Simplifying higher-order perturbation theory for ring-shaped Bose-Hubbard systems

TL;DR

This paper introduces a simplified higher-order perturbation theory tailored for ring-shaped Bose-Hubbard systems, reducing computational complexity while maintaining accuracy, and validates it through explicit calculations and comparisons.

Contribution

It develops a systematic simplification scheme for higher-order perturbation theory applicable to ring-shaped Bose-Hubbard systems, including a convergence criterion and explicit computational guide.

Findings

Simplified perturbation calculations up to ninth order.

Validated results against exact diagonalization.

Applicable to various Bose-Hubbard lattice geometries.

Abstract

In this paper, higher-order perturbation theory is applied and tailored to one-dimensional ring-shaped Bose-Hubbard systems. Spectral and geometrical properties are used to structurally simplify the contributions and reduce computational effort without sacrificing accuracy. For this, a guide for the computation of the individual perturbational orders up to order nine is provided, alongside a both system-specific and parametrization-dependent convergence criterion. The simplification scheme described is found to be applicable to a wider class of Bose-Hubbard systems with different lattice geometries. An exemplary validation of these findings is included in the form of explicit calculations of ground state energies of the three-site Bose-Hubbard system with repulsive on-site interactions. These calculations are successfully checked against numerical computations of exact diagonalization results.