A Davidson-Lanczos iteration method for computation of continued-fraction expansion of the Greens function at very low temperatures: Applications to the dynamical mean field theory

We present a combination method based on orignal version of Davidson algorithm for extracting few of the lowest eigenvalues and eigenvectors of a sparse symmetric Hamiltonian matrix and the simplest version of Lanczos technique for obtaining a tridiagonal representation of the Hamiltonian to compute the continued fraction expansion of the Greens function at a very low temperature. We compare the Davidson$+$Lanczos method with the full diagonalization on a one-band Hubbard model on a Bethe lattice of infinite-coordination using dynamical mean field theory.

Organization of the Hilbert Space for Exact Diagonalization of Hubbard Model

We present an alternative scheme to the widely used method of representing the basis of one-band Hubbard model through the relation $I=I_{uparrow}+2^{M}I_{downarrow}$ given by H. Q. Lin and J. E. Gubernatis [Comput. Phys. 7, 400 (1993)], where $I_{uparrow}$, $I_{downarrow}$ and $I$ are the integer equivalents of binary representations of occupation patterns of spin up, spin down and both spin up and spin down electrons respectively, with $M$ being the number of sites. We compute and store only $I_{uparrow}$ or $I_{downarrow}$ at a time to generate the full Hamiltonian matrix. The non-diagonal part of the Hamiltonian matrix given as ${cal{I}}_{downarrow}otimes{bf{H}_{uparrow}} oplus {bf{H}_{downarrow}}otimes{cal{I}}_{uparrow}$ is generated using a bottom-up approach by computing the small matrices ${bf{H}_{uparrow}}$(spin up hopping Hamiltonian) and ${bf{H}_{downarrow}}$(spin down hopping Hamiltonian) and then forming the tensor product with respective identity matrices ${cal{I}}_{downarrow}$ and ${cal{I}}_{uparrow}$, thereby saving significant computation time and memory. We find that the total CPU time to generate the non-diagonal part of the Hamiltonian matrix using the new one spin configuration basis scheme is reduced by about an order of magnitude as compared to the two spin configuration basis scheme. The present scheme is shown to be inherently parallelizable. Its application to translationally invariant systems, computation of Greens functions and in impurity solver part of DMFT procedure is discussed and its extention to other models is also pointed out.