Quantum Field Theory for the Three-Body Constrained Lattice Bose Gas -- Part I: Formal Developments

TL;DR

This paper introduces a quantum field theoretical approach to analyze the three-body constrained Bose-Hubbard model, extending beyond mean field approximations by mapping it to a coupled bosonic theory with polynomial interactions.

Contribution

It presents a novel formalism that maps the constrained model to a coupled bosonic theory, enabling analytical study beyond traditional approximations.

Findings

Tested the theory via scattering properties of few particles at low density

Analyzed excitations in the maximum filling limit involving holes and di-holes

Established a framework for future many-body applications in optical lattices

Abstract

We develop a quantum field theoretical framework to analytically study the three-body constrained Bose-Hubbard model beyond mean field and non-interacting spin wave approximations. It is based on an exact mapping of the constrained model to a theory with two coupled bosonic degrees of freedom with polynomial interactions, which have a natural interpretation as single particles and two-particle states. The procedure can be seen as a proper quantization of the Gutzwiller mean field theory. The theory is conveniently evaluated in the framework of the quantum effective action, for which the usual symmetry principles are now supplemented with a ``constraint principle'' operative on short distances. We test the theory via investigation of scattering properties of few particles in the limit of vanishing density, and we address the complementary problem in the limit of maximum filling, where the low lying excitations are holes and di-holes on top of the constraint induced insulator. This is the first of a sequence of two papers. The application of the formalism to the many-body problem, which can be realized with atoms in optical lattices with strong three-body loss, is performed in a related work [14].