Probing quantum phase transitions in a staggered bosonic Kitaev chain via Fock space resolved localization-delocalization transitions

The bosonic statistics, which allow for macroscopic multioccupancy of single-particle states, pose significant challenges for analyzing quantum phase transitions in interacting bosonic systems, both analytically and numerically. In this work, we systematically investigate the non-Hermitian Bloch core matrix of a Hermitian staggered bosonic Kitaev chain, formulated within the Nambu framework. We derive explicit analytic conditions for the emergence of exceptional points (EPs) in the $4\ifmmode\times\else\texttimes\fi{}4$ Bloch core matrix, with each EP marking the onset of complex-conjugate eigenvalue pairs. By mapping the full many-body Hamiltonian onto an effective tight-binding network in Fock space and introducing the layer-resolved inverse participation ratio, we demonstrate that these EPs coincide precisely with sharp localization-delocalization transitions of collective eigenstates. Comprehensive numerical analyses across hopping amplitudes, pairing strengths, and on-site potentials confirm that the EP of the effective Hamiltonian universally capture the global many-body phase boundaries. Our results establish an analytically tractable, EP-based criterion for detecting critical behavior in interacting bosonic lattices, with direct relevance to photonic and cold-atom experimental platforms.