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Green/Continuation is an Analytical continuation toolkit for Green Software Package

Green/Continuation comes in two forms

• C++ application
• Python package

Green/Continuation has the following required external dependencies

• HDF5 library version >= 1.10.2
• Message Passing Interface >= 3.1 (for C++ application)
• Eigen3 library >= 3.4.0
• GNU Multiprecision library
• pybind11 (optional to build python wrapper)

To build Green/Continuation CMake version 3.18 or above is required

The following example will build, test and install Green/Continuation to /path/to/weakcoupling/install/dir directory.

$ git clone https://github.com/Green-Phys/green-ac
$ cd green-ac
$ mkdir build && cd build
$ cmake -DCMAKE_BUILD_TYPE=Release \
        -DCMAKE_INSTALL_PREFIX=/path/to/install/dir ..
$ make
$ make test
$ make install

We provide pre-built binaries for major Linux distributions and recent MacOS version via pip. For installation using pip simply type pip install green-ac. If pre-built binaries can not be used, package will be built from sources.

After the Green/Continuation is built and installed, spectral function could be obtained by calling

<install dir>/bin/ac.exe  --BETA <Inverse temperature> --grid_file <grid file>    \
   --input_file <input file> --output_file <output file> --group <HDF5 group with data> \
   --e_min -5.0 --e_max 5.0 --n_omega 4000 --eta 0.01 \
   --kind Nevanlinna

• BETA -- the inverse temperature that was used to obtain results
• grid_file -- name of the grid file that was used to obtain results
• input_file -- name of the file that contains imaginary time data
• output_file -- name of the file to store results of the continuation
• group -- name of the group that contains data and mesh datasets with imaginary time data and grid
• e_min -- lowest frequency on the real axis
• e_max -- largest frequency on the real axis
• n_omega -- number of frequency points on the real axis
• eta -- broadening parameter
• kind -- type of continuation to be used (current version only supports Nevanlinna)

After the completetion, results will be stored in group HDF5 group in the output_file file.

In addition to C++ application, Green/Continuation provides a convinient Python package that can work directly with numpy arrays. It supports two types of parallelism, using ProcessPoolExecutor from concurrent.futures and MPI parallelization using mpi4py library.

To use Green/Continuation simply import it in your script as

import green_ac

and call solve function with the following parameters:

• Type of the continuation (currently we only provide Nevanlinna)
• Matsubara frequency grid
• Real frequency grid
• Data in Matsubara frequency domain
• Precision

Here is an example how to obtain real frequency Green's function for a simple two-pole non-interacting Green's function with precision at least 512 bits:

imgrid = (2*np.linspace(-50,49,100) + 1) * 1.j *np.pi/ 2
grid = np.linspace(-2,2,1001) + 0.01j
data = 0.5*(1/(imgrid + 0.5) + 1/(imgrid - 0.5))
data_out = green_ac.solve("Nevanlinna", imgrid, grid, data, 512)

Here we have 100 positive and negative Matsubara frequencies, define real frequency grid to have 1000 points and to be from -2 to 2 with broadening parameter 0.01, We define Green's function on Matsubara grid with poles at -0.5 and 0.5. Real frequency data will be stored in data_out array.

This work is supported by National Science Foundation under the award OCA-2310582