4.1.2. Parameters for the lattice

Chain [ Fig. 4.1 (a)]

  • L

    Type : Integer

    Description : The length of the chain is specified with this parameter.

    Schematic illustration of (a) one-dimensional chain lattice, (b) two-dimensional square lattice, and (c) two-dimensional triangular lattice. They have :math:`t`, :math:`V`, and :math:`J` as the nearest neighbor hopping, an offsite Coulomb integral, and a spin-coupling constant, respectively (magenta solid lines); they also have :math:`t'`, :math:`V'`, and :math:`J'` as the next nearest neighbor hopping, offsite Coulomb integral, and spin-coupling constant, respectively (green dashed line).

    Fig. 4.1 Schematic illustration of (a) one-dimensional chain lattice, (b) two-dimensional square lattice, and (c) two-dimensional triangular lattice. They have \(t\), \(V\), and \(J\) as the nearest neighbor hopping, an offsite Coulomb integral, and a spin-coupling constant, respectively (magenta solid lines); they also have \(t'\), \(V'\), and \(J'\) as the next nearest neighbor hopping, offsite Coulomb integral, and spin-coupling constant, respectively (green dashed line).

    Schematic illustration of the anisotropic honeycomb lattice. The first/second/third nearest neighbor hopping integral, spin coupling, and offsite Coulomb integral depend on the bond direction.

    Fig. 4.2 Schematic illustration of the anisotropic honeycomb lattice. The first/second/third nearest neighbor hopping integral, spin coupling, and offsite Coulomb integral depend on the bond direction.

    Schematic illustration of the Kagome lattice.

    Fig. 4.3 Schematic illustration of the Kagome lattice.

    Schematic illustration of the ladder lattice.

    Fig. 4.4 Schematic illustration of the ladder lattice.

Ladder ( Fig. 4.4 )

  • L

    Type : Integer

    Description : The length of the ladder is specified with this parameter.

  • W

    Type : Integer

    Description : The number of the ladder is specified with this parameter.

Shape of the numerical cell when :math:`{\boldsymbol a}_0 = (6, 2), {\boldsymbol a}_1 = (2, 4)` in the triangular lattice. The region surrounded by :math:`{\boldsymbol a}_0` (magenta dashed arrow) and :math:`{\boldsymbol a}_1` (green dashed arrow) becomes the cell to be calculated (20 sites).

Fig. 4.5 Shape of the numerical cell when \({\boldsymbol a}_0 = (6, 2), {\boldsymbol a}_1 = (2, 4)\) in the triangular lattice. The region surrounded by \({\boldsymbol a}_0\) (magenta dashed arrow) and \({\boldsymbol a}_1\) (green dashed arrow) becomes the cell to be calculated (20 sites).

Tetragonal lattice [ Fig. 4.1 (b)], triangular lattice [ Fig. 4.1 (c)], honeycomb lattice [ Fig. 4.2 ], Kagome lattice [ Fig. 4.3 ]

In these lattices, we can specify the shape of the numerical cell by using the following two methods.

  • W, L

Type : Integer

Description : The alignment of the original unit cells (dashed black lines in Fig. 4.1 - Fig. 4.3 ) is specified with this parameter.

  • a0W, a0L, a1W, a1L

    Type : Integer

    Description : We can specify two vectors (\({\boldsymbol a}_0, {\boldsymbol a}_1\)) that surround the numerical cell (Fig. 4.5 ). These vectors should be specified in the fractional coordinate.

If we use both these methods, \({\mathcal H}\Phi\) stops. When model=SpinGCCMA, we can use only the former.

We can check the shape of the numerical cell by using a file lattice.gp which is written in Standard mode. This file can be read by gnuplot as follows:

$ gnuplot lattice.gp