DiagHam_Stability

DiagHam version for the fractional Chern insulator (FCI) stability project.

1. How this repository was created
2. Getting started with this repository
3. Tutorials
4. References
5. Appendix: DiagHam tips
6. Appendix: Getting started with the latest DiagHam
7. Appendix: Debugging DiagHam compilation
8. Appendix: Boost Python tips

How this repository was created

This repository was created by taking the changes that David Bauer made (relative to revision 3304) and merging them with my current working branch (based on revision 4184).

1. Using the command svn info --show-item revision, I confirmed that David's repository is based on revision 3304 (whereas my working copy is based on revision 4184). For reference, I then used the command svn checkout -r 3304 https://www.nick-ux.org/diagham/svn/DiagHam DiagHam_r3304 to checkout the revision that David originally started with (or the revision closest to the one that he started with, if David took someone else's copy).
2. Performing a meld between two DiagHam directories is messy because the code formatting style is inconsistent. Specifically, there are often inconsequential file differences due to parenthesis placement, extra whitespace, etc. Therefore, I went to meld>Preferences>Text Filters and selected all of the text filters. Additionally, I added the patterns "^.*else.*$", "^.*if.*$", "["/+/"]*", to ignore lines containing if, else, and other miscellaneous whitespace. This gives a fairly readable comparison. From this, I can see that David's repository has the following files that are modified/new compared to r3304 (important files in bold):

• FQHE/src/Architecture/ArchitectureOperation/FQHESquareLatticeMultiBandSymmetrizeU1U1StateOperation.cc(.h)
• FQHE/src/Hamiltonian/ParticleOnTorusDeltaHamiltonian.cc(.h)
• FQHE/src/HilbertSpace/BosonOnSquareLatticeMomentumSpace(Long).h, BosonOnSquareLatticeMultiBandMomentumSpace.cc(.h)
• FQHE/src/Programs/FQHEOnTorus/FQHETorusBosonsDelta.cc, [data file]

~

• FTI/src/Hamiltonian/ParticleOnLatticeHofstadterMultiBandHamiltonian.cc(.h), ParticleOnLatticeHofstadterSingleBandHamiltonian.cc(.h), ParticleOnLatticeHofstadterSingleBandThreeBodyHamiltonian.cc(.h)
• FTI/src/Programs/FCI/ComputeHofstadterBands(2,Standalone).py, FCIHoftsadterModel.cc, FCIHofstadterWithAnyChernModel.cc, FCIRubyLatticeModel.cc, [data files]
• FTI/src/Programs/FTI/[data files]

~

• FTI/src/Tools/FTITightBinding/ TightBindingModelHofstadterSquare.cc(.h)

~

• src/Architecture/ArchitectureOperation/AbstractArchitectureOperation.h

In David's code, he built in the same directory as the source code. Since I try to keep the builds separate to the source (for debugging purposes), I have copied across David's changes with respect to r3304 to my working version (based on r4184).

3. The configure command that David listed in configure-command-db is...

./configure CXXFLAGS="-O3 -Wall -I/opt/local/include -I/opt/local/Library/Frameworks/Python.framework/Headers" LDFLAGS="-L/opt/local/lib -L/opt/local/Library/Frameworks/Python.framework/Versions/3.5/lib -flat_namespace -force_flat_namespace -lstdc++ -lboost_python3-mt -lpython3.5" --enable-lapack --with-blas-libs="-lopenblas -lgslcblas" --with-lapack-libs="-llapack -lf2c" --enable-fftw --enable-fqhe --enable-fti

...adapting this for my computer (UNIX) yields...

../configure CXXFLAGS="-O3 -Wall -I/usr/include -I/usr/include/python3.8" LDFLAGS="-L/usr/lib -L/usr/lib/python3.8 -L/usr/lib/x86_64-linux-gnu" LIBS="-lpython3.8 -lboost_python -lboost_system -lstdc++ -lfftw3" --enable-lapack --with-blas-libs="-lopenblas -lgslcblas" --with-lapack-libs="-llapack -lf2c" --enable-fftw --enable-fqhe --enable-fti

My library headers are located in /usr/include and my library object files are in /usr/lib. Occasionally, symbolic links need to be created for the library object files, so that their name matches the command. Boost python needs to be installed. The "-mt" suffix in the boost_python libraries has been deprecated, since they are now all multi-threading compatible by default. The flat_namespace flags are mac specific, so I have omitted those. Crucially, the library names need to be given in LIBS so that they are called after the object files -- the order of flags is important!

4. ParticleOnLatticeHofstadterSingleBandThreeBodyHamiltonian.cc was added to FTI/src/Hamiltonian/Makefile.am
5. FTI/src/Programs/FCI/FCIHofstadterCorrelation.cc and FTI/src/Programs/FCI/FCICheckerboardToHofstadterLatticeModel.cc, as well as mentions in corresponding Makefile.am file were removed, for now, due to conflicts

Getting started with this repository

In the ~/DiagHam_Stability/trunk directory:

1. ./bootstrap.sh
2. mkdir build; mkdir run; cd build
3. ../configure CXXFLAGS="-O3 -Wall -I/usr/include -I/usr/include/python3.8" LDFLAGS="-L/usr/lib -L/usr/lib/python3.8 -L/usr/lib/x86_64-linux-gnu" LIBS="-lpython3.8 -lboost_python -lboost_system -lstdc++ -lfftw3" --enable-lapack --with-blas-libs="-lopenblas -lgslcblas" --with-lapack-libs="-llapack -lf2c" --enable-fftw --enable-fqhe --enable-fti
4. make
5. cd ../run; ../build/FQHE/src/Programs/FQHEOnSphere/FQHESphereJackGenerator --help

Tutorials

The following tutorials are found in the ~/DiagHam_Stability/trunk/tutorials directory.

Before getting started, the following aliases can be defined in ~/.bash_aliases:

• alias FTIGetDimension=~/DiagHam_Stability/trunk/build/FTI/src/Programs/FTI/FTIGetDimension
• alias FCIHofstadterModel=~/DiagHam_Stability/trunk/build/FTI/src/Programs/FCI/FCIHofstadterModel
• alias GenericOverlap=~/DiagHam_Stability/trunk/build/src/Programs/GenericOverlap
• alias FindLatticeGap=~/DiagHam_Stability/trunk/scripts_bart/FindLatticeGap.pl
• alias PlotHofstadterSpectrum=~/DiagHam_Stability/trunk/scripts_bart/PlotHofstadterSpectrum.pl
• function vd() { cat $1 | sort -g -k3 | head }

And the following variable should be added to ~/.bashrc:

• export PYTHONPATH=$PYTHONPATH:~/DiagHam_Stability/trunk/FTI/src/Programs/FCI

After which, you need to either source ~/.bashrc, or restart the session, for the changes to take effect.

01_ener_spec

In this tutorial, we calculate the many-body energy spectrum for a FCI in the Hofstadter model.

1. We can check the dimension of the Hilbert space of our proposed configuration by running:

• FTIGetDimension -p 7 -x 3 -y 7

This shows us that for a fermionic system with 7 particles and 21 magnetic unit cells (yielding nu=7/21=1/3), the Hilbert space dimension is ~1e3, which is easily tractable on a personal computer (anything up to ~1e6 should be fine).

2. In order to generate the energy spectrum for the Hofstadter model, we run the command:

• FCIHofstadterModel -p 7 -x 3 -y 7 -X 7 -Y 3 -m 8000 -S --processors 4 -n 1 --lanczos-precision 1e-10 --eigenstate

We consider the parameters from above with a MUC of 7x3, with 8GB of RAM and 4 processors. Note that a larger MUC yields a physically more stable system, while square total system sizes are numerically the most stable. We can drop the --eigenstate flag if we're only interested in the energy spectrum. This flag generates the eigenvector(s) corresponding to the lowest eigenvalue(s) in each momentum sector. The -n 1 flag specifies the number of eigenvalues to compute in each momentum sector. Be careful, if you set -n 1 and all of the ground states happen to be in the same momentum sector, you will not find them!

3. Generate and plot the spectrum. (Note that for the DiagHam_Stability code, the spectrum does not need to be symmetry-extended with the -s flag, since we compute the spectrum for all momentum sectors, disregarding symmetries, akin to the --full-momentum flag in DiagHam.)

• PlotHofstadterSpectrum *0.dat

4. Check that the ground state degeneracy is correct in the spectrum. (Note that the magnitude of the energies in DiagHam is a factor of 2 larger than in DiagHam_Stability.)

• vd *0.dat

5. View the spectrum:

• evince *.ps

6. Determine the many-body gap (Delta):

• FindLatticeGap -d 3 *0.dat

7. Compute the overlap of two ground states:

• GenericOverlap -c *kx_0_ky_0.0.vec *kx_1_ky_0.0.vec

This tutorial is summarized in 01_ener_spec.sh.

02_gap_trace

In this tutorial, we plot the many-body gap (Delta) against the trace inequality saturation measure (TISM, denoted as <T>).

1. Using band_geometry.nb (dependent on BandGeometry.wl), we can generate the file geometry.csv, which tabulates t2 vs TISM.

• cd t2_trace
• mathematica band_geometry.nb
• cd ..

2. Using batch_diagham.py together with a parameter file batchfile.csv, we can generate the many-body energy spectra.

• cd t2_gap
• python batch_diagham.py batchfile.csv

3. Using batch_getgaps.py together with the parameter file batchfile.csv, we can generate the file outfile.csv, which tabulates t2 vs gap. Note that the -n flag should be large enough to capture the full ground-state degeneracy, in case all ground states are in the same momentum sector. Furthermore, the many-body gap is defined above the highest energy in the quasi-degenerate ground-state manifold. In this boson example, the degeneracy d=2 has not been resolved and so we take the gap above the lowest state.

• python batch_getgaps.py batchfile.csv outfile.csv
• cd ..

4. In the notebook final_plot.ipynb, we can merge the tables of t2 vs TISM vs gap, and plot the result.

• jupyter notebook &

This yields the final plot bosons_16_onsite.pdf, which reproduces Fig.9.(a) of [Bauer2022].

03_benchmark

In this tutorial, we reproduce the nine Delta vs TISM figures in [Bauer2022] (Figs.9-11).

Note: The t2_gap dat files in this tutorial are copied from DiagHam_David, with the following locations:

• Destination: bosons_onsite/*.dat
• Source: ~/DiagHam_Stability/trunk/FTI/src/Programs/FCI/quartic-bosons/*.dat
• Destination: fermions_NN/*.dat
• Source 1: ~/DiagHam_Stability/trunk/FTI/src/Programs/FCI/quartic-data-july/*.dat
• Source 2: ~/DiagHam_Stability/trunk/FTI/src/Programs/FCI/quartic-96/*.dat
• Destination: fermions_exp/*.dat
• Source: ~/DiagHam_Stability/trunk/FTI/src/Programs/FCI/rsquared-data/*.dat

While in principle, these dat files could be reproduced by running batch_diagham.py, small differences in DiagHam convergence means that the numbers do not precisely match up. Therefore, in order to reproduce exactly the same figures as [Bauer2022], we start with the same dat files.

0. [optional] Generate the many-body energy spectra (modify batch_diagham.py appropriately) e.g.

• cd t2_gap/bosons_onsite
• python ../batch_diagham.py bosons_onsite_batchfile.csv

1. Calculate the many-body gaps e.g.

• python ../batch_getgaps.py 1 bosons_onsite_batchfile.csv bosons_onsite_outfile.csv

The first argument is the degeneracy of the ground-state manifold. Note that, anomalously, a degeneracy of 1 is used for the boson examples above, since the d=2 degeneracy is not correctly resolved. We can run batch_diagham.py with a larger -n in order to resolve this degeneracy. For the fermion examples, the degeneracy is correctly resolved.

2. Generate the band geometry data for bosons (bosons_geometry.csv) and fermions (fermions_geometry.csv):

• cd ../../t2_trace
• mathematica band_geometry.nb

3. Evaluate the notebook final_plots.nb to plot the figures:

• jupyter notebook &

Note that for the fermions_exp figures, the many-body gap had to be further multiplied by a factor of 1/(2*Area) (in the notebook final_plots.nb), where Area is the area of the system in units of magnetic unit cells. This is for consistency with the other examples, since DiagHam includes a factor of 1/(2*Area) by default in the interaction factors, which was omitted for the custom exponential interaction.

04_int_sym

In this tutorial, we demonstrate the effect of interaction symmetry on the stability of FCIs.

The custom interactions are implemented in:

• FTI/src/Hamiltonian/ParticleOnLatticeHofstadterSingleBandHamiltonian.cc(.h)

For example, the non-zero elements of the exp(-r^4) interaction are illustrated below:

1. First, we can check for any discrepancies between DiagHam and DiagHam_Stability for the interactions that are implemented in both, using an example for bosons (onsite and NN) and fermions (NN and NNN). For example, we can run 04_int_sym.sh and then compare the spectra, as follows:

• ./04_int_sym.sh
• vd bosons/diagham_stability/onsite/*.dat; vd bosons/diagham/onsite/*ext.dat
• vd bosons/diagham_stability/NN/*.dat; vd bosons/diagham/NN/*ext.dat
• vd fermions/diagham_stability/NN/*.dat; vd fermions/diagham/NN/*ext.dat
• vd fermions/diagham_stability/NNN/*.dat; vd fermions/diagham/NNN/*ext.dat

Here we can see that the magnitude of the energies in DiagHam is a factor of 2 larger than in DiagHam_Stability but the spectra otherwise agree.

2. Next, we can recover the fermions NN and NNN results above by modifying the hard-coded exp(-r^4) interaction.

• mkdir fermions_test
• cd fermions_test
• mkdir NN; mkdir NNN

For NN fermions:

• cd NN
• mkdir ref
• FCIHofstadterModel -p 6 -x 3 -y 6 -X 6 -Y 3 -m 8000 -S --processors 4 -n 10 --lanczos-precision 1e-10 > /dev/null
• cd ..

Now we can edit ~/DiagHam_Stability/trunk/FTI/src/Hamiltonian/ParticleOnLatticeHofstadterSingleBandHamiltonian.cc and then run make in the directory ~/DiagHam_Stability/trunk/build/FTI/src. Once we have made our changes, we can continue by running:

• FCIHofstadterModel -p 6 -x 3 -y 6 -X 6 -Y 3 -m 8000 -S --processors 4 -n 10 --lanczos-precision 1e-10 > /dev/null
• vd *.dat; vd ref/*.dat

Here we can see that the NN fermion results can be recovered by modifying the hard-coded exp(-r^4) interaction. The same can be repeated for the NNN fermions.

3. Finally, we can go to the exp(-r^4) interaction for bosons and modify it to recover the boson onsite and NN results.

• cd ~/DiagHam_Stability/trunk/tutorials/04_int_sym
• mkdir bosons_test
• cd bosons_test
• mkdir onsite; mkdir NN

For onsite bosons:

• cd onsite
• mkdir ref
• FCIHofstadterModel --boson -p 6 -x 3 -y 4 -X 4 -Y 3 -m 8000 -S --processors 4 -n 10 --lanczos-precision 1e-10 > /dev/null
• cd ..

In a separate tab, we can edit ~/DiagHam_Stability/trunk/FTI/src/Hamiltonian/ParticleOnLatticeHofstadterSingleBandHamiltonian.cc and then run make in the directory ~/DiagHam_Stability/trunk/build/FTI/src. Once we have made our changes, we can continue by running:

• FCIHofstadterModel --boson -p 6 -x 3 -y 4 -X 4 -Y 3 -m 8000 -S --processors 4 -n 10 --lanczos-precision 1e-10 > /dev/null
• vd *.dat; vd ref/*.dat

Here we can see that the onsite boson results can be recovered by modifying the hard-coded exp(-r^4) interaction. The same can be repeated for the NN bosons.

References

[Bauer2016] "Quantum geometry and stability of the fractional quantum Hall effect in the Hofstadter model", by David Bauer, Tom Jackson, and Rahul Roy, PRB 93, 235133 (2016).

[Andrews2018] "Stability of fractional Chern insulators in the effective continuum limit of Harper-Hofstadter bands with Chern number |C|>1", by Bartholomew Andrews and Gunnar Moller, PRB 97, 035159 (2018).

[Bauer2022] "Fractional Chern insulators with a non-Landau level continuum limit", by David Bauer et al., PRB 105, 045144 (2022).

Appendix: DiagHam tips

• DiagHam wiki: https://nick-ux.org/diagham/index.php/Main_Page
• DiagHam website: http://www.phys.ens.fr/~regnault/diagham/

1. Makefile.am is user modified, whereas Makefile.in is created by the compiler
2. You can head config.log to view the configure command used in the last build
3. You can ../configure --help for useful info about the configure command
4. configure.in is deprecated in favour of configure.ac
5. always build the code in a separate build directory

Appendix: Getting started with the latest DiagHam

These instructions are in addition to those listed on the wiki.

1. Intel libraries need to be installed for optimal performance on Intel processors / for MKL workflows. This software is now free to use and no longer requires an academic licence. At the time of writing, you need the Intel oneAPI Base Toolkit for the C/C++ compiler and MKL library and the Intel oneAPI HPC Toolkit for the Fortran compiler and MPI library. After the installation, you can remove the installer if desired, which is in e.g. /tmp/root/ or ~/Downloads/. You can also add the following two lines to your ~/.bashrc: source /opt/intel/oneapi/compiler/latest/env/vars.sh; source /opt/intel/oneapi/mkl/latest/env/vars.sh. This saves some time when starting a shell, compared to sourcing the entire /opt/intel/oneapi/setvars.sh.

2. The configure command to automatically use the latest instruction set (-xHost) is:

../configure --enable-fqhe --enable-fti --enable-lapack --enable-gmp --enable-lapack-only --with-lapack-libs="" --with-blas-libs="-mkl" CC=icc CXX=icpc --enable-debug CFLAGS="-O3 -xHOST" CXXFLAGS="-O3 -xHOST"

...or for using both AVX and AVX2 instruction sets...

../configure --enable-fqhe --enable-fti --enable-lapack --enable-gmp --enable-lapack-only --with-lapack-libs="" --with-blas-libs="-mkl" CC=icc CXX=icpc --enable-debug CFLAGS="-O3 -xAVX -axCORE-AVX2" CXXFLAGS="-O3 -xAVX -axCORE-AVX2"

NB: BLAS will always be called when LAPACK is called, and it contains LAPACK, so no need to duplicate mkl flags. Since MKL is provided by both BLAS and LAPACK, it’s sufficient to give one or the other – but both options are required so one can also cope with separate libraries. The -mkl flag used to be called -lmkl, and it will soon be changed to -qmkl.

A guide to Intel compiler flags can be found here: https://www.bu.edu/tech/support/research/software-and-programming/programming/compilers/intel-compiler-flags/

3. There are several packages from the configure script that may need to be installed before proceeding e.g. f2c, gsl, b2z, etc. Please go through the output of the configure script to check what may be missing, before building.

Appendix: Debugging DiagHam compilation

The following hello world examples are found in the test directory.

boost-helloworld

• g++ -I/usr/include -I/usr/include/python3.8 -L/usr/lib -L/usr/lib/python3.8 -L/usr/lib/x86_64-linux-gnu PyInitTest.cpp -lpython3.8 -lboost_python -lboost_system
• ./a.out

autotools-helloworld-c

Repository found at: https://www.gnu.org/software/automake/manual/html_node/Creating-amhello.html

• autoreconf --install
• ./configure
• make
• src/hello

autotools-helloworld-cpp

Repository found at: https://github.com/jmlamare/autotools-helloworld-cpp

• autoreconf --install
• ./configure
• make
• src/hello

Appendix: Boost Python tips

1. Make sure that the ~/user_config.jam is pointing to the correct python interpreter
2. You can install additional packages either via pip or globally e.g. sudo apt-get install python3-scipy
3. Allow python to load modules from the correct directory by setting PYTHONPATH before calling Py_Initialize()...

• #include <cstdlib> // setenv followed by setenv("PYTHONPATH", "/home/bart/DiagHam_Stability/trunk/FTI/src/Programs/FCI", 1); before Py_Initialize()

...or by appending to the PYTHONPATH variable.