Exact diagonalization techniques are used to study spin Hamiltonians
such as the Heisenberg model, many-body Hamiltonians such as the Hubbard
model, and single-particle models, such as a particle in a disordered
potential (Anderson localization). The method aims to solve the Schroedinger
equation exactly for a finite system, i.e., a linear eigenvalue problem
and delivers good results for the ground state and low-lying excited states.
Unfortunately, the dimensionality of the Hilbert space grows exponentially,
hence the method can only be applied to small system sizes. Nevertheless exact
diagonalization is often used for problems where other generally more
efficient methods fail, such as in the case of frustrated quantum magnets.
References:
- C. Lanczos, J. Res. Natl. Bur. Stand. 45, 255 (1950).
- E. Gagliano, et al., Phys. Rev. B 34, 1677 (1986).
- J. C. Bonner and M. E. Fisher, Phys. Rev. 135, 640 (1964).
- M. Troyer, Computational Physics II, lecture notes (2004).
- N. Laflorencie and D. Poilblanc, Simulations of pure and doped
low-dimensional spin-1/2 gapped systems, Lect. Notes Phys. 645, 227-252
(2004) - cond-mat/0408363.
- Z. Bai, et al., Templates for the Solution of Algebraic Eigenvalue Problems
Further reading can be found at the
ALPS webpages, in particluar the talk
by Andreas Laeuchli (Introduction to Exact Diagonalization)
which can be found
here.
|