Organization of the Hilbert Space for Exact Diagonalization of Hubbard Model

TL;DR

This paper introduces a new basis organization scheme for the Hubbard model that reduces computational time and memory usage, enabling more efficient exact diagonalization and applications in DMFT and other models.

Contribution

The authors propose an alternative basis representation that simplifies Hamiltonian matrix construction and significantly speeds up computations compared to previous methods.

Findings

CPU time reduced by about an order of magnitude

Scheme is inherently parallelizable

Applicable to translationally invariant systems and DMFT

Abstract

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 Green's functions and in impurity solver part of DMFT procedure is discussed and its extention to other models is also pointed out.