Laughlin state

This page describes different ways to generate the Laughlin state with DiagHam on different geometries

Sphere geometry

We will exemplify here how to get the Laughlin state for 8 fermions at filling 1/3 on the sphere geometry. The fermionic Laughlin state can be obtained from exact diagonalization of the hollow core interaction. First we need to create a pseudopotential file pseudopotential_laughlin3_2s_21.dat that contains

   Pseudopotentials = 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Now we can rely on QHEFermionsTwoBodyGeneric to perform the exact diagonalization of this interaction

$PATHTODIAGHAM/build/FQHE/src/Programs/FQHEOnSphere/QHEFermionsTwoBodyGeneric -p 8 -l 21 --interaction-name laughlin --interaction-file pseudopotential_laughlin3_2s_21.dat --use-lapack --eigenstate -n 1 --nbr-lz 1

One can check that we have a zero energy state looking at the output spectrum fermions_laughlin_n_8_2s_21_lz.dat

   # Lz E
   0 1.4210854715202e-14

The Laughlin state is stored in the binary vector file fermions_laughlin_n_8_2s_21_lz_0.0.vec .

A more accurate way to generate the Laughlin state relies on the Jack generator. First we need to provide a squeezed basis description

   NbrParticles=8
   LzMax=21
   ReferenceState=1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1

then one should use the Jack generator in this way

$PATHTODIAGHAM/build/FQHE/src/Programs/FQHEOnSphere/FQHESphereJackGenerator -a -2 -o fermions_haldane_laughlinjack_n_8_2s_21_lz_0.0.vec -n --reference-file laughlin_n_8_2s_21.dat --fermion

The binary vector fermions_haldane_laughlinjack_n_8_2s_21_lz_0.0.vec is defined within the squeezed basis. To convert it to the usual n-body basis, we have to use FQHESphereConvertHaldaneBasis

$PATHTODIAGHAM/build/FQHE/src/Programs/FQHEOnSphere/FQHESphereConvertHaldaneBasis fermions_haldane_laughlinjack_n_8_2s_21_lz_0.0.vec --reference-file laughlin_n_8_2s_21.dat -o fermions_laughlinjack_n_8_2s_21_lz_0.0.vec

The output of the two methods can b compared as a crossed check computing the overlap between the two binary vectors with GenericOverlap

$PATHTODIAGHAM/build/src/Programs/GenericOverlap fermions_laughlinjack_n_8_2s_21_lz_0.0.vec fermions_laughlin_n_8_2s_21_lz_0.0.vec

   File 0  fermions_laughlinjack_n_8_2s_21_lz_0.0.vec
   File 1  fermions_laughlin_n_8_2s_21_lz_0.0.vec
   Overlap |<0|1>|^2 = 0.99999999999999