tem _alpha _ _ dat G_opt_tem _alpha dat error_G_opt_tem _alpha dat moments_optimal_spectrum_tem _alpha dat auto_corr_error_G_opt_tem _alpha dat spectral function at Aopt w Aopt w fermions w Aopt w w bosons spectral functions around Qopt Aopt w Aopt w Fopt w Topi w output Green function at Qopr I
43. s only to yes in PREPRO CESSING EXECUTION OPTIONS to prevent the actual calculation from starting after the preprocessing It is also better to set display preprocessing figures to yes to ver ify convergence on the figure of Mp Once norm of spectral function is set you can disable preprocess only and continue the execution If you do not know the norm and there is not enough frequencies in your data to allow the program to extract it namely there is no plateau in the figure of Mo just tell the program not to use moments at all by setting maximum moment to 1 or less 2 4 3 Providing moments If you know some moments of the spectral function with good precision you can provide them with their respective errors in subsection DATA PARAMETERS If you provide enough moments first and second for fermions first for bosons the program will not try to extract them from your data In that case however it is recommended to set evaluate moments and display preprocessing figures to yes in subsection COMPUTATION OPTIONS when running QMaxEnt for the first time with a given set of data This way provided that your data has some frequencies in the asymptotic part of G if so there will be a plateau in the figures of My and M 15 you can verify if the moments you provided have values close to the ones extracted which will be printed in the terminal In addition the program will al
44. so print where the asymptotic regime sets in This verification step is important to make sure you do not impose wrong moments to the spectrum which is worse than not imposing any When making that verification set preprocess only to yes to prevent the actual calculation from starting Once you provided moments if your Matsubara frequency cutoff is far in the asymp totic part of G there are long plateaus in the figures of Mo and M when evaluate moments set Matsubara frequency cutoff in subsection FREQUENCY GRID PA RAMETERS to the onset value given by the program Note that this is done automat ically if evaluate moments is enabled If the asymptotic regime is not reached at the cutoff you can make sure that all the frequencies are used whatever the value of eval uate moments is by setting Matsubara frequency cutoff to a value larger than the cutoff 2 4 4 G r as input data If you have imaginary time data the first two finite moments M and M gt are necessary to Fourier transform G r before computing the spectrum from the resulting G iw with which 2MaxEnt works internally Thus in that case either you provide those moments or the program extracts them from G r In the latter case it does so using a polynomial fit to G T around the boundaries 7 0 and 5 The condition for that fit to give the correct moments is that Ar lt 1 W see introduction of section 2 4 for the definition
45. spectively The overall sign on the right hand side of those expressions can also be different the program will detect which convention you are using Those expressions are spectral representations which apply very generally to Green and correlation functions They imply that G iw G iw and that Re G iw and Im G iw behave asymptotically as 1 w and 1 iw respectively as wn gt 00 The positivity of A w or o w is an additional constraint necessary for the max imum entropy method to be applicable Assuming the same sign convention as in 1 1 it implies that sign Im G iw sign w in the case of fermions and that Re G iw lt 0 for bosons QMaxEnt can treat directly any data satisfying those conditions This includes a wide variety of Green or correlation functions but also the dynamical part of a self energy X iwn In the latter case it is better to know in advance the value of the static part if it is not already subtracted from the data OMaxEnt takes column wise data files with only spaces between the columns 1 3 What the program does Finding the spectrum A w corresponding to a given G w by numerically inverting expression 1 1 or 1 2 does not work The errors on A w resulting from the ones on G iw would be greatly amplified and the result would not make sense This is what is called an ill conditioned problem Instead the program uses the maximum entropy approach which cons
46. st basic execution If you do not have more information than the Green function itself you can execute QMaxEnt now See section 2 7 for more details on the execution If you have more information or want to learn about other functionalities follow the next steps or see 11 section 3 2 3 Set errors As mentioned in section 1 3 the data s covariance matrix C is used during the calcula tion For a given set of data G iw the accuracy of C actually controls how well the information contained in G can be transferred into the resulting spectrum A w For the remaining of the guide unless mentioned otherwise the term error is used in a broad sense sometimes meaning the standard deviation of data points or more generally re ferring to the covariance matrix The program can treat the case of a general non diagonal covariance matrix but you can simply provide the standard deviation in the case of diagonal covariance 2 3 1 Uncorrelated errors If the errors on the elements of G are uncorrelated define the parameter error file in the subsection INPUT FILES PARAMETERS and the correct column numbers on the lines Re G column in error file and Im G column in error file if the input is G iw or on the line column of G r error in error file if the input is G 7 unless the column number s correspond to the default value s 2 3 2 Correlated errors If the errors are correlated between elements
47. stop the calculation before those condition are satisfied with parameters minimum value of alpha subsection COMPUTATION PARAMETERS or number 30 of values of alpha computed in one execution subsection MINIMIZATION EXECU TION OPTIONS In those cases the calculation is only paused however as discussed in section 2 7 unless the two conditions above are satisfied In any case the calculation is not over until the lowest a is deep in the noise fitting regime This can be determined easily from the figure of log y versus log a on which you should see a plateau at high a an intermediate region of large slope and a quasi plateau a region of very small slope at low a The latter quasi plateau should be wide enough to be sure that it is not just a smaller slope region in the information fitting regime To verify that the lowest a is indeed in the noise fitting region see that the function AG versus wp or covariance eigenvector index contains only noise 2 14 How to improve the results Typically the first results you obtain with a given set of data are not the best you can obtain This can be because the default grid is not optimal for the spectrum or the default model needs some adjustments or because the errors used are not accurate The first step to improve the results is to define a grid more adapted to the spectrum using subsection FREQUENCY GRID PARAMETERS For example if some peaks are defined by only a few poi
48. t this parameter to yes to display figures during the preprocessing display advanced preprocessing figures Set this parameter to yes to display the advanced figures during preprocess ing Those figures show some intermediate quantities of different preprocessing operations 39 e print other parameters 3 2 3 2 1 Set this parameter to yes to display the parameters of file OmegaMaxEnt_other_params dat OPTIONAL MINIMIZATION TIME PARAMETERS OUTPUT FILES PARAMETERS output directory Use this parameter if you want to put the two output directories OmegaMax Ent_output_files and OmegaMaxEnt_final_result in another directory than the one provided at input directory or the current directory otherwise output file names suffix Use this parameter to add a suffix to the names of the output files see section 2 12 For example you can add parameter values specific to your data to archive your files more clearly The suffix you define will be added after the default descriptive part of the file names before tem P maximum alpha for which results are saved Use this parameter to choose the range of a for which the results are saved differ ently than how it is done by default as described in section 2 12 minimum alpha for which results are saved Use this parameter to choose the range of a for which the results are saved differ ently than how it is done by default as described in section 2 12
49. that parameter to provide the Re G Im G part of the matrix There must be an equal number of rows and columns in the file column of G tau in data file For imaginary time data use that parameter if the function value is not on the second column of your data file column of G tau error in error file For imaginary time data if an error file is provided use that parameter if the error is not on the second column imaginary time covariance file For imaginary time data use that parameter if you have a covariance matrix 36 3 1 3 FREQUENCY GRID PARAMETERS Matsubara frequency cutoff Use this parameter to set the maximum Matsubara frequency to be used during the calculation It is however recommended to read subsections 2 4 1 and 2 4 3 to learn how to use it properly spectral function width You can use this parameter to set the width of the main spectral region namely the dense part of the real frequency grid spectral function center Use this parameter to set the center of the main spectral region real frequency grid origin Use this parameter to ensure the grid contains a specific frequency real frequency step If you are using a uniform grid in the main spectral region you can use this pa rameter to set the step in that region real frequency grid file Use that parameter to provide a user defined frequency grid use parameterized real frequency grid To use grid parameters on the next line set this p
50. tions having the spectral representation 1 1 It is recommended to set evaluate moments to yes subsection COM PUTATION OPTIONS to verify that your value is close to the one extracted from your data and printed in the terminal 1 moment error If you provided the first moment use that parameter to provide its absolute error Otherwise the value default_error_M Aw tq will be used where default_error_M is a parameter in file OmegaMaxEnt_other_params dat and Awsig is the standard deviation of the spectrum 2d moment If you know the second moment of the spectral function use that parameter to pro vide it The second moment corresponds to the coefficient of 1 iw in the high 33 frequency expansion of of Matsubara functions having the spectral representation 1 1 It is recommended to set evaluate moments to yes subsection COM PUTATION OPTIONS to verify that your value is close to the one extracted from your data and printed in the terminal 21d moment error If you provided the second moment use that parameter to provide its absolute er ror Otherwise the value default_error_M defined in OmegaMaxEnt_other_params dat will be used as the relative error 3 moment If you know the third moment of the spectral function use that parameter to pro vide it The third moment corresponds to the coefficient of 1 w in the high frequency expansion of of Matsubara functions having the sp
51. ubsection COMPUTATION PARAMETERS e number of values of alpha computed in one execution Na IMIZATION EXECUTION OPTIONS subsection MIN max The parameter preprocess only can be used to do some verifications with input parameters and try different values before starting the actual calculation For example you can verify the values of moments with the ones extracted from the data or check that your grid has a reasonable number of frequencies If display preprocessing figures is enabled subsection PREPROCESSING EXECUTION OPTIONS the preprocessing stops when a group of figures is displayed and you have to close them to resume it This pause is not the right time to modify the parameter files however since they will not be read when execution resumes once the figures are closed Once you are satisfied with the input parameters you can disable preprocess only to start computing the spectrum The parameters minimum value of alpha and number of values of alpha computed in one execution can be used to look at the results at intermediate stages during mini mization time where the spectrum is computed as a function of a By default at each pause during that part of the calculation the program displays figures showing the results at the optimal a if it has been found and at the last lowest value of a computed A list of the output figures is given in section 2 8 Once you have closed all the figu
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