  In this paper, we employ the bootstrap method, a technique that relies on
consistency relations instead of direct diagonalization, to determine the
expectation values in quantum many-body systems. We then use these values to
assess the entanglement content of the system. Our work extends the bootstrap
approach to quantum many-body systems, rather than single-body or few-body
systems, concentrating on the well-known Lipkin-Meshkov-Glick (LMG) model with
both transverse and longitudinal external magnetic fields. In the bootstrap
method we solve the LMG model with up to 16 sites. Unlike previous studies that
have focused mainly on ground-state properties, our methodology allows for the
calculation of a broad range of properties, including energy spectrum, angular
momentum, concurrence, tangle, residual tangle, and quantum Fisher information
(QFI), for all eigenstates or a particular sector of the eigenstates, without
referring to the explicit wavefunctions of these states. We show that this
approach offers not only a new computational methodology but also a
comprehensive view of both bipartite and multipartite entanglement properties
across the entire spectrum of eigenstates. Specifically, we demonstrate that
states typically found in the central region of the spectrum exhibit greater
multipartite entanglement, as indicated by larger QFI values, compared to
states at the edges of the spectrum. In contrast, concurrence displays the
opposite trend. This observed behavior is in line with the monogamy principle
governing quantum entanglement.
