# FQHETorusFermionsWithTranslationsNBodyHollowCore

FQHETorusFermionsWithTranslationsNBodyHollowCore diagonalizes the shortest (i.e. hollowcore) n-body interaction for a fermionic system on the torus geometry. It is restricted to rectangular tori (for a generic n-body interaction, see FQHETorusFermionsWithTranslationsGenericNBody). For example, to generate the <math>Z_4</math> Read-Rezayi state for 8 particles, we should use

*$PATHTODIAGHAM/build/FQHE/src/Programs/FQHEOnTorus/FQHETorusFermionsWithTranslationsNBodyHollowCore -p 8 -l 12 --nbr-nbody 5 --full-reducedbz --use-lapack*

The spectrum is stored in the file fermions_torus_5body_hollowcore_n_8_2s_12_ratio_1.000000.dat whose content should look like

# Kx Ky E 0 0 6.981082378843e-13 0 0 1.3428147482841e-12 0 0 1.3770501376445 0 0 2.1259921043698 ...

Using the option --full-reducedbz forces the program to look at all the momentum sectors of the reduced Brillouin zone, i.e. for <math>k_x,k_y</math> going from 0 to <math>{\rm GCD}(N,N_{\Phi}) - 1</math>. For a square torus, not using the --full-reducedbz forces to look only at the sectors that are not related by the <math>C_4</math> or inversion symmetries. As usual, all the parameters for the diagonalization (such the number for eigenstates, the threshold between full-diag and Lanczos,..) can be tuned through the options described in QHEFermionsTwoBodyGeneric or in Lanczos.

Computing the interaction matrix elements is time consuming. For that reason the code creates a file 5body_hollowcore_interactioncoefficient_2s_12_ratio_1.0000000000.dat. If this file is in the current directory, any diagonalization for 5-body hollow core with 12 flux quanta will re-read this file instead of generating the interaction matrix elements. If you want to regenerate the interaction file, you can either erase it or add the --regenerate-interactionelements option.

The n-body hollowcore interaction is normalized such that the typical energy scale when considering n particles interacting on a large torus is one. Note that the normalization has NOT been obtained from an analytical calculation but rather by numerically solving the n particles spectrum on a very large square torus.

A two-body defined through pseudo-potentials can be added via the --interaction-file option. Its usage is similar to the one of FQHEFermionsTorusWithTranslation, including the input file syntax. It might be useful to tune the strength of the n-body interaction, this can be done through the option --nbody-strength. By default, this coefficient is set to one.

A linear interpolation between the n-body interaction and any two-body interaction can be achieved thanks to the --linear-interpolation option. Let's say we want to interpolate between the 5-body hollowcore interaction and the <math>V_1</math> pseudo-potentials. First we need to write a pseudo-potential file pseudopotentials_torus_v1.dat

Name = v1 Pseudopotentials = 0.0 1.0

Then, to diagonalize the hamiltonian <math>(1-\lambda) H_{\rm 5b} + \lambda V_1</math> , we just use the command

*$PATHTODIAGHAM/build/FQHE/src/Programs/FQHEOnTorus/FQHETorusFermionsWithTranslationsNBodyHollowCore -p 12 -l 18 --nbr-nbody 5 --use-lapack --full-diag 2000 --interaction-file pseudopotentials_torus_v1.dat --linear-interpolation --linear-lambda 0.25*

The linear interpolation parameter <math>\lambda</math> is set via --linear-lambda (here <math>\lambda =0.25</math>) . The spectrum is stored in fermions_torus_5body_hollowcore_v1_lambda_0.250000_n_12_2s_18_ratio_1.000000.dat , the output file name being created from the type of n-body interaction, the name provided in pseudopotentials_torus_v1.dat and the linear interpolation parameter <math>\lambda</math>.