# FQHESphereLLLProjectedLLLTimesManyLL

FQHESphereLLLProjectedLLLTimesManyLL allows to compute the product of a fermionic state in the two (or three) first Landau levels with either a fermionic or bosonic state in the lowest Landau level. In particular, it allows to generate Jain's composite fermion states for both fermionic and bosonic systems.

## Basic usage, fermionic CF states

As a example, let's see how this program can be used to generate the 2/5 fermionic Jain state for 8 fermions.

$PATHTODIAGHAM/build/FQHE/src/Programs/FQHEOnSphere/FQHESphereLLLProjectedLLLTimesManyLL --fermion fermions_l_1_n_8_2s_2_lz_0.0.vec --lll-state bosons_laughlin_n_8_2s_14_lz_0.0.vec --projection --normalize -o fermions_jain_n_8_2s_16_lz_0.0.vec

The vector that follows the option --fermion is the fermionic vector in the two first Landau levels. In the present case, as the Hilbert dimension is 1, it had been obtained by putting a 1 in an Ascii file and then converted in a binary file using VectorAscii2Binary.

The vector that follows the option --lll-state is the bosonic or fermionic vector that belongs to the first Landau level. In this case, it is the 1/2 bosonic laughlin state that can be computed using the FQHESphereJackGenerator . The bosonic state should be unnormalized (integer coefficients) and if it was obtained via JackGenerator, the usual options "--haldane --reference-file " need to be given.

The option --projection indicates that the state generates by the product will be projected into the lowest Landau level. This option is compulsory when using a 3 Landau level state.

The option --normalize indicates that the state will be normalized on the sphere.

The usual vector output options are available.

If the fermionic state belongs to the three first Landau levels, the option --3-ll must be used.

## Bosonic CF states

Generation of bosonic CF states is similar their fermionic analogues. In the simplest case, we want to perform a single flux attachment. For that purpose, the fermionic LLL that we need is just the filled LLL, i.e. a normalized vector with a single compoment. For a system with 10 particles, we create such a vector using VectorAscii2Binary as above and name it fermions_lll_n_10_2s_9_lz_0.0.vec (notice that the number of flux quanta should be equal to the number of particles minus one). If we want to generate the <math>\nu=\frac{2}{3}</math> state for this number of particles, we also need a vector that corresponds to two filled Landau levels. Once again, it is a vector with a single component. So we can just copy fermions_lll_n_10_2s_9_lz_0.0.vec and name the copy fermions_2ll_n_10_2s_3_lz_0.0.vec (the number of flux quanta should satisfy <math>N=2 N_\Phi+4</math>).

*$PATHTODIAGHAM/build/FQHE/src/Programs/FQHEOnSphere/FQHESphereLLLProjectedLLLTimesManyLL --fermion fermions_2ll_n_10_2s_3_lz_0.0.vec --lll-state fermions_lll_n_10_2s_9_lz_0.0.vec --projection --normalize --interaction-name jaincf_2ll*

The standard output should look like

Number of Lambda Landau level = 2 memory requested for Hilbert space = 552ko memory requested for Hilbert space = 12 memory requested for lookup table = 96ko memory requested for Hilbert space = 11Mo 0 1 : 0.565733s

The code will generate one vector bosons_jaincf_2ll_n_10_2s_12_lz_0.0.vec whose name is set by the option --interaction-name, and is both projected onto the lowest Landau level (--projection) and normalized on the sphere geometry (normalize).

Similarly, the bosonic jain at 3/4 for 8 bosons (involving three Landau levels) is generated by the following command :

*$PATHTODIAGHAM/build/FQHE/src/Programs/FQHEOnSphere/FQHESphereLLLProjectedLLLTimesManyLL --fermion fermions_3ll_n_9_2s_0_lz_0.0.vec --lll-state fermions_slater_n_9_2s_8_lz_0.0.vec --projection --normalize -o bosons_jain_n_9_2s_8_lz_0.0.vec --3-ll*

Ihe two vectors are also vectors with a single component.