HubbardKitaevHeisenbergModel

<math>H_{SH}= -t \sum_{ <i,j>} [c_{i\sigma}^{\dagger}d_{j\sigma}+d_{j\sigma}^{\dagger}c_{i\sigma}] -t'\sum_{ <i,j> } [c_{i\sigma}^{\dagger}d_{j\sigma'}\sigma_{\sigma\sigma'}^{\alpha}+d_{i\sigma}^{\dagger}c_{j\sigma'}\sigma_{\sigma\sigma'}^{\alpha}] +J_1 \sum_{ <i,j>} \vec{S}_i\cdot\vec{S}_j +J_2 \sum_{ <i,j>} [S_i^{\alpha}S_j^{\alpha}-S_i^{\beta}S_j^{\beta}-S_i^{\gamma}S_j^{\gamma}] +U \sum_{i} n_{i\uparrow} n_{i\downarrow} </math>

$PATHTODIAGHAM/build/FTI/src/Programs/HubbardModels/HubbardKitaevHeisenbergModel -p 6 -x 6 --gutzwiller --j1 0.49 --j2 0.51 --use-lapack --stripe --eigenstate -n 1

Exact diagonalisation program for a system of spin 1/2 fermions on the Honeycomb lattice Kitaev-Heisenberg model. The number of particles and the number of sites are set through the -p option and -x respectively. The Gutzwiller projection is turned on via the --gutzwiller option. In that case, a reduced Hilbert space forbidding the occupation of a site by two fermions is used. The parameters of the Hamiltonian are set using :

• --isotropic-t for the spin isotropic nearest neighbor hopping amplitude i.e. <math>t</math>
• --anisotropic-t for the spin anisotropic nearest neighbor hopping amplitude i.e. <math>t'</math>
• --j1 for the neareast neighbor Heisenberg interaction i.e. <math>J_1</math>
• --j2 for the neareast neighbor anisotropic spin-spin interaction i.e. <math>J_2</math>
• --u-potential for the on-site Hubbard potential i.e. <math>U</math>. This option is useless if the Gutzwiller projection is turned on.

For the parallelization options, they are similiar to those of many diagonalization codes such as FCICheckerboardLatticeModel.

Open parallelogram configuration

The --open option sets the geometry of the system to be a parallelogram with open boundary conditions (zigzag alignment in both x and y directions). The options --nbrsites-x and --nbrsites-y set the number of hexagons in the x and y directions.

The figure below gives an example of the bond types and sites indices for an open system of length 3 and of width 2. The bond type is given by the following color code: red (<math>x</math>), green (<math>y</math>), blue (<math>z</math>)

Stripe configuration

The --stripe option sets the geometry of the system to be a stripe: the boundary conditions are periodic in the x direction, and open in the y direction.

• The option --nbrsites-x sets the number of unit cells (number or hexagons in the direction, or length of the stripe).
• The option --nbrsites-y sets the number of hexagons in the y direction (width of the stripe).

The figure below gives an example of the bond types and sites indices for a stripe of length 3 and of width 2. The bond type is given by the following color code: red (<math>x</math>), green (<math>y</math>), blue (<math>z</math>)

Torus configuration

The --torus option sets the geometry of the system to a torus (periodic boundary conditions in both directions).

• The number of unit cells in the x direction is specified using the option --nbrsites-x
• The number of unit cells in the y direction is specified using the option --nbrsites-y

The total number of sites (set using the option --nbr-sites) should be equal to twice the product of the two.

For instance: $PATHTODIAGHAM/build/FTI/src/Programs/HubbardModels/HubbardKitaevHeisenbergModel -p 12 -x 12 --use-lapack --gutzwiller --j1 0.3 --j2 0.25 --isotropic-t 1.0 --anisotropic-t 0.5 --full-diag 10000 --torus --nbrsites-x 3 --nbrsites-y 2

The sites are labeled in a way that is consistent with translational invariance. For instance, for 3 unit cells in the x directions and 2 unit cells in the y direction:

Generic geometry

When the --stripe and --torus options are not activated, one can specify the geometry of the system by providing a three-column text file through the option --geometry-file. The two first columns give two site indices, and the third column gives the nature of the link between them (0 if link <math>x</math>, 1 if link <math>y</math>, 2 if link <math>z</math>). For instance, the following input geometry file corresponds to a single hexagon:

0 1 2
0 2 1
1 3 0
2 4 0
3 5 1
4 5 2

The name of the output files (spectrum and vectors) is specified using the option --geometry-name. For instance

$PATHTODIAGHAM/build/FTI/src/Programs/HubbardModels/HubbardKitaevHeisenbergModel -p 4 -x 6 --j1 1 --j2 1 --isotropic-t 0 --anisotropic-t 0 --use-lapack --eigenstate -n 1 --geometry-file hexagon.dat --geometry-name hexagon

will generate a file fermions_kitaev_heisenberg_hexagon_n_4_x_6_t_0_tK_0_j1_1_j2_1.dat

Periodic boundary conditions

The current version can resolve momentum in one or two directions for a system with periodic boundary conditions. We define the real space vectors a₁, a₂ and reciprocal space vectors a' ₁, a' ₂ in the following figure. When the periodic boundary conditions are implemented, a translation by a vector N_x a₁ or N_y a₂ leaves the system invariant (where N_x and N_y are defined by the --nbrsites-x and --nbrsites-y options).

For one direction, this is enabled with the --xperiodic-boundary option. The periodicity is expressed using the site index convention. For example, in the stripe geometry with boundary conditions, the periodicity has to be set to 4 since the site <math>n</math> and <math>n+4</math> are related by translation. This can be done via the --max-xmomentum option which sets the number of possible momentum values in the x direction. For the stripe geometry, this should be the numbers of sites divided by 4. A typical usage would be

$PATHTODIAGHAM/build/FTI/src/Programs/HubbardModels/HubbardKitaevHeisenbergModel -p 10 -x 12 --gutzwiller --stripe --isotropic-t 1.0 --anisotropic-t 1.0 --j1 1.0 --j2 1.0 --use-lapack --memory 500 -n 3 --block-lanczos --block-size 3 --force-reorthogonalize --xperiodic-boundary --max-xmomentum 3 --nbrsites-x 3 --nbrsites-y 1

A specific momentum sector can be computed by setting to the --x-momentum option to the corresponding momentum sector. Note that these options are not restricted to the stripe geometry. Any geometry where the periodity can be written as a relation between site <math>n</math> and <math>n+T_x</math> where <math>T_x</math> is the periodicity (that has to be provided through the --max-xmomentum option) can use these options.

For two (decoupled) directions, one needs to enable the --2dperiodic-boundaries. In that case both --max-xmomentum and --max-ymomentum options have to be set. Note that the site numbering should satisfy the above mentioned condition for the x direction. For the y direction the site numbering should be such that two sites related by a single translation in the y direction satisfy <math>n</math> and <math>n+T_Y</math>, with <math>N_Y T_Y=T_X</math>, <math>N_Y</math> being the number of unit cells in the y direction. Here is an example for a torus having 3x3 hexagons and periodic boundary conditions

$PATHTODIAGHAM/build/FTI/src/Programs/HubbardModels/HubbardKitaevHeisenbergModel -p 18 -x 18 --use-lapack --geometry-file torus_nx_3_ny_3.dat --geometry-name torus --gutzwiller --j1 0.3 --j2 0.25 --isotropic-t 1.0 --anisotropic-t 0.5 --full-diag 7000 --max-ymomentum 3 --max-xmomentum 3 --2dperiodic-boundaries --memory 6000

If using the --torus option instead of specifying the geometry by an external file, the values of --nbrsites-x and --nbrsites-y are automatically set to those of --max-xmomentum and --max-ymomentum.