Quantum Basis

Basis of condensed matter quantum lattice problems, for usage in exact diagonalization (ED). The code is designed for any general bosonic or fermionic problem (or a mix of both), as long as the user can provide the matrix form of the elementary operators (see Chapter 2 for details) of the Hamiltonian.

Human-friendly usage (Heisenberg model as example)

Note: the grammars below may differ slightly from the acutal code, please refer to the folder "examples" for exact usage.

• Writing a two-site spin-1/2 Heisenberg model can be as easy as:

auto Heisenberg = Sx1 * Sx2 + Sy1 * Sy2 + Sz1 * Sz2;

• Add a site "3"?

Heisenberg += Sx2 * Sx3 + Sy2 * Sy3 + Sz2 * Sz3;

• Add more exotic interactions?

Heisenberg += (Sx1 * Sx2 + Sy1 * Sy2 + Sz1 * Sz2) * (Sx1 * Sx2 + Sy1 * Sy2 + Sz1 * Sz2);

• Eigen problem?

Heisenberg.locate_E0_lanczos();

• Measurement?

Heisenberg.measure(Sx1*Sx2);

• More featuers (good quantum numbers, translation symmetry, dynamical response, etc.)? Explore the folder examples!
• Need design very different Hamiltonians? Explore the folder "examples" for ideas!

Examples

To learn how to use this library to design ED code for your own models, please refer to the folder "examples":

• Chain
  • Heisenberg spin-1/2
  • Heisenberg spin-1
  • Kondo Lattice model
  • t-J model
• Honeycomb lattice
  • Spinless fermion
• Kagome lattice
  • Heisenberg spin-1/2
  • t-J model
• Square lattice
  • Bose-Hubbard model
  • Fermi-Hubbard model
  • Kondo Lattice model
• Triangular lattice
  • Heisenberg spin-1/2

Install from source (Linux):

0. Create a local folder to keep the manually installed libraries (optional for experts):

   cd ${HOME}; mkdir ~/installs; mkdir ~/installs/lib; mkdir ~/installs/include

   Then add the following to the end of your ~/.bashrc file and restart the terminal:

   export LD_LIBRARY_PATH=${HOME}/installs/lib:${HOME}/installs/lib64:$LD_LIBRARY_PATH
   if [ -d ${HOME}/installs/lib/pkgconfig ]; then
       export PKG_CONFIG_PATH=${HOME}/installs/lib/pkgconfig:$PKG_CONFIG_PATH
   fi
   if [ -d ${HOME}/installs/lib64/pkgconfig ]; then
       export PKG_CONFIG_PATH=${HOME}/installs/lib64/pkgconfig:$PKG_CONFIG_PATH
   fi
   export OMP_NUM_THREADS=1
   

1. Install MKL. There are two options:

   • It is recommended to download from Intel, and use the Offline Installer option.

   Then add the following to the end of your ~/.bashrc file and restart the terminal:

   if [ -f /opt/intel/oneapi/setvars.sh ]; then
       source /opt/intel/oneapi/setvars.sh intel64
       export MKL_INC_DIR=${MKLROOT}/include
       export MKL_LIB_DIR=${MKLROOT}/lib/intel64
   fi
   unset PYTHONPATH
   

   Notes:

   • The variable intel64 and the destination folders may vary depending on the system, should change the above lines accordingly.
   • If you need install MKL on a cluster by hand without root permission, should designate the installation location to your home folder, and change the lines added to bashrc accordingly.

2. Install the following dependencies (the names may appear slightly different for different Linux distros):

   • gfortran
   • g++
   • cpptoml
   • pkg-config
   • boost-dev

   For Ubuntu, a single command should suffice:

   sudo apt install gfortran g++ libcpptoml-dev pkg-config libboost-dev
   

   For other Linux distros, cpptoml can be downloaded easily:

   cd /tmp
   wget https://github.com/skystrife/cpptoml/archive/refs/tags/v0.1.1.tar.gz
   tar xf v0.1.1.tar.gz
   cd cpptoml-0.1.1
   cp include/cpptoml.h ~/installs/include
   

3. Install ARPACK-NG:

   cd /tmp
   wget https://github.com/opencollab/arpack-ng/archive/refs/tags/3.9.0.tar.gz
   tar xf 3.9.0.tar.gz
   cd arpack-ng-3.9.0
   sh bootstrap
   FFLAGS="-m64 -I${MKL_INC_DIR}" FCFLAGS="-m64 -I$MKL_INC_DIR" CFLAGS="-DMKL_ILP64 -m64 -I${MKL_INC_DIR}" CXXFLAGS="-DMKL_ILP64 -m64 -I${MKL_INC_DIR}" LIBS="-L${MKL_LIB_DIR} -Wl,--no-as-needed -lmkl_gf_ilp64 -lmkl_tbb_thread -lmkl_core -lpthread -ltbb -lstdc++ -lm -ldl" LIBSUFFIX="ILP64" INTERFACE64="1" ./configure --with-blas=mkl_gf_ilp64 --with-lapack=mkl_gf_ilp64 --enable-icb --prefix=$HOME/installs
   make check
   make install
   

4. Test Quantum Basis:

   Create a directory "bin" that sits at the same level of "src": mkdir bin, then go inside "src": cd src, then type

   make clean; make test

   Then execute the file ./test.x in folder bin. 🍺 if you see "All tests passed!" 🍺

5. Install Quantum Basis as a library:

   make clean; make install

6. To try out the examples, go to folders:

   • examples/trans_absent/platform_linux (without using translational symmetry)
   • examples/trans_symmetric/platform_linux (using translational symmetry)

   and use the makefiles therein.

Doxygen documentation:

https://wztzjhn.github.io/quantum_basis/

Restrictions on lattice:

When using translational symmetry, at least one of the dimensions (Lx, Ly, Lz, or number of sublattices) has to be an even number (current implementation of the generalized Lin Table).