Exact diagonalization solver: scipy/sparse¶
scipy/sparse computes the Green’s function by the exact diagonalization (ED) method. Small matrices are diagonalized by full diagonalization method implemented in scipy.linalg, while large mtrices are treated by sparse solvers implemented in scipy.sparse.linalg.
Features¶
Arbitrary temperature
All interactions available in
DCoreare supported.Support thread parallelization by OpenMP (via NumPy and SciPy).
Install¶
The following Python library needs to be installed:
SciPy
Input parameters¶
Parameters for this solver are as follows:
[impurity_solver]
name = scipy/sparse
n_bath{int} = 0
# fit_gtol{float} = 1e-5
# dim_full_diag{int} = 10000
# particle_numbers{str} = [1,2,3]
# weight_threshold{float} = 1e-6
# n_eigen{int} = 100
# eigen_solver{str} = eigsh
# gf_solver{str} = bicgstab
# check_n_eigen{bool} = False
# check_orthonormality{bool} = False
The first two parameters are mandatory. The remaining parameters, which are commented out, are optional.
The table below shows the detailed description of the parameters.
Name |
Type |
Default |
Description |
|---|---|---|---|
n_bath |
int |
0 |
Number of bath sites. See Pomerol solver for details. |
fit_gtol |
float |
1e-5 |
Tolerance for the fitting of the hybridization function. See Pomerol solver for details. |
dim_full_diag |
int |
10000 |
Maximum dimension of the matrix to be diagonalized by full diagonalization method. If the matrix is larger than this value, it will be treated by sparse solver. |
particle_numbers |
str |
auto |
Particle numbers to be considered. Allowed inputs are ‘auto’, ‘all’, or ‘[1,2,3]’ etc. By default (‘auto’), a minimal set of particle numbers is evaluated by prescreening calculations. If the set is known in advance, one can specify it in a list format, e.g., [1,2,3]. All particle numbers are considered if ‘all’ is chosen. Note that all states necessary to the Green’s function calculation, namely, thermally occupied N-particle states and N±1 states, should be included. |
weight_threshold |
float |
1e-6 |
Threshold for the Boltzmann factor. States with Boltzmann factor smaller than this value are ignored in the Green’s function calculation. |
n_eigen |
int |
100 |
Number of eigenvalues to be computed in each \(N\)-particle subspace by the sparse solver. |
eigen_solver |
str |
eigsh |
Name of the eigenvalue solver to be used. Available option is ‘eigsh’ only. |
gf_solver |
str |
bicgstab |
Name of the linear equation solver to be used for Green’s function calculation. Available options are ‘spsolve’, ‘bicg’, ‘bicgstab’, ‘cg’, ‘cgs’, ‘gmres’, ‘lgmres’, ‘minres’, ‘qmr’, ‘gcrotmk’, ‘tfqmr’. See SciPy official document for details. |
check_n_eigen |
bool |
True |
If True (default), calculation stops if |
check_orthonormality |
bool |
True |
If True (default), orthonormality of the eigenvectors is checked. If the check fails, calculation stops. |
Standard output¶
The standard output of the solver is saved in a file stdout.log in directory work/imp_shell0_ite1, where shell0 and ite1 stand for the 0-th shell and the 1-st iteration, respectively.
Some relevant output are explained below.
Particle-number conservation:
N dim[N]
0 1
1 10
2 45
3 120
4 210
5 252
6 210
7 120
8 45
9 10
10 1
This shows the dimension of the Hamiltonian matrix for each particle number.
When particle_numbers='auto' (default), a prescreening calculation is done to find a minimal set of particle numbers.
Prescreening -- Solving the eigenvalue problem...
N = 0 (dim = 1)
Time: 0m0.000s
N = 1 (dim = 10)
Time: 0m0.001s
...
Finish the eigenvalue problem
Time: 0m0.006s
Summary of lowest energy state in each N block:
N dim E_min weight_rel
------------------------------------
0 1 0.00e+00 1.4e-87
1 10 -4.00e+00 3.3e-70
2 45 -8.00e+00 7.7e-53
3 120 -1.20e+01 1.8e-35
4 210 -1.60e+01 4.2e-18
5 252 -2.00e+01 1.0e+00
6 210 -1.60e+01 4.2e-18
7 120 -1.20e+01 1.8e-35
8 45 -8.00e+00 7.7e-53
9 10 -4.00e+00 3.3e-70
10 1 1.82e-09 1.4e-87
Particle numbers of thermally occupied states:
[5]
Particle numbers to be considered:
[4 5 6]
The result of the prescreening is shown above. In this example, only N=5 includes thermally occupied states. For Green’s function calculation, particle numbers N plus/minus 1 should be included as shown in the final line.
Next, the eigenvalue problems are solved for the specific particle numbers.
Solving the eigenvalue problem...
N = 4 (dim = 210)
full diagonalization
Time: 0m0.009s
N = 5 (dim = 252)
full diagonalization
Time: 0m0.010s
N = 6 (dim = 210)
full diagonalization
Time: 0m0.006s
Finish the eigenvalue problem
Time: 0m0.216s
The time taken for solving the eigenvalue problem is shown for each particle number. The solver uses full diagonalization method for matrices smaller than dim_full_diag, while it switches to sparse solver specified by eigen_solver for larger matrices. In this example, only full diagonalization is used.
Total eigenvalues computed: 672
Save eigenvalues and eigenvectors in
'eigenvalues.dat'
'eigenvectors.dat'
Number of initial states: 32
N=4: 0 / 210
N=5: 32 / 252
N=6: 0 / 210
The first line shows the total number of eigenvalues computed. The last block shows the number of initial states that have the Boltzmann weight larger than weight_threshold. As expected from the prescreening, only N=5 contains initial states.
Then proceed to the calculation of the impurity Green’s function.
Calculating impurity Green's function...
Initial state 1/32 (N = 5)
particle excitation: N + 1 = 6
Use the Lehmann representation
hole excitation: N - 1 = 4
Use the Lehmann representation
Initial state 2/32 (N = 5)
particle excitation: N + 1 = 6
Use the Lehmann representation
hole excitation: N - 1 = 4
Use the Lehmann representation
Lehmann representation is used when the dimension of \(N \pm 1\)-particle states is smaller than dim_full_diag. Otherwise, linear equations are solved by the sparse solver specified by gf_solver. The output in this case is as follows:
Calculating impurity Green's function...
Initial state 1/32 (N = 5)
particle excitation: N + 1 = 6
Solve linear equations
Time: 0m3.816s
hole excitation: N - 1 = 4
Solve linear equations
Time: 0m3.774s
Initial state 2/32 (N = 5)
particle excitation: N + 1 = 6
Solve linear equations
Time: 0m3.794s
hole excitation: N - 1 = 4
Solve linear equations
Time: 0m3.782s
Output file¶
eigenvalues.dat
# dim = 1024 # n_eigen = 100 (for each n) # N E_i Boltzmann_weight 5 -2.00000000e+01 3.12500e-02 5 -2.00000000e+01 3.12500e-02 5 -2.00000000e+01 3.12500e-02 5 -2.00000000e+01 3.12500e-02 5 -2.00000000e+01 3.12500e-02 5 -2.00000000e+01 3.12500e-02 5 -2.00000000e+01 3.12500e-02 ...
This file contains the eigenvalues computed and the corresponding Boltzmann weights in ascending order. The numbers from left to right show the particle number, the eigen-energy, and the Boltzmann weight.
eigenvectors.dat
# The i-th digit from the left of the 10-dimensional Fock state # indicates whether (spin, site) state is occupied (1) or empty (0). # 'site' includes bath sites (b) after Wannier orbitals (w). # # w1+ w2+ ... b1+ b2+ ... w1- w2- ... b1- b2- ... # # E=-2.00000000e+01, N=5 |0000011111> 1.00000000e+00 # E=-2.00000000e+01, N=5 |0000111110> 1.00000000e+00 # E=-2.00000000e+01, N=5 |0001011101> 1.00000000e+00 # E=-2.00000000e+01, N=5 |0001111100> 1.00000000e+00 # E=-2.00000000e+01, N=5 |0110010011> 1.00000000e+00 # E=-2.00000000e+01, N=5 |0110110010> 1.00000000e+00 ...
This file contains the eigenvectors. The non-zero coefficients and the corresponding Fock states are shown for each eigenvector.
Benchmark¶
to be updated.