Multipartite entanglement in a non-Hermitian quadruple-well potential with synthetic gauge fields

Multipartite entanglement lies at the heart of quantum many-body physics and quantum information science, serving as a key resource for quantum computation, quantum simulation, and quantum metrology. While its behavior in Hermitian systems has been extensively studied, its fate in non-Hermitian systems, especially those with synthetic gauge fields, remains largely unexplored. In this work we investigate multipartite entanglement dynamics in a quadruple-well system under the influence of synthetic gauge fields and nonreciprocal coupling. By constructing a many-body Hamiltonian that incorporates synthetic gauge fluxes, interaction-induced nonlinearity, and parity-time ($\mathcal{PT}$) symmetry-breaking terms, we reveal that the interplay between non-Hermiticity and gauge fields provides unprecedented control over entanglement structures and their evolution. Notably, the synthetic gauge field acts as a tunable switch for entanglement configurations, enabling transitions between separable, bipartite, tripartite, and globally entangled configurations. Furthermore, we uncover a striking odd-even effect of particle number on both $\mathcal{PT}$-symmetry phase transitions and entanglement patterns, highlighting the role of particle-number parity in non-Hermitian many-body systems. These findings not only deepen our understanding of many-body quantum correlations in non-Hermitian systems but also open alternative avenues for designing entanglement-resolved quantum sensors and preparing topologically nontrivial many-body states in synthetic quantum platforms.