FCIKagomeLatticeModel

FCIKagomeLatticeModel handles interacting fermions or bosons on the Kagome lattice. This exact diagonalization code offers similar calculation features to those of FCICheckerboardLatticeModel.

How to produce a bosonic Laughlin like state

We will show how to produce a Laughlin bosonic state at filling <math>\nu=1/2</math>. We will use only the on-site interaction and turn to zero the next nearest neighbor hopping term. To get the energy spectrum for 2 bosons on a 2x2 lattice, we just have to run

$PATHTODIAGHAM/build/FTI/src/Programs/FCI/FCIKagomeLatticeModel -p 2 -x 2 -y 2 --flat-band --use-lapack --boson --t1 1.0 --l1 1.0 --t2 0.0 --l2 0.0

This produces an ASCII text file bosons_singleband_kagomelattice_n_2_x_2_y_2_t1_1_t2_0_l1_1_l2_0_gx_0_gy_0.dat which contains

   # eigenvalues
   # kx ky  E
   0 0 -2.7755575615629e-17
   0 0 0.03349364905389
   0 0 0.41666666666667
   0 0 0.46650635094611
   0 1 0.25
   0 1 0.33333333333333
   1 0 0.25
   1 0 0.33333333333333
   1 1 0.25
   1 1 0.33333333333333

For this particular choice of the lattice aspect ratio (a 2x2 lattice), the almost twofold degenerate groundstate is located in the momentum sector <math>(k_x,k_y)=(0,0)</math>. To get the two eigenstates corresponding to these two low energies, we can run the code as followed

$PATHTODIAGHAM/build/FTI/src/Programs/FCI/FCIKagomeLatticeModel -p 2 -x 2 -y 2 --flat-band --use-lapack --boson --t1 1.0 --l1 1.0 --t2 0.0 --l2 0.0 --eigenstate -n 2 --only-kx 0 --only-ky 0

The --eigenstate option requires to compute the eigenstates and -n 2 indicates that we want only the two first eigenstates. Since we want to focus on the (0,0) sector, we have added the two options --only-kx 0 --eigenstates only-ky 0 . Running this command line will generate a new spectrum file and two binary vectors bosons_singleband_kagomelattice_n_2_x_2_y_2_t1_1_t2_0_l1_1_l2_0_gx_0_gy_0_kx_0_ky_0.0.vec and bosons_singleband_kagomelattice_n_2_x_2_y_2_t1_1_t2_0_l1_1_l2_0_gx_0_gy_0_kx_0_ky_0.1.vec. The extension .x.vec indicates this the the x-th eigenstate (zero being the groundstate in the given momentum sector).

The eigenstates are binary vectors that can be managed with the corresponding tools. Their decomposition onto the n-body basis can be read using FTIShowBasis.

If more than one eigenvalue or eigenstate per momentum are needed and the Hilbert space requires to use the Lanczos algorithm, it is safer to use the options --block-lanczos --block-size x --force-reorthogonalize . Here x should match the number of eigenvalues set by the -n option i.e.

$PATHTODIAGHAM/build/FTI/src/Programs/FCI/FCIKagomeLatticeModel -p 8 -x 4 -y 4 --flat-band --use-lapack --boson --t1 1.0 --l1 1.0 --t2 0.0 --l2 0.0 -n 3 --block-lanczos --block-size 3 --force-reorthogonalize

How to produce a fermionic Laughlin like state

We will show how to produce a Laughlin fermionic state at filling <math>\nu=1/3</math>. We will use only the nearest neighbor interaction and turn to zero the next nearest neighbor hopping term. To get the energy spectrum for 4 bosons on a 4x3 lattice, we just have to run

$PATHTODIAGHAM/build/FTI/src/Programs/FCI/FCIKagomeLatticeModel -p 4 -x 4 -y 3 --t1 1.0 --l1 1.0 --t2 0.0 --l2 0.0 --use-lapack --flat-band -n 1 --eigenstate

According to the folding principle, the three almost degenerate states corresponding to the Laughlin groundstate manifold are located in the momentum sectors (kx,ky)=(2,0), (2,1) and (2,2). The three corresponding eigenstates have the following names :

   fermions_singleband_kagomelattice_n_4_x_4_y_3_t1_1_t2_0_l1_1_l2_0_gx_0_gy_0_kx_2_ky_0.0.vec
   fermions_singleband_kagomelattice_n_4_x_4_y_3_t1_1_t2_0_l1_1_l2_0_gx_0_gy_0_kx_2_ky_1.0.vec
   fermions_singleband_kagomelattice_n_4_x_4_y_3_t1_1_t2_0_l1_1_l2_0_gx_0_gy_0_kx_2_ky_2.0.vec

Their decomposition onto the n-body basis can be read using FTIShowBasis and the particle entanglement spectrum can be computed using FTIEntanglementEntropyParticlePartition.