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FQHEQuantumSpinHallCheckerboardModelTwoBands provides an exact diagonalization code for FQSH based on two copies of the checkerboard lattice model. Most of the running options are similar to those of FQHECheckerboardLatticeModel.

Decoupled layers

In its simplest version, FQHEQuantumSpinHallCheckerboardModelTwoBands involved two decoupled copies of the checkerboard lattice model (spin up and down). The typical usage is

$PATHTODIAGHAM/build/FTI/src/Programs/FTI/FQSHCheckerboardModelTwoBands -p 4 -x 3 -y 2 --use-lapack --flat-band --project-fourbands --decoupled

The eigenvalue output file looks like

   # eigenvalues
   # kx ky Sz E
   0 0 -4 1.7063718059659
   0 0 -4 2.1234062994611
   0 0 -4 2.1434774759191
   0 0 -2 0.80880078458526
   0 0 -2 0.84834788180414
   0 0 -2 0.85652103153087
   0 0 -2 0.86641822256772
   0 0 -2 0.915790768175
   0 0 -2 1.0302577225926
   0 0 -2 1.0356483557114

Compared to FCICheckerboardLatticeModel, there is an additional column (the third one) which is two times the Sz value (Sz being a good quantum number in the case of two decoupled copies).

The interaction can be tuned through three paramaters : --u-potential sets the repulsive nearest neighbor potential strength between two identical spins (note that this option is inactive in flat-band mode, as U sets the energy scale of the system), --v-potential is the repulsive on-site potential strength between opposite spins and --w-potential allows to change the repulsive nearest neighbor potential strength between opposite spins.

Tilted periodic boundary conditions

Tilted periodic boundary conditions are available for the spin conserved (decoupled) mode. The typical usage is

$PATHTODIAGHAM/build/FQHE/src/Programs/FQHETopInsulator/FQHEQuantumSpinHallCheckerboardModelTwoBands -p 6 -x 9 -y 1 --use-lapack --flat-band --project-fourbands --decoupled --nx1 3 --ny1 0 --nx2 1 --ny2 3 --offset 3

nx1 and ny1 (respectively nx2 and ny2) set the coordinates of T1 (respectively T2), the periodicity vectors (a translation by either of these vectors leaves the system invariant). They must verify the condition nx1*ny2 - nx2*ny1 = Ns where Ns is the total number of unit cells. Nx and Ny must be set (using the -x and -y options) to

<math>Nx = Ns/GCD(Ns, nx2, ny2)</math>

<math>Ny = GCD(Ns, nx2, ny2)</math>

--offset is an integer that must verify the relations

<math>(offset*ny2 - ny1) mod Nx = 0</math>

<math>(nx1 - offset*nx2) mod Nx = 0</math>

For nx1 = Nx, ny2 = Ny, ny1 = nx2 = 0, and offset = 0, one recovers the non-tilted periodic boundary conditions.

In tilted mode, one can also look at a time-reversal breaking system of two Checkerboard Chern insulator models. The --break-timereversal option activates the bilayer mode.

Coupled layers

The coupled layer model is defined through the following hamiltonian

<math>H=\left( {\begin{array}{cc}

H_0(k) & \Delta \; \sigma_y^*  \\
\Delta^* \; \sigma_y &  H_0^*(-k) \\
\end{array} } \right)</math>

where <math>H_0(k)</math> is the one body hamiltonian of the checkerboard lattice and <math>\sigma_y=\left( {\begin{array}{cc}

0 & i  \\
-i &  0 \\
\end{array} } \right)</math>.

The code can be run using only the two lowest energy bands or the full four bands using the --four-bands option. The coupling parameter <math>\Delta</math> is set through the two options --mixing-norm <math>\left(\left|\Delta\right|\right)</math> and --mixing-arg <math>\left(arg(\Delta) / 2 \pi\right)</math>. The typical usage is

$PATHTODIAGHAM/build/FQHE/src/Programs/FQHETopInsulator/FQHEQuantumSpinHallCheckerboardModelTwoBands -p 4 -x 3 -y 2 --flat-band --use-lapack

The eigenvalue output file looks like

   # eigenvalues
   # kx ky  E
   0 0 0.27058366707565
   0 0 0.27271072155586
   0 0 0.38540586407147
   0 0 0.53925025231446
   0 0 0.58915542718289
   0 0 0.61372286112607
   0 0 0.70253212436423
   0 0 0.74470356767789

The --four-bands is much more demanding than the default two band mode. As a safety check, one can use the combination of --four-bands and --project-fourbands options to compare the result of the four band calculation projected onto the two lower band model.