Parameters

A comprehensive dictionary about parameters.

The official configuration file for the ACFlow toolkit is case.toml. This page contains all the valid parameters that can appear in case.toml. As for the format of case.toml, please look at case.toml.

Contents

• Parameters
• • Contents
  • [BASE] Block
  • • finput
    • solver
    • ktype
    • mtype
    • grid
    • mesh
    • ngrid
    • nmesh
    • wmax
    • wmin
    • beta
    • offdiag
    • fwrite
    • pmodel
    • pmesh
    • exclude
  • [MaxEnt] Block
  • • method
    • stype
    • nalph
    • alpha
    • ratio
    • blur
  • [BarRat] Block
  • • atype
    • denoise
    • epsilon
    • pcut
    • eta
  • [NevanAC] Block
  • • pick
    • hardy
    • hmax
    • alpha
    • eta
  • [StochAC] Block
  • • nfine
    • ngamm
    • nwarm
    • nstep
    • ndump
    • nalph
    • alpha
    • ratio
  • [StochSK] Block
  • • method
    • nfine
    • ngamm
    • nwarm
    • nstep
    • ndump
    • retry
    • theta
    • ratio
  • [StochOM] Block
  • • ntry
    • nstep
    • nbox
    • sbox
    • wbox
    • norm
  • [StochPX] Block
  • • method
    • nfine
    • npole
    • ntry
    • nstep
    • theta
    • eta

[BASE] Block

finput

Definition:

Filename for the input data. The input data should be stored in a column-wised and formated (CSV-like) text file.

Type:

String.

Example:

finput = "gtau.data"

Comment:

This parameter is mandatory.

solver

Definition:

This parameter specifies the solvers that used to solve the analytic continuation problem. Now the ACFlow toolkit supports seven different solvers. They are as follows:

• MaxEnt
• BarRat
• NevanAC
• StochAC
• StochSK
• StochOM
• StochPX

Here, MaxEnt means the maximum entropy method. The MaxEnt solver can be used to treat the correlators in Matsubara frequency or imaginary time axis. If solver = "MaxEnt", then the [MaxEnt] block must be available in the configuration file.

BarRat means the barycentric rational function approximation. The BarRat solver can be used to treat the correlators in Matsubara frequency axis only. If solver = "BarRat", then the [BarRat] block must be available in the configuration file.

NevanAC means the Nevanlinna analytical continuation. The NevanAC solver can be used to treat the fermionic correlators in Matsubara frequency. Note that this solver is extremely sensitive to the noise level of the input data.

StochAC means the stochastic analytic continuation method (K. S. D. Beach's algorithm). The StochAC solver can be used to treat the correlators in Matsubara frequency or imaginary time axis. If solver = "StochAC", then the [StochAC] block must be available in the configuration file.

StochSK means the stochastic analytic continuation method (A. W. Sandvik's algorithm). The StochSK solver can be used to treat the correlators in Matsubara frequency or imaginary time axis. If solver = "StochSK", then the [StochSK] block must be available in the configuration file.

StochOM means the stochastic optimization method. The StochOM solver can be used to treat the correlators in Matsubara frequency or imaginary time axis. If solver = "StochOM", then the [StochOM] block must be available in the configuration file.

StochPX means the stochastic pole expansion method. The StochPX solver can be used to treat the correlators in Matsubara frequency axis only. If solver = "StochPX", then the [StochPX] block must be available in the configuration file.

Type:

String.

Example:

solver = "MaxEnt"

Comment:

This parameter is mandatory. Overall, we recommend the MaxEnt, BarRat, and StochPX solvers.

ktype

Definition:

It denotes the type of kernel functions. Now the ACFlow toolkit supports three types of kernel functions. They are:

• fermi
• boson
• bsymm

Here, fermi means fermionic kernel function, which reads

\[K(\tau,\omega) = \frac{e^{-\tau\omega}}{1 + e^{-\beta\omega}},\]

and

\[K(\omega_n,\omega) = \frac{1}{i\omega_n - \omega}.\]

boson means bosonic kernel function, which reads

\[K(\tau,\omega) = \frac{\omega e^{-\tau\omega}}{1 - e^{-\beta\omega}},\]

and

\[K(\omega_n,\omega) = \frac{\omega}{i\omega_n - \omega}.\]

bsymm means symmetric bosonic kernel function, which reads

\[K(\tau,\omega) = \frac{\omega [e^{-\tau\omega} + e^{-(\beta - \tau)\omega}]} {1 - e^{-\beta\omega}},\]

and

\[K(\omega_n, \omega) = \frac{-2\omega^2}{\omega_n^2 + \omega^2}.\]

As for detailed formula for these kernel functions, please refer to the comments in src/kernel.jl.

Type:

String.

Example:

ktype = "fermi"

Comment:

This parameter is mandatory. It must be compatible with the grid parameter.

mtype

Definition:

It denotes the type of default model functions. Now the ACFlow toolkit supports the following choices:

• flat
• gauss
• 1gauss
• 2gauss
• lorentz
• 1lorentz
• 2lorentz
• risedecay
• file

Here, flat means the flat model (i.e., constant), gauss means the Gaussian model, 1gauss means the Shifted Gaussian model, 2gauss means the Two Gaussians model, lorentz means the Lorentzian model, 1lorentz means the Shifted Lorentzian model, 2lorentz means the Two Lorentzians model, and risedecay means the Rise-And-Decay model.

Besides flat and file, all the other model functions need additional parameters to customize them (Of course, the ACFlow toolkit will supplement default parameters). The parameters can be specified by the pmodel parameter.

Especially, if mtype = "file", then the default model function is encoded in model.inp. ACFlow will read this file and initialize the default model function automatically. Be careful, the mesh for this model function must be consistent with the one used in the analytic continuation calculations.

As for detailed formula for these models, please refer to the comments in src/model.jl.

Type:

String.

Example:

mtype = "flat"

Comment:

This parameter is mandatory. Only the MaxEnt solver need these model functions. The StochAC solver only supports the flat model. The StochSK, StochOM, and StochPX solvers are free of model functions.

grid

Definition:

This parameter specifies the grid's type for input data in imaginary axis. Now the ACFlow toolkit supports the following choices:

• ftime
• fpart
• btime
• bpart
• ffreq
• ffrag
• bfreq
• bfrag

Here, ftime means fermionic and imaginary time, btime means bosonic and imaginary time, ffreq means fermionic and Matsubara frequency, and bfreq means bosonic and Matsubara frequency. fpart means fermionic and imaginary time as well, but the grid of imaginary time might be incomplete. bpart is similar to fpart, but it is for the bosonic case. ffrag means fermionic and Matsubara frequency as well, but the grid of Matsubara frequency might be incomplete. bfrag is similar to ffrag, but it is for the bosonic case.

Type:

String.

Example:

grid = "ftime"

Comment:

This parameter is mandatory. It must be compatible with the ktype parameter. If grid is "bfrag", the first Matsubara frequency point, i.e. $i\omega_0 = 0$, should be kept. See also ngrid.

mesh

Definition:

This parameter specifies the mesh's type for output data (usually the spectral functions) in real axis. Now the ACFlow toolkit supports the following choices:

• linear
• tangent
• lorentz
• halflorentz

Here, linear means the Linear mesh, tangent means the Tangent mesh, lorentz means the Lorentzian mesh, and halflorentz means the Half-Lorentzian mesh.

Notes that only the linear mesh is uniform, the other three meshes are non-uniform. And the halflorentz mesh is defined on the positive-half axis only.

Type:

String.

Example:

mesh = "linear"

Comment:

This parameter is mandatory. See also nmesh.

ngrid

Definition:

Number of grid points. The parameter, together with the beta and grid parameters, controls the generation of grid for input data.

Type:

Integer.

Example:

ngrid = 10

Comment:

This parameter is mandatory. It must be compatible with the input data. See also grid.

nmesh

Definition:

Number of mesh points. The parameter, together with the wmax, wmin, and mesh parameters, controls the generation of mesh for output data.

Type:

Integer.

Example:

nmesh = 501

Comment:

This parameter is mandatory. See also mesh.

wmax

Definition:

Right boundary (maximum value) of mesh. Note that wmax should be always greater than wmin.

Type:

Float.

Example:

wmax = 10.0

Comment:

This parameter is mandatory.

wmin

Definition:

Left boundary (minimum value) of mesh. Note that wmax should be always greater than wmin.

Type:

Float.

Example:

wmin = -10.0

Comment:

This parameter is mandatory.

beta

Definition:

Inverse temperature $\beta$. It is equal to $1/T$.

Type:

Float.

Example:

beta = 10.0

Comment:

This parameter is mandatory. This parameter must be compatible with the input data and grid. Specifically, for the imaginary time axis, the last grid point should be $\beta$. As for the Matsubara frequency axis, the difference between two successive grid points should be $\pi/\beta$.

offdiag

Definition:

Is the input correlator the offdiagonal part in matrix-valued function? As for the offdiagonal correlator, the corresponding spectral function might be not positive-definite. Some tricks have been implemented to cure this issue.

Type:

Bool.

Example:

offdiag = false

Comment:

This parameter is mandatory. This parameter is useful for the MaxEnt and StochPX solvers only.

fwrite

Definition:

Are the analytic continuation results written into external files? If it is false, then only the terminal output is retained and all the other outputs are disable. By default (if this parameter is missing or true), the files should be generated.

Type:

Bool.

Example:

fwrite = false

Comment:

This parameter is optional.

pmodel

Definition:

Additional parameters for customizing the model functions. Note that the gauss, lorentz, and risedecay models need one parameter $\Gamma$. The 1gauss and 1lorentz models need two parameters, $\Gamma$ and $s$. The 2gauss and 2lorentz models need three parameters, $\Gamma$, $s_1$, and $s_2$.

The pmodel parameter is used to define these parameters. If there is only one element in pmodel, then $\Gamma$ = pmodel[1]. If there are two elements in pmodel, then $\Gamma$ = pmodel[1] and $s$ = pmodel[2]. If there are three elements in pmodel, then $\Gamma$ = pmodel[1], $s_1$ = pmodel[2], and $s_2$ = pmodel[3].

Type:

Array.

Example:

pmodel = [1.0]

Comment:

This parameter is optional. The default values for $\Gamma$, $s$, $s_1$, and $s_2$ are 2.0, 2.0, -2.0, and 2.0, respectively.

pmesh

Definition:

Additional parameters for customizing the mesh. The tangent mesh needs the $f_1$ parameter. The lorentz and halflorentz meshes need the cut parameter. The pmesh parameter can be used to setup the two parameters. If pmesh contains one element or more than one elements, then $f_1 \equiv$ cut $\equiv$ pmesh[1].

Type:

Array.

Example:

pmesh = [2.1]

Comment:

This parameter is optional. The default values for $f_1$ and cut are 2.1 and 0.01, respectively. See also mesh.

exclude

Definition:

Restriction of the energy range of the calculated spectral functions. This features is implemented by the StochAC, StochSK, StochOM, and StochPX solvers. In these solvers, the $\delta$ or box functions, which are used to mimic the spectral functions, are restricted to live out of the given energy ranges. For example, exclude = [[8.0,16.0]] means that the energy range [8.0,16.0] is strictly forbidden.

Type:

Array.

Example:

exclude = [[-8.0,-4.0],[4.0,8.0]]

Comment:

This parameter is optional. If you are using the MaxEnt solver, this parameter will be ignored. If solver = StochPX and offdiag = true, this parameter is mandatory. In this case, it is used to restrict the regions that the poles with positive weights can survive (or equivalently, the regions that the poles with negative weights can survice are also determined). For example, if exclude = [[-3.0,3.0]], wmin = -5.0, and wmax = 5.0, then the regions for poles with negative weights are [-3.0,3.0], while the regions for poles with positive weights are [-5.0,-3.0] U [3.0,5.0].

[MaxEnt] Block

method

Definition:

How to determine the optimized $\alpha$ parameter? The MaxEnt solver supports four different algorithms. They are

• historic
• classic
• bryan
• chi2kink

Usually, the chi2kink algorithm is preferred.

Type:

String.

Example:

method = "bryan"

Comment:

This parameter is mandatory. As for the underlying principles of these algorithms, please see Maximum Entropy Method.

stype

Definition:

Type of the entropic factor. The MaxEnt solver supports two schemes. They are

• sj
• br

Here, sj means the Shannon-Jaynes entropy, while br means the Bayesian Reconstruction entropy. Usually, the Shannon-Jaynes entropy is preferred, since with it the positivity of the generated spectrum is always guaranteed. The Bayesian Reconstruction entropy tends to yield sharp features.

Type:

String.

Example:

stype = "sj"

Comment:

This parameter is mandatory. As for the underlying principles of these entropic factors, please see Maximum Entropy Method.

nalph

Definition:

Total number of the chosen $\alpha$ parameters.

Type:

Integer.

Example:

nalph = 12

Comment:

This parameter is mandatory. Only the chi2kink algorithm needs this parameter to control the number of $\alpha$ parameters.

alpha

Definition:

Starting value for the $\alpha$ parameter. The MaxEnt solver always starts with a huge $\alpha$ parameter, and then decreases it gradually.

Type:

Float.

Example:

alpha = 1e9

Comment:

This parameter is mandatory. It should be a very large number, such as $10^9 \sim 10^{13}$.

ratio

Definition:

Scaling factor for the $\alpha$ parameter. The next $\alpha$ is equal to the current $\alpha$ divided by ratio.

Type:

Float.

Example:

ratio = 10.0

Comment:

This parameter is mandatory. It muse be larger than 1.0.

blur

Definition:

Sometimes, the kernel functions and spectral functions can be preblurred to obtain smoother results. Shall we preblur them? If blur is larger than zero, then it means the blur parameter. If blur is smaller than zero, then it means that the preblur feature is disable.

Type:

Float.

Example:

blur = -1.0

Comment:

This parameter is mandatory.

[BarRat] Block

atype

Definition:

Possible type of the spectrum.

• cont
• delta

If it is cont, it means that the spectrum should be board and continuous. If it is delta, it means that the spectrum consists a few $\delta$-like peaks. The BarRat solver will deduce the positions of the poles from the barycentric rational function approximation, and then the BFGS algorithm is used to determine the weights / amplitudes of these poles. The original and real-frequency Green's function are then reconstructed by using the pole representation.

Type:

String.

Example:

atype = "cont"

Comment:

This parameter is mandatory. If atype is "delta", then the pcut and eta parameters will take effect. On the contrary, if atype is "cont", then the pcut and eta parameters will be ignored.

denoise

Definition:

This parameter specifies how to denoise the input data.

• none
• prony_s
• prony_o

The BarRat solver will adopt the Prony approximation to approximate the Matsubara data and suppress the noise. The denoise parameter is used to control whether the Prony approximation is actived. If it is "none", the Prony approximation is disabled. If it is "prony_s", the Prony approximation will run once, and its accuracy is controlled by the epsilon parameter. If it is "prony_o", an optimal Prony approximation is automatically determined. In such a case, the epsilon parameter is nonsense.

If the Prony approximation is activated, the grid parameter should not be ffrag or bfrag.

Type:

String.

Example:

denoise = "none"

Comment:

This parameter is mandatory. If the noise level is obvious, please set denoise to "prony_s" or "prony_o". If the noise level is small (such as $\epsilon < 10^{-8}$), Prony approximation could lead to worse results.

epsilon

Definition:

Threshold for the Prony approximation. It is used to control the accuracy of the Prony approximation. It can be considered as a measurement of the noise level of the input Matsubara data.

Type:

Float.

Example:

epsilon = 1e-10

Comment:

This parameter is mandatory. But it is only useful when denoise is not "none". See denoise for more details.

epsilon should be set to the noise level of the input Matsubara data. It should not be too small or too large. In principle, $\sigma[1] < \rm{epsilon} < \sigma[\rm{end}]$, where $\sigma$ are the singular values.

pcut

Definition:

Cutoff for unphysical poles. This parameter is used to filter the unphysical poles generated by the AAA algorithm. Given a pole, if the imaginary part of its location or weight is larger than pcut, it should be discarded or removed.

Sometimes if pcut is too small, all of the poles are consiered as unphysical and removed. The ACFlow will throw an error and stop. To cure this problem, please increase pcut and redo the calculation.

Type:

Float.

Example:

pcut = 1e-3

Comment:

This parameter is mandatory. But it is only useful when atype = "delta". See atype for more details.

eta

Definition:

Tiny distance from the real axis. It is used to construct the retarded Matsubara Green's function within the pole representation.

If eta is smaller than 1.0, then

\[G(\omega) = \sum_j \frac{\operatorname{Re} A_j}{\omega - \operatorname{Re} P_j + i\eta }\]

If eta is larger than 1.0, then

\[G(\omega) = \sum_j \frac{A_j}{\omega - \operatorname{Re} P_j + i(\eta - 1) }\]

Type:

Float.

Example:

eta = 1e-2

Comment:

This parameter is mandatory. But it is only useful when atype = "delta". See atype for more details.

Well, for normal Green's function, the imaginary parts of A and P must be quite small. So please let eta < 1.0. But for anormal Green's function, the imaginary parts of A and P might be quite large. You can set eta > 1.0. Anyway, more tests are essential.

[NevanAC] Block

pick

Definition:

Check the Pick criterion or not. If pick is true, ACFlow will try to figure out the optimal number of the input data (i.e., how many data points are retained for further postprocessing) by using the Pick criterion.

Type:

Bool.

Example:

pick = true

Comment:

This parameter is mandatory.

hardy

Definition:

Perform Hardy basis optimization or not. The spectrum obtained by the Nevanlinna analytical continuation is usually wiggly. So, the Hardy basis optimization can help us smooth the spectrum.

Type:

Bool.

Example:

hardy = true

Comment:

This parameter is mandatory. See also hmax.

hmax

Definition:

Upper cut off of Hardy order. In principle, the larger the Hardy order is, the smoother the obtained spectrum is. Usually hmax = 20 is enough to get smooth spectrum.

Type:

Integer.

Example:

hmax = 50

Comment:

This parameter is mandatory. See also hardy.

alpha

Definition:

Regulation parameter for smooth norm. If the alpha parameter is too large, the detailed features in the spectra could be smeared out.

Type:

Float.

Example:

alpha = 1e-4

Comment:

This parameter is mandatory.

eta

Definition:

Tiny distance from the real axis. It is used to construct the Hardy matrix, instead of the Green's function.

Type:

Float.

Example:

eta = 1e-2

Comment:

This parameter is mandatory.

[StochAC] Block

nfine

Definition:

Number of points of a very fine linear mesh. This mesh is for the $\delta$ functions.

Type:

Integer.

Example:

nfine = 10000

Comment:

This parameter is mandatory.

ngamm

Definition:

Number of $\delta$ functions. Their superposition is used to mimic the spectral functions.

Type:

Integer.

Example:

ngamm = 512

Comment:

This parameter is mandatory.

nwarm

Definition:

Number of Monte Carlo thermalization steps.

Type:

Integer.

Example:

nwarm = 4000

Comment:

This parameter is mandatory.

nstep

Definition:

Number of Monte Carlo sweeping steps.

Type:

Integer.

Example:

nstep = 4000000

Comment:

This parameter is mandatory.

ndump

Definition:

Intervals for monitoring Monte Carlo sweeps. For every ndump steps, the StochAC solver will try to output some useful information to help diagnosis.

Type:

Integer.

Example:

ndump = 40000

Comment:

This parameter is mandatory.

nalph

Definition:

Total number of the chosen $\alpha$ parameters.

Type:

Integer.

Example:

nalph = 20

Comment:

This parameter is mandatory.

alpha

Definition:

Starting value for the $\alpha$ parameter. The StochAC solver always starts with a small $\alpha$ parameter, and then increases it gradually.

Type:

Float.

Example:

alpha = 1.0

Comment:

This parameter is mandatory.

ratio

Definition:

Scaling factor for the $\alpha$ parameter. It should be larger than 1.0.

Type:

Float.

Example:

ratio = 1.2

Comment:

This parameter is mandatory.

[StochSK] Block

method

Definition:

How to determine the optimized $\Theta$ parameter? The StochSK solver supports two different algorithms. They are

• chi2min
• chi2kink

Usually, the chi2min algorithm is preferred. This algorithm is suggested by Shao and Sandvik et al. See Stochastic Analytic Continuation 1 for more details.

Type:

String.

Example:

method = "chi2min"

Comment:

This parameter is mandatory.

nfine

Definition:

Number of points of a very fine linear mesh. This mesh is for the $\delta$ functions.

Type:

Integer.

Example:

nfine = 100000

Comment:

This parameter is mandatory.

ngamm

Definition:

Number of $\delta$ functions. Their superposition is used to mimic the spectral functions.

Type:

Integer.

Example:

ngamm = 1000

Comment:

This parameter is mandatory.

nwarm

Definition:

Number of Monte Carlo thermalization steps.

Type:

Integer.

Example:

nwarm = 1000

Comment:

This parameter is mandatory.

nstep

Definition:

Number of Monte Carlo sweeping steps.

Type:

Integer.

Example:

nstep = 20000

Comment:

This parameter is mandatory.

ndump

Definition:

Intervals for monitoring Monte Carlo sweeps. For every ndump steps, the StochSK solver will try to output some useful information to help diagnosis.

Type:

Integer.

Example:

ndump = 200

Comment:

This parameter is mandatory.

retry

Definition:

How often to recalculate the goodness-of-fit function (it is actually $\chi^2$) to avoid numerical deterioration.

Type:

Integer.

Example:

retry = 10

Comment:

This parameter is mandatory.

theta

Definition:

Starting value for the $\Theta$ parameter. The StochSK solver always starts with a huge $\Theta$ parameter, and then decreases it gradually.

Type:

Float.

Example:

theta = 1e+6

Comment:

This parameter is mandatory.

ratio

Definition:

Scaling factor for the $\Theta$ parameter. It should be less than 1.0.

Type:

Float.

Example:

ratio = 0.9

Comment:

This parameter is mandatory.

[StochOM] Block

ntry

Definition:

Number of attempts to figure out the solution.

Type:

Integer.

Example:

ntry = 2000

Comment:

This parameter is mandatory.

nstep

Definition:

Number of Monte Carlo steps per try.

Type:

Integer.

Example:

nstep = 1000

Comment:

This parameter is mandatory.

nbox

Definition:

Number of boxes. Their superposition is used to construct the spectral functions.

Type:

Integer.

Example:

nbox = 100

Comment:

This parameter is mandatory.

sbox

Definition:

Minimum area of the randomly generated boxes.

Type:

Float.

Example:

sbox = 0.005

Comment:

This parameter is mandatory.

wbox

Definition:

Minimum width of the randomly generated boxes.

Type:

Float.

Example:

wbox = 0.02

Comment:

This parameter is mandatory.

norm

Definition:

Is the norm calculated? If norm is larger than 0.0, it denotes the normalization factor. If norm is smaller than 0.0, it means that the normalization condition is ignored.

Type:

Float.

Example:

norm = -1.0

Comment:

This parameter is mandatory.

[StochPX] Block

method

Definition:

How to evaluate the final spectral density? The StochPX solver supports two different algorithms. They are

• mean
• best

If method = "mean", then the solver will try to calculate an averaged spectrum from some selected good solutions. If method = "best", then the solver will pick up the best solution (which should exhibit the smallest goodness-of-fit functional $\chi^2$).

Type:

String.

Example:

method = "mean"

Comment:

This parameter is mandatory. Note that the "mean" method is suitable for the condensed matter cases (broad and smooth peaks), while the "best" method is useful for the molecule cases (sharp peaks).

nfine

Definition:

Number of points of a very fine linear mesh. This mesh is for the poles.

Type:

Integer.

Example:

nfine = 100000

Comment:

This parameter is mandatory.

npole

Definition:

Number of poles on the real axis, which is used to mimic the Matsubara Green's function.

Type:

Integer.

Example:

npole = 200

Comment:

This parameter is mandatory. For condensed matter cases, npole should be quite large. While for molecule cases, npole should be small.

ntry

Definition:

Number of attempts to figure out the solution.

Type:

Integer.

Example:

ntry = 1000

Comment:

This parameter is mandatory.

nstep

Definition:

Number of Monte Carlo sweeping steps per attempt / try.

Type:

Integer.

Example:

nstep = 1000000

Comment:

This parameter is mandatory. This parameter is related to the npole parameter. If npole is large, nstep could be small. If npole is small, nstep should be large.

theta

Definition:

Artificial inverse temperature $\Theta$. When it is increased, the transition probabilities of Monte Carlo updates will decrease.

Type:

Float.

Example:

theta = 1e+6

Comment:

This parameter is mandatory. The users can check the stat.data file to judge whether the theta parameter is reasonable.

eta

Definition:

Tiny distance from the real axis $\eta$, which is used to reconstruct the retarded Green's function and the spectral density. When it is increased, the spectral density will be become more and more smooth.

Type:

Float.

Example:

eta = 1e-4

Comment:

This parameter is mandatory.