sac - A Julia program to run the Stochastic Analytic Continuation Method

Currently supports unconstrained sampling, the $\delta$-function edge parameterization, and the power-law edge constrained parameterization for fermionic or bosonic spectral functions.

The Stochastic Analytic Continuation (SAC) Method

This library contains code for solving the challenging inverse problem of numerical analytic continuation using the Stochastic Analytic Continuation (SAC) method. Provided with QMC-generated imaginary time correlation function data G(τ), the SAC method will generate a spectral function S(ω) satisfying the integral

$$G(\tau) = \int_{\infty}^{\infty} K(\tau, \omega) S(\omega),$$

where $K(\tau, \omega)$ is the kernel, specific to the type of spectral function being resolved.

Where SAC differs from many other numerical analytic contination methods is that it represents the spectrum as a sum of a large number of $\delta$-functions, and relies on the stochastic averaging of $\delta$-function configurations to produce a smooth $S(\omega)$. This also allows for the resolution of spectra with sharp features, by imposing appropriate constraints on the sampling space of the $\delta$-functions.

The full formalism, and many examples, can be found in the Physics Reports article Progress on stochastic analytic continuation of quantum Monte Carlo data. This library is based heavily off the procedures laid out in this article, but also includes some extensions and more recent developments.

Prepare G($\tau$) data

To run the SAC program, you first need to prepare your QMC-generated G(τ) data. The program reads in a file t.in containing the covariance (in imaginary time) matrix for your G(τ) data (eigenvectors and eigenvalues). To generate this file, use make_tin.jl in the folder sac/process_G, which reads in your G(τ) in the form of a full list of G(τ) bins (not bin averages). The two input files for make_tin.jl are cor.dat and tgrid.dat (see examples of these files in the process_G folder). Note: you need to edit the inverse temperature β set in the run function in make_tin.jl to reflect the inverse temperature used in your specific data set

cor.dat

The file cor.dat contains the raw G(τ) bins. The value of G(τ_i) for each bin is listed (one value per row), and each bin is seperated by "1," which is used a an indicator to seperate the bins. This will look like:

cor.dat
1
G_1(τ_0)
G_1(τ_1)
...
G_1(τ_N-1)
G_1(β)
1
G_2(τ_0)
G_2(τ_1)
...
G_2(τ_N-1)
G_2(β)
1
...

For proper normalization of the finite temperature fermionic spectral function, your G(τ) data must include both G(τ = 0) and G(τ = β).

If you generate your G(τ) data using either

• a zero temperature QMC method and the spectral function is for a fermionic operator
• a finite temperature QMC method and the spectral function is for a bosonic operator

then use make_tin_zeroT.jl, which treats the normalization of A(ω) differently. In this case, the only required τ point is G(τ = 0). Make sure to change the the inverse temperature β set in the run function in make_tin_zero.jl (if bosonic, for zero temperature fermionic spectra this value ignored).

tgrid.dat

The file tgrid.dat contains the τ grid that G(τ) is evaluated on. For the file, each value of τ_i is listed, one value per row. This will look like:

tgrid.dat
τ_0
τ_1
...
τ_N-1
τ_N

Running make_tin.jl will generate a t.in file containing G(τ) averaged over all bins, the corresponding error bars, and the covariance matrix. You can move this file into the sac/in_files directory.

After generating, renaming, and moving the t.in file, navigate to the folder corresponding to the SAC paramterization you want to use (sac/free, (sac/peak, or sac/edge), and follow the instructions in the README.md files (follow hyperlinks above).

This library also contains code to run cross validation as a tool for model selection. This can be used to select the most appropriate SAC paramaterization when multiple can produce spectra with acceptable $\chi^2$ values. You can find details on the implmentation here.