Exact diagonalization: the Bose-Hubbard model as an example

TL;DR

This paper demonstrates how to apply exact diagonalization techniques to the Bose-Hubbard model, emphasizing basis generation, Hamiltonian setup, and eigenstate computation, serving as a pedagogical guide for quantum many-body problems.

Contribution

It provides a detailed, step-by-step methodology for exact diagonalization of the Bose-Hubbard model, highlighting basis enumeration and matrix construction techniques.

Findings

Illustrates basis vector generation and hashing for Hamiltonian setup

Shows how to use Lanczos algorithm for low-lying eigenstates

Provides general techniques applicable to other quantum models

Abstract

We take the Bose-Hubbard model to illustrate exact diagonalization techniques in a pedagogical way. We follow the road of first generating all the basis vectors, then setting up the Hamiltonian matrix with respect to this basis, and finally using the Lanczos algorithm to solve low lying eigenstates and eigenvalues. Emphasis is placed on how to enumerate all the basis vectors and how to use the hashing trick to set up the Hamiltonian matrix or matrices corresponding to other quantities. Although our route is not necessarily the most efficient one in practice, the techniques and ideas introduced are quite general and may find use in many other problems.