  We study Krylov complexity of a one-dimensional Bosonic system, the
celebrated Bose-Hubbard Model. The Bose-Hubbard Hamiltonian consists of
interacting bosons on a lattice, describing ultra-cold atoms. Apart from
showing superfluid-Mott insulator phase transition, the model also exhibits
both chaotic and integrable (mixed) dynamics depending on the value of the
interaction parameter. We focus on the three-site Bose Hubbard Model (with
different particle numbers), which is known to be highly mixed. We use the
Lanczos algorithm to find the Lanczos coefficients and the Krylov basis. The
orthonormal Krylov basis captures the operator growth for a system with a given
Hamiltonian. However, the Lanczos algorithm needs to be modified for our case
due to the instabilities instilled by the piling up of computational errors.
Next, we compute the Krylov complexity and its early and late-time behaviour.
Our results capture the chaotic and integrable nature of the system. Our paper
takes the first step to use the Lanczos algorithm non-perturbatively for a
discrete quartic bosonic Hamiltonian without depending on the auto-correlation
method.
