SEET Example

Prerequsite

System libraries:

1. MPI
2. HDF5
3. BLAS
4. CMake (version >= 3.27)
5. C++17 compatible compiler

Third party libraries:

1. Eigen3
2. Boost

Install dependencies

First, clone required repositories:

git clone https://github.com/ALPSCore/ALPSCore
git clone https://github.com/opencollab/arpack-ng
git clone https://github.com/Q-Solvers/EDLib
git clone https://github.com/Green-Phys/seet_solvers

1. ALPSCore – Core libraries of ALPS software package

cmake -S ALPSCore -B build --install-prefix `pwd`/install/ALPSCore
cmake --build build -j 32
cmake --build build -t test install
rm -rf build

2. ARPACK – Arnoldi/Lancsoz factorization library

cmake -S arpack-ng -B build \
  --install-prefix `pwd`/install/arpack \
  -DCMAKE_BUILD_TYPE=Release
cmake --build build -j 32
cmake --build build -t test install
rm -rf build

3. EDLib – Exact diagonalization library for quantum electron problems

cmake -S EDLib -B build --install-prefix `pwd`/install/EDLib \
  -DCMAKE_BUILD_TYPE=Release \
  -DALPSCore_DIR=`pwd`/install/ALPSCore/share/ALPSCore \
  -DARPACK_DIR=`pwd`/install/arpack
cmake --build build -j 32
cmake --build build -t install
rm -rf build

4. seet_solver – Exact diagonalization solver for SEET impurity problem

cmake -S seet_solvers -B build \
   --install-prefix `pwd`/install/seet_solvers \
   -DCMAKE_BUILD_TYPE=Release         \
   -DALPSCore_DIR=`pwd`/install/ALPSCore/share/ALPSCore  \
   -DARPACK_DIR=`pwd`/install/arpack \
   -DEDLib_DIR=`pwd`/install/EDLib/share/EDLib/cmake
cmake --build build -j 32
cmake --build `pwd`/build -t install
rm -rf build

SEET framework build

git clone https://github.com/Green-Phys/green-mbpt
cd green-mbpt
git checkout SEET
cd ..
cmake -S green-mbpt -B build \
   --install-prefix `pwd`/install/green-mbpt \
   -DCMAKE_BUILD_TYPE=Release
cmake --build build -j 32
cmake --build build -t test install
rm -rf build

First, we assume that we have solved weak-coupling problem for the input file input.h5 and the results are in the sim.h5 file. The input data for SEET can be prepared with the help of init_seet.py file which is available in the SEET installation directory:

python <installation path>/python/init_seet.py --orth true --gf2_input_file sim.h5 \
       --transform_file transform.h5 --active_space 0 1 --active_space 2 3   \
       --from_ibz true --orth_method symmetrical_orbitals --input_file input.h5

The various parameters used in this step primarily control the orbitals used in the active space and the type of orthogonalization:

• --orth=true – requests orthogonalization of orbitals,
• --input_file – path to the input file for weak-coupling (e.g., GW, GF2) solution,
• --gf2_input_file – path to the weak-coupling output file,
• --transform_file – path to the output file containing transformation matrices,
• --active_space – multiple sets of of orbital indices to define the active space for impurities,
• --from_ibz – set to true if the sim.h5 is in the reduced Brillouin Zone,
• --orth_method – type of orthogonalization (available options: natural_orbitals, symmetrical_orbitals, and canonical_orbitals),

Selecting active-space orbitals

One of the most important step in SEET is to select the correct orbitals to form the active space, also known as the impurity. The process generally involves heuristics and analysis of the weak-coupling result. For instance, one can look at occupation number of symmetrized atomic orbitals (SAO) and identify half-filled $d$-orbitals as impurity. For further details, please see the SEET theory page and references therein.

In the example above, we construct orthogonal transformation matrices using symmetrical orthogonalization. We choose two active subspaces of correlated orbitals ({0, 1} and {2, 3}). Note that these are orbital numbers in the chosen orthogonal basis. Since Green results are obtained in the reduced Brillouin zone we also specify --from_ibz=true.

As the result we will get transform.h5 file with orthogonal trasformation matrices and projection matrices.

The next preprocessing step for SEET solver is the transformation of the active-space two-electron integrals to the orthonormal basis-set selected and prepared in the previous step. This is done by calling

<installation path>/bin/int-transform.exe --input_file transform.h5 --in_file input.h5 --in_int_file df_int --transform 1

Here the following parameters have to be provided:

• --input_file – file with the trasformation matrices that has been obtained at the previous step
• --in_file – input file for the weak-coupling problem, i.e., input.h5 from the previous step
• --in_int_file – path to two body integrals that will be used for impurity problem. Note that Green has two sets of integrals, one to be used for Hartree-Fock solution, and one that is used for GW calculations. We strongly advice using GW integrals as they contain additional finite-size correction.

This procedure is time consuming and we advice to submit it as job on cluster.

After all the preparations are done, SEET can be run as

<installation path>/bin/embedding.exe --scf_type=GW --BETA 100  \
  --grid_file ir/1e4.h5 --itermax 1 --results_file sim.seet.h5 --weak_results sim.h5 --embedding_type SEET  \
  --mixing_type CDIIS --diis_start 2 --diis_size 5 --mixing_weight 0.3 \
  --seet_input transform.h5 --bath_file bath.txt  \
  --impurity_solver_exec <path to seet ED solver> \
  --impurity_solver_params " --arpack.NEV=8 --arpack.NCV=20 --lanc.NOMEGA=1000 --FREQ_FILE=<installation path>/share/ir/1e4.h5 --FREQ_PATH=/fermi/ngrid " \
  --dc_data_prefix "dc_int" \
  --seet_root_dir "./seet" \
  --spin_symm true

The following parameters in additional to regular Green parameters are used

• --weak_results – initial results obtained from weak-coupling solver
• --embedding_type – type of SEET, FC_SEET for full self-consistency, SEET for inner self-consistency
• --seet_input – transformation matricies for impurity problems
• --bath_file – initial guess for impurity bath parameters
• --impurity_solver_exec – path to seet_solvers exact diagonalization executable
• --impurity_solver_params – parameters for Exact diagonalization solver
• --dc_data_prefix – prefix for double-counting directories (has to be dc_int)
• --seet_root_dir – root directory for SEET intermediate input/outputs
• --spin_symm – whether the bath spin-symmetrization is needed during bath-fitting

Bath file

The bath file provides an initial guess for constructing the impurity-bath interaction in the Anderson impurity model for the active space. This interaction is constructed by fitting the hybridization to the form

$$ \Delta_{ij \in A_\lambda}(\omega_n) = \sum\limits_{b}\frac{V_{ib}V^*_{bj}}{\omega_n - \epsilon_b}, $$

where the interaction $V$ and energy $\epsilon$’s are both unknowns (see Theory for more details).

For our example, the bath file may be initialized as:

2 6
3 3
0.5 0.5 0.5 -1.0 0 1.0
0.5 0.5 0.5 -1.0 0 1.0
2 6
3 3
0.5 0.5 0.5 -1.0 0.0 1.0
0.5 0.5 0.5 -1.0 0.0 1.0

The format for bath file is as follows (example below):

• For each impurity, we first specify two numbers: number of impurity orbitals and total number of bath sites. Here, we have 2 impurity orbitals and 6 bath sites for each impurity.
• In the next line, we specify the number of bath sites for each impurity orbital, which we choose as {3, 3}.
• Finally, for each impurity orbital we specify the initial values for $V_k$ and for $\epsilon_k$. In our example, we initialize all the $V_k$’s to 0.5 and the $\epsilon_k$’s to {-1.0, 0.0, 1.0}.