Exact diagonalization of bilayer quantum Hall (BLQH) systems in the torus geometry
This program performs exact diagonalization and calculation the excitation spectrum of the bilayer/monolayer quantum Hall system (BLQH/MLQH) in the torus geometry and in the lowest-Landau-level (LLL). The bilayer quantum Hall system contains two layers of two-dimensional electron gas (2DEG), with intralayer and interlayer Coulomb interaction. Including the spin degrees of freedom, the noninteracting hamiltonian with bias voltage, Zeeman splitting energy and interlayer tunneling (symmetric-antisymmetric energy gap) can be included. Exact diagonalizing the full hamiltonian matrix, we could get all the eigenstates and analyze the results. To shrink the Hilbert space, and simplify the calculation, we utilize the magnetic translation operators in x and y directions, and then apply a three-pass Lanczos algorithm to calculate the lowest eigenstate at each K-point.
Final version of the program, which performs calculation with electron spin degree of freedom included for the BLQH system, realize all the functions as described above. This program is universal and with upper layer electrons or spin-up electrons set to zero, it could be easily reduced to spinless or monolayer system.
2nd version of the program, performs calculation for BLQH with electron spin degree of freedom quenched.
1st version of the program, performs monolayer fractional quantum Hall (FQHE) system with electron spin degree of freedom quenched.
Environment:linux
Libraries:
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icc,mkl: https://software.intel.com/en-us/parallel-studio-xe
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g++,liblapack-dev,libblas-dev
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GNU Scientific Library(GSL): https://www.gnu.org/software/gsl/
ubuntu:
sudo apt-get install g++ g++-multilib build-essential liblapack-dev libblas-dev libgsl-dev
make blqh
with icc,mkl installed:
make blqh_mkl
./blqh -h
./blqh: option requires an argument -- 'h'
Usage: ./blqh [Options]
Options:
-l nLL
-n nphi
-e Total No. of electrons in two layers
-u No. of electrons in upper-layer
-k kx in upper-layer
-S Delta_SAS: tunnelling amplitude
-v Delta_V: bias voltage
-j total J in upper-layer or down-layer
-g gamma=lx/ly aspect ratio
-d interlayer distance
-m Lambda
-t nthread
-s seed
Default: (l,n,u,d,lambda) = (4,4,2,1,200)
We calculate 8-electrons with 4-electrons in the upper-layer and 4-electrons in
the down layer at layer distance d/l=0.0. The ground state energy is at K-point (j,k)=(4,4), we use four threads to calculate the hamiltonian.
./blqh -n 8 -e 8 -u 4 -j 4 -k 4 -t 4 -d 0.0
----------- ED results --------------
nHilbert: =79
E_gs:= -0.636707
----------- Lanczos results ---------
E_gs:= -0.6367069257
# ground state wave function
71 : | _ 1 2 3 4 _ _ _)| 0 _ _ _ _ 5 6 7) 000000001110000100011110 0.3380617019
6 : | 0 _ _ _ 4 _ 6 7)| _ 1 2 3 _ 5 _ _) 000000000010111011010001 0.3380617019
48 : | 0 _ _ _ 4 5 6 _)| _ 1 2 3 _ _ _ 7) 000000001000111001110001 0.3380617019
29 : | 0 _ 2 _ _ 5 _ 7)| _ 1 _ 3 4 _ 6 _) 000000000101101010100101 0.3380617019
20 : | 0 _ _ _ 4 5 _ 7)| _ 1 2 3 _ _ 6 _) 000000000100111010110001 0.3380617019
10 : | 0 _ _ 3 _ _ 6 7)| _ 1 2 _ 4 5 _ _) 000000000011011011001001 0.3380617019
54 : | 0 _ 2 _ _ 5 6 _)| _ 1 _ 3 4 _ _ 7) 000000001001101001100101 0.3380617019
25 : | 0 _ _ 3 _ 5 _ 7)| _ 1 2 _ 4 _ 6 _) 000000000101011010101001 0.3380617019
66 : | 0 1 _ _ 4 5 _ _)| _ _ 2 3 _ _ 6 7) 000000001100110000110011 0.2390457219
57 : | 0 _ 2 _ 4 _ 6 _)| _ 1 _ 3 _ 5 _ 7) 000000001010101001010101 0.1690308509
19 : | 0 _ _ _ 4 _ 6 7)| 0 _ 2 3 _ _ 6 _) 000000000100110111010001 1.306913788e-13
The ground state energies of ED and Lanczos approaches are the same within accuracy of 1E-5.
The first column of the wave function (WF) is the ID of the basis, the |**)|**) columns are
electrons which occupy the orbitals ('_' stands for not occupied,left for upper-layer and right for down-layer). the next column is
the electron occupaptions in binary representation ('1' for occupied and '0' for empty), and the last column are the coefficients in the many-body basis.
As we can see, at d/l=0.0, all the basis with nonzero coefficients are Haplerin "111 state".
At distance d/l=3.0, we could see that the all the basis with nonzero coefficients are two decoupled composite Fermi liquids.
./blqh -n 8 -e 8 -u 4 -j 4 -k 4 -t 4 -d 3.0
----------- ED results --------------
nHilbert: =79
E_gs:= -1.23588
----------- Lanczos results ---------
E_gs:= -1.235882219
# ground state wave function
66 : | 0 1 _ _ 4 5 _ _)| _ _ 2 3 _ _ 6 7) 000000001100110000110011 0.5388040068
52 : | 0 _ _ 3 4 _ _ 7)| 0 _ _ 3 4 _ _ 7) 000000001001100110011001 0.530402778
51 : | _ 1 _ 3 4 _ 6 _)| 0 _ _ 3 4 _ _ 7) 000000001001100101011010 0.3033881081
33 : | 0 _ 2 _ _ 5 _ 7)| _ 1 2 _ _ 5 6 _) 000000000110011010100101 0.3033881081
5 : | _ _ 2 3 _ _ 6 7)| 0 _ 2 3 _ 5 _ _) 000000000010110111001100 0.3033881081
68 : | 0 1 _ _ 4 5 _ _)| _ 1 _ _ 4 _ 6 7) 000000001101001000110011 0.3033881081
29 : | 0 _ 2 _ _ 5 _ 7)| _ 1 _ 3 4 _ 6 _) 000000000101101010100101 0.1269396409
55 : | 0 _ _ 3 _ 5 6 _)| 0 _ 2 _ _ 5 _ 7) 000000001010010101101001 0.1215007838
35 : | 0 _ 2 _ _ 5 _ 7)| 0 _ _ 3 _ 5 6 _) 000000000110100110100101 0.1215007838
34 : | 0 _ _ 3 _ 5 6 _)| 0 _ _ 3 _ 5 6 _) 000000000110100101101001 0.1173494879
3 : | 0 1 _ _ 4 5 _ _)| _ 1 2 3 4 _ _ _) 000000000001111000110011 0.009704390136
Pull requests are welcome. For major changes, please open an issue first to discuss what you would like to change.