Input Files

The input files for the ACFlow toolkit can be divided into two groups: data files and configuration files.

Data Files

The input data should be stored in some text-based files, which adopt the space-separated data format. For imaginary time Green's function, the data file should contain three columns. They represent $\tau$, $\bar{G}(\tau)$, and standard deviation of $\bar{G}(\tau)$. For fermionic Matsubara Green's function, the data file should contain five columns. They represent $\omega_n$, Re$G(i\omega_n)$, Im$G(i\omega_n)$, standard deviation of Re$G(i\omega_n)$, and standard deviation of Im$G(i\omega_n)$. For bosonic correlation function $\chi(i\omega_n)$, the data file should contain three columns. They represent $\omega_n$, Re$\chi(i\omega_n)$, and standard deviation of Re$\chi(i\omega_n)$.

Configuration Files

The configuration file adopts the TOML format. It is used to customize the computational parameters. It consists of one or more blocks. Possible blocks (or sections) of the configuration file include [BASE], [MaxEnt], [BarRat], [NevanAC], [StochAC], [StochSK], [StochOM], and [StochPX]. The [BASE] block is mandatory, while the other blocks are optional. A schematic configuration file (ac.toml) is listed as follows:

[BASE]
finput = "giw.data"
solver = "StochOM"
...

[MaxEnt]
method = "chi2kink"
...

[BarRat]
atype  = "cont"
...

[NevanAC]
pick   = true
...

[StochAC]
nfine  = 10000
...

[StochSK]
method = "chi2min"
...

[StochOM]
ntry   = 100000
...

[StochPX]
method = "mean"
...

In the [BASE] block, the analytic continuation problem is defined. The solver used to solve the problem must be assigned. The types of mesh, grid, default model function, and kernel function are also determined. The [MaxEnt], [BarRat], [NevanAC], [StochAC], [StochSK], [StochOM], and [StochPX] blocks are used to customize the corresponding analytic continuation solvers further. In Table 1-Table 8, all the possible input parameters for these blocks are collected and summarized. As for detailed explanations of these parameters, please see Parameters.

Parameter	Type	Default	Description
`finput`	string	''green.data''	Filename for input data.
`solver`	string	''MaxEnt''	Solver for the analytic continuation problem.
`ktype`	string	''fermi''	Type of kernel function.
`mtype`	string	''flat''	Type of default model function.
`grid`	string	''ffreq''	Grid for input data (imaginary axis).
`mesh`	string	''linear''	Mesh for output data (real axis).
`ngrid`	integer	10	Number of grid points.
`nmesh`	integer	501	Number of mesh points.
`wmax`	float	5.0	Right boundary (maximum value) of mesh.
`wmin`	float	-5.0	Left boundary (minimum value) of mesh.
`beta`	float	10.0	Inverse temperature.
`offdiag`	bool	false	Treat the off-diagonal part of matrix-valued function?
`fwrite`	bool	N/A	Are the analytic continuation results written into files?
`pmodel`	array	N/A	Additional parameters for customizing the default model.
`pmesh`	array	N/A	Additional parameters for customizing the mesh.
`exclude`	array	N/A	Restriction of energy range of the spectrum.

Table 1 | Possible parameters for the [BASE] block (See [BASE] Block).

Parameter	Type	Default	Description
`method`	string	''chi2kink''	How to determine the optimized $\alpha$ parameter?
`stype`	string	''sj''	Type of the entropic factor.
`nalph`	integer	12	Total number of the used $\alpha$ parameters.
`alpha`	float	1e9	Starting value for the $\alpha$ parameter.
`ratio`	float	10.0	Scaling factor for the $\alpha$ parameter.
`blur`	float	-1.0	Shall we preblur the kernel and spectrum?

Table 2 | Possible input parameters for the [MaxEnt] block, which are used to configure the solver based on the maximum entropy method (See [MaxEnt] Block).

Parameter	Type	Default	Description
`atype`	string	''cont''	Possible type of the spectrum.
`denoise`	string	''prony''	How to denoise the input data.
`epsilon`	integer	1e-10	Threshold for the Prony approximation.
`pcut`	float	1e-3	Cutoff for unphysical poles.
`eta`	float	1e-2	Tiny distance from the real axis.

Table 3 | Possible input parameters for the [BarRat] block, which are used to configure the solver based on the Barycentric rational function approximation (See [BarRat] Block).

Parameter	Type	Default	Description
`pick`	bool	false	Check the Pick criterion or not.
`hardy`	bool	false	Perform Hardy basis optimization or not.
`hmax`	integer	50	Upper cut off of Hardy order.
`alpha`	float	1e-4	Regulation parameter for smooth norm.
`eta`	float	1e-2	Tiny distance from the real axis.

Table 4 | Possible input parameters for the [NevanAC] block, which are used to configure the solver based on the Nevanlinna analytical continuation (See [NevanAC] Block).

Parameter	Type	Default	Description
`nfine`	integer	10000	Number of points of a very fine linear mesh.
`ngamm`	integer	512	Number of $\delta$ functions.
`nwarm`	integer	4000	Number of Monte Carlo thermalization steps.
`nstep`	integer	4000000	Number of Monte Carlo sweeping steps.
`ndump`	integer	40000	Intervals for monitoring Monte Carlo sweeps.
`nalph`	integer	20	Total number of the used $\alpha$ parameters.
`alpha`	float	1.0	Starting value for the $\alpha$ parameter.
`ratio`	float	1.2	Scaling factor for the $\alpha$ parameter.

Table 5 | Possible input parameters for the [StochAC] block, which are used to configure the solver based on the stochastic analytic continuation (Beach's algorithm. See [StochAC] Block).

Parameter	Type	Default	Description
`method`	string	''chi2min''	How to determine the optimized $\Theta$ parameter?
`nfine`	integer	100000	Number of points of a very fine linear mesh.
`ngamm`	integer	1000	Number of $\delta$ functions.
`nwarm`	integer	1000	Number of Monte Carlo thermalization steps.
`nstep`	integer	20000	Number of Monte Carlo sweeping steps.
`ndump`	integer	200	Intervals for monitoring Monte Carlo sweeps.
`retry`	integer	10	How often to recalculate the goodness-of-fit function.
`theta`	float	1e6	Starting value for the $\Theta$ parameter.
`ratio`	float	0.9	Scaling factor for the $\Theta$ parameter.

Table 6 | Possible input parameters for the [StochSK] block, which are used to configure the solver based on the stochastic analytic continuation (Sandvik's algorithm. See [StochSK] Block).

Parameter	Type	Default	Description
`ntry`	integer	2000	Number of attempts to figure out the solution.
`nstep`	integer	1000	Number of Monte Carlo sweeping steps per try.
`nbox`	integer	100	Number of rectangles to used construct the spectrum.
`sbox`	float	0.005	Minimum area of the randomly generated rectangles.
`wbox`	float	0.02	Minimum width of the randomly generated rectangles.
`norm`	float	-1.0	Is the norm calculated?

Table 7 | Possible input parameters for the [StochOM] block, which are used to configure the solver based on the stochastic optimization method (See [StochOM] Block).

Parameter	Type	Default	Description
`method`	string	''mean''	How to evaluate the final spectral density?
`nfine`	integer	100000	Number of points of a very fine linear mesh.
`npole`	integer	200	Number of poles.
`ntry`	integer	1000	Number of attempts to figure out the solution.
`nstep`	integer	1000000	Number of Monte Carlo sweeping steps per attempt / try.
`theta`	float	1e+6	Artificial inverse temperature $\Theta$.
`eta`	float	1e-4	Tiny distance from the real axis $\eta$.

Table 8 | Possible input parameters for the [StochPX] block, which are used to configure the solver based on the stochastic pole expansion (See [StochPX] Block).