exact diagonalization

• Python and Julia codes of exact diagonalization

• Use sparse matrices

• Sz conserved / Sz nonconserved

• Sz conserved case: algorithm based on titpack2 and HPhi

  • Use the snoob function for finding the next higher number after a given number that has the same number of 1-bits (down spins)

    • See Hacker's Delight Hacker's Delight (web archive)

    • See also HPhi tips HPhi tips (mirror)

    • For related bitwise operations, see Count bits Anothr count bits

  • Use two-dimensional search

    • See DOI:10.1103/PhysRevB.42.6561

    • See also manual of titpack2 (en) manual of titpack2 (jp)

  • Without generating matrix elements to save memory (test version, not so efficient...)

    • Use scipy.sparse.linalg.LinearOperator for Python

    • Use LinearMap for Julia

      • Tutorials in Japanese https://qiita.com/cometscome_phys/items/9f9cbd4f2ffa75e09f5e https://qiita.com/cometscome_phys/items/643f56d365c095884b89

Models

• 1D J1-J2 Heisenberg model

  • See, for example, numerical results in DOI:10.1016/0375-9601(92)90823-5 DOI:10.1103/PhysRevB.54.R9612

  • See also analytical results in DOI:10.1063/1.1664978 DOI:10.1063/1.1664979 DOI:10.1103/PhysRevB.25.4925

• 2D J1-J2 Heisenberg model on a square lattice

  • See, for example, numerical results in DOI:10.1103/PhysRevLett.63.2148 DOI:10.1051/jp1:1996236

  • See also DOI:10.1016/j.cpc.2018.08.014 (spin correlations at J2=0)

Useful references

• Papers

  • HPhi DOI:10.1016/j.cpc.2017.04.006 (arXiv)

  • Sz and momentum conservation, Lanczos DOI:10.1063/1.3518900 (arXiv)

  • 2D search of states DOI:10.1103/PhysRevB.42.6561

  • search by Pascal's triangle arXiv:1912.11240

• Codes / Lecture notes

  • Anders W. Sandvik http://physics.bu.edu/~sandvik/vietri/index.html http://physics.bu.edu/~sandvik/vietri/vietri.pdf http://physics.bu.edu/~sandvik/vietri/dia.pdf

    • Use a bisectional search instead of making a 2D search list
    • Implementation may be easier with momentum conservation

  • Frank Pollmann http://tccm.pks.mpg.de/?page_id=871 https://www.pks.mpg.de/~frankp/comp-phys/ https://www.pks.mpg.de/~frankp/comp-phys/exact_diagonalization_conserve.py

  • Guillaume Roux http://lptms.u-psud.fr/wiki-cours/index.php/Lectures_on_Exact_Diagonalization http://lptms.u-psud.fr/membres/groux/Test/ED/ED_Lecture1.html http://lptms.u-psud.fr/membres/groux/Test/ED/ED_Lecture2.html

  • Alexander Wietek https://github.com/alexwie/ed_basics

    • State search by a python built-in function index() for list (np.where for numpy array)

  • Glen Evenbly https://www.tensors.net/exact-diagonalization

  • Ryan Levy https://ryanlevy.github.io/physics/Heisenberg1D-ED/

  • Junya Otsuki (in Japanese) http://www.physics.okayama-u.ac.jp/~otsuki/lecture/CompPhys2/eigen/spin.html