FQHESphereBosonEntanglementEntropy

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FQHESphereBosonEntanglementEntropy (for bosons) and FQHESphereFermionEntanglementEntropy (for fermions) generate the reduced density matrix, including its eigenvalues and eigenstates. They also compute the entanglement entropy

Basic usage

Let's say you have generate the Gaffnian state using the Jack generator. So you have a bosons_haldane_gaffnian_n_14_2s_18_lz_0.0.vec and the description of the root configuration in gaffnian_n_14_2s_18.dat

$PATHTODIAGHAM/build/FQHE/src/Programs/FQHEOnSphere/FQHESphereBosonEntanglementEntropy bosons_haldane_gaffnian_n_14_2s_18_lz_0.0.vec --haldane --reference-file gaffnian_n_14_2s_18.dat --use-lapack

This will compute the orbital entanglement entropy of the Gaffnian state for a number of orbitals going from 1 to 10 (i.e. up to the hemisphere cut). The result is stored in a text file named bosons_partial_unnormalized_haldane_gaffnian_n_14_2s_18_lz_0.0.ent

If you want the eigenvalues of the reduced density matrix, just add the --density-matrix option with the name of file where you want to store them (here bosons_partial_unnormalized_haldane_gaffnian_n_14_2s_18_lz_0.0.full.ent)

$PATHTODIAGHAM/build/FQHE/src/Programs/FQHEOnSphere/FQHESphereBosonEntanglementEntropy bosons_haldane_gaffnian_n_14_2s_18_lz_0.0.vec --haldane --reference-file gaffnian_n_14_2s_18.dat --density-matrix bosons_partial_unnormalized_haldane_gaffnian_n_14_2s_18_lz_0.0.full.ent --use-lapack

The data in bosons_partial_unnormalized_haldane_gaffnian_n_14_2s_18_lz_0.0.full.ent will look like

   # l_a    N    Lz    lambda
   1 0 0 0.38469657408456
   1 1 0 0.49376474656819
   1 2 0 0.12153867934773

The file is column formatted (using a single space as separator) :

  • the first column is the number of orbitals (l_a)
  • the second column the number of particles (N)
  • the third column is the projection of the angular momentum (Lz)
  • the forth column is the eigenvalue (lambda)

Lines in the file are sorted by increasing l_a, N, Lz and decreasing lambda.

The bosons_partial_unnormalized_haldane_gaffnian_n_14_2s_18_lz_0.0.full.ent file can be processed using FQHESphereEntanglementSpectrum to extract the entanglement spectrum.


You can restrict the calculation to a given range of l_a values using the --min-la and --max-la options. In the following example

$PATHTODIAGHAM/build/FQHE/src/Programs/FQHEOnSphere/FQHESphereBosonEntanglementEntropy bosons_haldane_gaffnian_n_14_2s_18_lz_0.0.vec --haldane --reference-file gaffnian_n_14_2s_18.dat --density-matrix bosons_partial_unnormalized_haldane_gaffnian_n_14_2s_18_lz_0.0.full.ent --use-lapack --min-la 9 --max-la 9

the calculation will be done only for l_a=9.

To generate the spectrum when A has 2 edges, add the option --stripe-subsystem while calling FQHESphereBosonEntanglementEntropy. The option --shift-la then indicates the position of the edge of A closest to the north pole. For example,

$PATHTODIAGHAM/build/FQHE/src/Programs/FQHEOnSphere/FQHESphereBosonEntanglementEntropy bosons_haldane_gaffnian_n_14_2s_18_lz_0.0.vec --haldane --reference-file gaffnian_n_14_2s_18.dat --density-matrix bosons_partial_unnormalized_haldane_gaffnian_n_14_2s_18_lz_0.0.full.ent --use-lapack --stripe-subsystem --shift-la 5

will generate the eigenvalue spectrum for all possible sizes l_a, for A being the subsystem between orbitals 5 and 5+l_a.

Advanced usage

Eigenstates of the reduced can be computed using the --density-eigenstate flag. Since this can generate a huge number of vectors, you restrict to a given Na and Lza sector with the --na-eigenstate and --lza-eigenstate options. You may also restrict the calculation to the x first largest eigenvalues using --nbr-eigenstates x . For example,

$PATHTODIAGHAM/build/FQHE/src/Programs/FQHEOnSphere/FQHESphereBosonEntanglementEntropy bosons_haldane_gaffnian_n_14_2s_18_lz_0.0.vec --haldane --reference-file gaffnian_n_14_2s_18.dat --density-matrix bosons_partial_unnormalized_haldane_gaffnian_n_14_2s_18_lz_0.0.full.ent --use-lapack --min-la 9 --max-la 9 --na-eigenstate 7 --lza-eigenstate 0 --nbr-eigenstates 2 --density-eigenstate

Two vectors are generated : bosons_sphere_density_n_14_2s_18_lz_0_la_9_na_7_lza_0.0.vec and bosons_sphere_density_n_14_2s_18_lz_0_la_9_na_7_lza_0.1.vec

Beware that the --lza-eigenstate uses the lz counting convention from the density matrix eigenvalue file and not the convention from the entanglement spectrum. As usual, --lza-eigenstate is twice the lz value to avoid half integer number.

You can store the full density matrix using the --full-densitymatrix option

$PATHTODIAGHAM/build/FQHE/src/Programs/FQHEOnSphere/FQHESphereBosonEntanglementEntropy bosons_haldane_gaffnian_n_14_2s_18_lz_0.0.vec --haldane --reference-file gaffnian_n_14_2s_18.dat --density-matrix bosons_partial_unnormalized_haldane_gaffnian_n_14_2s_18_lz_0.0.full.ent --use-lapack --min-la 9 --max-la 9 --na-eigenstate 7 --lza-eigenstate 0 --nbr-eigenstates 2 --full-densitymatrix matrix.dat

This will store the reduced density matrix in the ASCII file matrix.dat for the sector la=9, Na=7 Lza=0. Indices of the matrix are associated to n-body using the standard DiagHam convention that can be obtained using FQHESphereShowBasis with 7 bosons, 8 flux quanta (i.e. la-1) and total lz=0.