interaction parameter

Model Hamiltonian is defined as

$$\hat{H}=\sum _{ij}\sum _{\alpha \beta }^{N_{\mathrm{b}\mathrm{a}\mathrm{n}\mathrm{d}}}\sum _{\sigma =\uparrow ,\downarrow }{t}_{i\alpha j\beta }{c}_{i\alpha \sigma }^{†}{c}_{j\beta \sigma }+h.c.+{\hat{H}}_{\mathrm{i}\mathrm{n}\mathrm{t}},$$

where

(1)$${\hat{H}}_{\mathrm{i}\mathrm{n}\mathrm{t}}=\frac{1}{2}\sum _{i,\alpha \beta \gamma \delta ,\sigma {\sigma }^{\prime }}{U}_{\alpha \beta \gamma \delta }^i{c}_{i\alpha \sigma }^{†}{c}_{i\beta {\sigma }^{\prime }}^{†}{c}_{i\delta {\sigma }^{\prime }}{c}_{i\gamma \sigma }.$$

The interaction matrix $U_{\alpha \beta \gamma \delta }^i$ is specified by the parameter interaction.

interaction = kanamori

In this case, the Kanamori-type interaction is used, i.e.

$$\begin{array}{r}\begin{array}{rl}{U}_{\alpha \alpha \alpha \alpha }& =U,\\ U_{\alpha \beta \alpha \beta }& =U^{\prime }(\alpha \ne \beta ),\\ U_{\alpha \beta \beta \alpha }& =J(\alpha \ne \beta ),\\ U_{\alpha \alpha \beta \beta }& =J(\alpha \ne \beta ),\end{array}\end{array}$$

where $U,U^{\prime },J$ at each correlated shell are specified by the parameter kanamori as

interaction = kanamori
kanamori = [(U_1, U'_1, J_1), (U_2, U'_2, J_2), ... ]

For example, if there are two correlated shells that have $(U,U^{\prime },J)=(4,2,1)$ and $(U,U^{\prime },J)=(6,3,1.5)$, respectively, you need to set the input parameters as

interaction = kanamori
kanamori = [(4.0, 2.0, 1.0), (6.0, 3.0, 1.5)]

interaction = slater_f

In this case, the fully rotationally invariant Slater interaction is used. The interaction matrix is constructed by the effective Slater integrals $F_0,F_2,F_4,F_6$. These Slater integrals and the angular momentum at each correlated shell are specified by the parameter slater_f as follows

interaction = slater_f
slater_f = [(angular_momentum, F_0, F_2, F_4, F_6), ... ]

For example, if there are two correlated shells, one has d-orbital with $(F_0,F_2,F_4)=(2,1,0.5)$ and the other has p-orbital with $(F_0,F_2)=(3,1.5)$, you need to set the input parameter as

interaction = slater_f
slater_f = [(2, 2.0, 1.0, 0.5, 0.0), (1, 3.0, 1.5 0.0, 0.0)]

Note

You must specify all of $F_0,F_2,F_4,F_6$.

The basis can be specified by slater_basis parameter. See slater_basis parameter for details.

interaction = slater_uj

The Slater interaction is used as in the case with interaction = slater_f, but a conventional intuitive parameterization is implemented. The effective Slater integrals $F_0,F_2,F_4,F_6$ are evaluated from parameters $U$ and $J$. The explicit formulas are given as follows:

• $l=1$

$$F_0=U,F_2=5J$$

• $l=2$

$$F_0=U,F_2=\frac{14J}{1.0+0.63},F_4=0.63F_2$$

• $l=3$

$$F_0=U,F_2=\frac{6435J}{286+195\times 451/675+250\times 1001/2025},F_4=\frac{451F_2}{675},F_6=\frac{1001F_2}{2025}$$

The $U$, $J$ and the angular momentum at each correlated shell are specified by the parameter slater_uj as

interaction = slater_uj
slater_uj = [(angular_momentum1, U1, J1), (angular_momentum2, U2, J2), ... ]

The basis can be specified by slater_basis parameter. See slater_basis parameter for details.

interaction = respack

Use the output by RESPACK. Under construction.

interaction = file

One can set the U tensor $U_{\alpha \beta \gamma \delta }^i$ from an external file. The input parameters for this are

interaction = file
interaction_file = ['file1.npy', 'file2.npy',]

The filenames for all inequivalent shells are listed in interaction_file parameter. The suppored formats are

• NumPy .npy format (binary file): Four dimensional NumPy array corresponding to $U_{\alpha \beta \gamma \delta }^i$ for shell $i$ is save by numpy.save command.

• text file: $U_{\alpha \beta \gamma \delta }^i$ for each shell $i$ is written in a one dimansional manner as follows (example for norb=3 without spin-orbit coupling)

  Re(U[0,0,0,0])
  Re(U[0,0,0,1])
  Re(U[0,0,0,2])
  Re(U[0,0,1,0])
  ...
  

  or

  Re(U[0,0,0,0]) Im(U[0,0,0,0])
  Re(U[0,0,0,1]) Im(U[0,0,0,1])
  Re(U[0,0,0,2]) Im(U[0,0,0,2])
  Re(U[0,0,1,0]) Im(U[0,0,1,0])
  ...
  

  No blank line should be put between consecutive elements. Lines beginning with # are skipped.

The definition of the U tensor depends on the spin_orbit parameter. For spin_orbit=False, $U_{\alpha \beta \gamma \delta }^i$ is defined by Eq. (1). The shape of the U tensor is (norb, norb, norb, norb).

For spin_orbit=True, on the other hand, $U_{\alpha \beta \gamma \delta }^i$ is defined by

$${\hat{H}}_{\mathrm{i}\mathrm{n}\mathrm{t}}=\frac{1}{2}\sum _{i,\alpha \beta \gamma \delta }{U}_{\alpha \beta \gamma \delta }^i{c}_{i\alpha }^{†}{c}_{i\beta }^{†}{c}_{i\delta }{c}_{i\gamma }.$$

In this case, the indices $\alpha ,\beta ,\gamma ,\delta$ include spin. The shape of the U tensor is therefore (2*norb, 2*norb, 2*norb, 2*norb).

density_density option

If you want to treat only the density-density part

$${\hat{H}}_{\mathrm{i}\mathrm{n}\mathrm{t}}=\frac{1}{2}\sum _{i,\alpha ,\sigma {\sigma }^{\prime }}{U}_{\alpha \alpha \alpha \alpha }^i{c}_{i\alpha \sigma }^{†}{c}_{i\beta {\sigma }^{\prime }}^{†}{c}_{i\beta {\sigma }^{\prime }}{c}_{i\alpha \sigma }+\frac{1}{2}\sum _{i,\alpha \ne \beta ,\sigma {\sigma }^{\prime }}{U}_{\alpha \beta \alpha \beta }^i{c}_{i\alpha \sigma }^{†}{c}_{i\beta {\sigma }^{\prime }}^{†}{c}_{i\beta {\sigma }^{\prime }}{c}_{i\alpha \sigma }+\frac{1}{2}\sum _{i,\alpha \ne \beta ,\sigma }{U}_{\alpha \beta \beta \alpha }^i{c}_{i\alpha \sigma }^{†}{c}_{i\beta \sigma }^{†}{c}_{i\alpha \sigma }{c}_{i\beta \sigma },$$

you specify the parameter density_density as

density_density = True

Note

It can not be used in conjunction to the Hubbard-I solver or the double-counting correction.