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 DCore are 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	None	Particle numbers to be considered. Specify in a list format, e.g., [1,2,3]. All particle numbers are considered by default. This parameter is useful to reduce the calculation time. 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 n_eigen is not sufficient. Note that, even if this check is passed, n_eigen may be insufficient. One should check the convergence of the Green’s function by increasing n_eigen.
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 stands for the 0-th shell and the 1-st iteration.

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

Particle numbers to be considered:
 [0 1 2]

This shows the dimension of the Hamiltonian matrix for each particle number. When particle_numbers option is given, the particle numbers to be considered are shown.

Solving the eigenvalue problem...

N = 0  (dim[N] = 1)
 full diagonalization
 Time: 0m0.000s

N = 1  (dim[N] = 10)
 full diagonalization
 Time: 0m0.000s

N = 2  (dim[N] = 45)
 full diagonalization
 Time: 0m0.001s

N = 3  (dim[N] = 120)
 Iterative solver: n_eigen=100 eigenvalues are computed.
 Time: 0m0.009s

This shows the time taken for solving the eigenvalue problem for each particle number. The solver uses full diagonalization method for matrices smaller than dim_full_diag (set to 100 in this example), while it switches to sparse solver specified by eigen_solver for larger matrices.

Total eigenvalues computed:  1024

Eigenvalues:
[-2.00000000e+01 -2.00000000e+01 -2.00000000e+01 ...  1.45984558e-09
  1.45984558e-09  1.82480875e-09]

Weights (Boltzmann factors / Z):
[3.12500000e-02 3.12500000e-02 3.12500000e-02 ... 4.32467662e-89
 4.32467662e-89 4.32467661e-89]

Number of initial states: 32

The first line shows the total number of eigenvalues computed. The last line shows the number of initial states that have the Boltzmann weight larger than weight_threshold.

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

This shows the calculation of the impurity Green’s function. 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.

Benchmark

to be updated.