4.1.4. Parameters for the Hamiltonian

A default value is \(0\) unless a specific value is written in the description. Table 4.1 shows the parameters for each models. In the case of a complex type, a file format is “a real part, an imaginary part ” while in the case of a real type, only “a real part “.

Local terms

  • mu

    Type : Real

    Description : It is available only for the Hubbard and Kondo lattice model. The chemical potential \(\mu\) (including the site potential) is specified with this parameter.

  • U

    Type : Real

    Description : It is available only for the Hubbard and Kondo lattice model. The onsite Coulomb integral \(U\) is specified with this parameter.

  • Jx, Jy, Jz, Jxy, Jyx, Jxz, Jzx, Jyz, Jzy

    Type : Real

    Description : It is available only for the Kondo lattice model. The spin-coupling constant between the valence and the local electrons is specified with this parameter. If the exchange coupling J is specified in the input file, instead of Jx, Jy, Jz, the diagonal exchange couplings, Jx, Jy, Jz, are set as Jx = Jy = Jz = J. When both the set of exchange couplings (Jx, Jy, Jz) and the exchange coupling J are specified in the input file, HPhi++ will stop.

  • h

Type : Real

Description : The longitudinal magnetic field is specified with this parameter.

  • Gamma, D

Type : Real

Description : (Spin model) The transverse magnetic field, and the single-site anisotropy parameter are specified with these parameters. The single-site anisotropy parameter is not available for model=SpinGCCMA.

The non-local terms described below should be specified differently according to the lattice structure: For lattice=Ladder, the non-local terms are specified differently from those for the lattice=Chain Lattice, Square Lattice, Triangular Lattice, Honeycomb Lattice, Kagome. Below, the available parameters for each lattice are shown in Table 4.1 .

Table 4.1 Interactions for each model defined in an input file. We can define spin couplings as a matrix format.
Interactions 1D chain 2D square 2D triangular Honeycomb Kagome Ladder
J, t, V(simplified) \({\circ}\) \({\circ}\) \({\circ}\) \({\circ}\) \({\circ}\) -
J0, t0, V0 \({\circ}\) \({\circ}\) \({\circ}\) \({\circ}\) \({\circ}\) \({\circ}\)
J1, t1, V1 - \({\circ}\) \({\circ}\) \({\circ}\) \({\circ}\) \({\circ}\)
J2, t2, V2 - - \({\circ}\) \({\circ}\) \({\circ}\) \({\circ}\)
J’, t’, V’(simplified) \({\circ}\) \({\circ}\) \({\circ}\) \({\circ}\) \({\circ}\) -
J0’, t0’, V0’ \({\circ}\) \({\circ}\) \({\circ}\) \({\circ}\) \({\circ}\) -
J1’, t1’, V1’ - \({\circ}\) \({\circ}\) \({\circ}\) \({\circ}\) \({\circ}\)
J2’, t2’, V2’ - - \({\circ}\) \({\circ}\) \({\circ}\) \({\circ}\)
J’‘, t’‘, V’‘(simplified) \({\circ}\) \({\circ}\) \({\circ}\) \({\circ}\) - -
J0’‘, t0’‘, V0’‘ \({\circ}\) \({\circ}\) \({\circ}\) \({\circ}\) - -
J1’‘, t1’‘, V1’‘ - \({\circ}\) \({\circ}\) \({\circ}\) - -
J2’‘, t2’‘, V2’‘ - - \({\circ}\) \({\circ}\) - -

Non-local terms[ for Ladder ( Fig. 4.4 )]

  • t0, t1, t1', t2, t2'

    Type : Complex

    Description : (Hubbard and Kondo lattice model) Hopping integrals in the ladder lattice (see Fig. 4.4 ) are specified with this parameter.

  • V0, V1, V1', V2, V2'

    Type : Real

    Description : (Hubbard and Kondo lattice model) Offsite Coulomb integrals on the ladder lattice ( Fig. 4.2 ) are specified with these parameters.

  • J0x, J0y, J0z, J0xy, J0yx, J0xz, J0zx, J0yz, J0zy

  • J1x, J1y, J1z, J1xy, J1yx, J1xz, J1zx, J1yz, J1zy

  • J1'x, J1'y, J1'z, J1'xy, J1'yx, J1'xz, J1'zx, J1'yz, J1'zy

  • J2x, J2y, J2z, J2xy, J2yx, J2xz, J2zx, J2yz, J2zy

  • J2'x, J2'y, J2'z, J2'xy, J2'yx, J2'xz, J2'zx, J2'yz, J2'zy.

    Type : Real

    Description : (Spin model) Spin-coupling constants in the ladder lattice (see Fig. 4.4 ) are specified with these parameters. If the simplified parameter J0 is specified in the input file instead of the diagonal couplings, J0x, J0y, J0z, these diagonal couplings are set as J0x = J0y = J0z = J0. If both J0 and the set of the couplings (J0x, J0y, J0z) are specified, HPhi++ will stop. The above rules are also valid for the simplified parameters, J1, J1', J2, and J2'.

Non-local terms [other than Ladder ( Fig. 4.1 , Fig. 4.2 , Fig. 4.3 )]

  • t, t0, t1, t2

    Type : Complex

    Description : (Hubbard and Kondo lattice model) The nearest neighbor hoppings for each direction (see Fig. 4.1 - Fig. 4.3 ) are specified with these parameters. If there is no bond dependence of the hoppings, the simplified parameter t is available to specify t0, t1, and t2 as t0 = t1 = t2 = t. If both t and the set of the hoppings (t0, t1, t2) are specified, HPhi++ will stop.

  • t', t0', t1', t2'

    Type : Complex

    Description : (Hubbard and Kondo lattice model) The second nearest neighbor hoppings for each direction (see Fig. 4.1 - Fig. 4.3 ) are specified with these parameter. If there is no bond dependence of the hoppings, the simplified parameter t' is available to specify t0', t1', and t2' as t0' = t1' = t2' = t'. If both t' and the set of the hoppings (t0', t1', t2') are specified, HPhi++ will stop.

  • t'', t0'', t1'', t2''

    Type : Complex

    Description : (Hubbard and Kondo lattice model) The third nearest neighbor hoppings for each direction (see Fig. 4.1 - Fig. 4.3 ) are specified with these parameter. If there is no bond dependence of the hoppings, the simplified parameter t'' is available to specify t0'', t1'', and t2'' as t0'' = t1'' = t2'' = t''. If both t'' and the set of the hoppings (t0'', t1'', t2'') are specified, HPhi++ will stop.

  • V, V0, V1, V2

    Type : Real

    Description : (Hubbard and Kondo lattice model) The nearest neighbor offsite Coulomb integrals \(V\) for each direction (see Fig. 4.1 - Fig. 4.3 ) are specified with these parameters. If there is no bond dependence of the offsite Coulomb integrals, the simplified parameter V is available to specify V0, V1, and V2 as V0 = V1 = V2 = V. If both V and the set of the Coulomb integrals (V0, V1, V2) are specified, HPhi++ will stop.

  • V', V0', V1', V2'

    Type : Real

    Description : (Hubbard and Kondo lattice model) The second nearest neighbor-offsite Coulomb integrals \(V'\) for each direction (see Fig. 4.1 - Fig. 4.3 ) are specified with this parameter. If there is no bond dependence of the offsite Coulomb integrals, the simplified parameter V' is available to specify V0', V1', and V2' as V0' = V1' = V2' = V'. If both V' and the set of the Coulomb integrals (V0', V1', V2') are specified, HPhi++ will stop.

  • V'', V0'', V1'', V2''

    Type : Real

    Description : (Hubbard and Kondo lattice model) The third nearest neighbor-offsite Coulomb integrals \(V'\) for each direction (see Fig. 4.1 - Fig. 4.3 ) are specified with this parameter. If there is no bond dependence of the offsite Coulomb integrals, the simplified parameter V'' is available to specify V0'', V1'', and V2'' as V0'' = V1'' = V2'' = V''. If both V'' and the set of the Coulomb integrals (V0'', V1'', V2'') are specified, HPhi++ will stop.

  • J0x, J0y, J0z, J0xy, J0yx, J0xz, J0zx, J0yz, J0zy

  • J1x, J1y, J1z, J1xy, J1yx, J1xz, J1zx, J1yz, J1zy

  • J2x, J2y, J2z, J2xy, J2yx, J2xz, J2zx, J2yz, J2zy

    Type : Real

    Description : (Spin model) The nearest neighbor exchange couplings for each direction are specified with these parameters. If the simplified parameter J0 is specified, instead of J0x, J0y, J0z, the exchange couplings, J0x, J0y, J0z, are set as J0x = J0y = J0z = J0. If both J0 and the set of the exchange couplings (J0x, J0y, J0z) are specified, HPhi++ will stop. The above rules are valid for J1 and J2.

    If there is no bond dependence of the exchange couplings, the simplified parameters, Jx, Jy, Jz, Jxy, Jyx, Jxz, Jzx, Jyz, Jzy, are available to specify the exchange couplings for every bond as J0x = J1x = J2x = Jx. If any simplified parameter (Jx-Jzy) is specified in addition to its counterparts (J0x-J2zy), HPhi++ will stop. Below, examples of parameter sets for nearest neighbor exchange couplings are shown.

    • If there are no bond-dependent, and no anisotropic and offdiagonal exchange couplings (such as \(J_{x y}\)), please specify J in the input file.
    • If there are no bond-dependent and offdiagonal exchange couplings but there are anisotropic couplings, please specify the non-zero couplings in the diagonal parameters, Jx, Jy, Jz.
    • If there are no bond-dependent exchange couplings but there are anisotropic and offdiagonal exchange couplings, please specify the non-zero couplings in the nine parameters, Jx, Jy, Jz, Jxy, Jyz, Jxz, Jyx, Jzy, Jzx.
    • If there are no anisotropic and offdiagonal exchange couplings, but there are bond-dependent couplings, please specify the non-zero couplings in the three parameters, J0, J1, J2.
    • If there are no anisotropic exchange couplings, but are bond-dependent and offdiagonal couplings, please specify the non-zero couplings in the nine parameters, J0x, J0y, J0z, J1x, J1y, J1z, J2x, J2y, J2z.
    • If there are bond-dependent, anisotropic, and offdiagonal exchange couplings, please specify the non-zero couplings in the twenty-seven parameters from J0x to J2zy.

  • J'x, J'y, J'z, J'xy, J'yx, J'xz, J'zx, J'yz, J'zy

  • J0'x, J0'y, J0'z, J0'xy, J0'yx, J0'xz, J0'zx, J0'yz, J0'zy

  • J1'x, J1'y, J1'z, J1'xy, J1'yx, J1'xz, J1'zx, J1'yz, J1'zy

  • J2'x, J2'y, J2'z, J2'xy, J2'yx, J2'xz, J2'zx, J2'yz, J2'zy

    Type : Real

    Description : (Spin model) The second nearest neighbor exchange couplings are specified. However, for lattice = Honeycomb Lattice and lattice = Kagome with model=SpinGCCMA, the second nearest neighbor exchange couplings are not available in the \(Standard\) mode. If the simplified parameter J' is specified, instead of J'x, J'y, J'z, the exchange couplings are set as J'x = J'y = J'z = J'. If both J' and the set of the couplings (J'x, J'y, J'z) are specified, HPhi++ will stop.

  • J''x, J''y, J''z, J''xy, J''yx, J''xz, J''zx, J''yz, J''zy

  • J0''x, J0''y, J0''z, J0''xy, J0''yx, J0''xz, J0''zx, J0''yz, J0''zy

  • J1''x, J1''y, J1''z, J1''xy, J1''yx, J1''xz, J1''zx, J1''yz, J1''zy

  • J2''x, J2''y, J2''z, J2''xy, J2''yx, J2''xz, J2''zx, J2''yz, J2''zy

    Type : Real

    Description : (Spin model) The third nearest neighbor exchange couplings are specified. However, for lattice = Honeycomb Lattice and lattice = Kagome with model=SpinGCCMA, the third nearest neighbor exchange couplings are not available in the \(Standard\) mode. If the simplified parameter J'' is specified, instead of J''x, J''y, J''z, the exchange couplings are set as J''x = J''y = J''z = J''. If both J'' and the set of the couplings (J''x, J''y, J''z) are specified, HPhi++ will stop.

  • phase0, phase1

    Type : Double (0.0 as defaults)

    Description : We can specify the phase for the hopping through the cell boundary with these parameter (unit: degree). These factors for the \(\boldsymbol{a}_0\) direction and the \(\boldsymbol{a}_1\) direction can be specified independently. For the one-dimensional system, only phase0 can be used. For example, a fopping from \(i\)-th site to \(j\)-th site through the cell boundary with the positive direction becomes as

    \[\exp(i \times {\rm phase0}\times\pi/180) \times t {\hat c}_{j \sigma}^\dagger {\hat c}_{i \sigma} + \exp(-i \times {\rm phase0}\times\pi/180) \times t^* {\hat c}_{i \sigma}^\dagger {\hat c}_{j \sigma}\]