Many-Body Entanglement Properties of Finite Interacting Fermionic Hamiltonians

We analyze many-body entanglement in interacting fermionic systems by using the $M$-body reduced density matrix. We demonstrate that if a particle number conserving fermionic Hamiltonian contains only up to $M$-body interaction terms, then its $N$-particle ground state cannot be maximally $M$-body entangled. As a key step in the proof, we show that the energy expectation value of a maximally $M$-body mixed state is equal to the spectral mean of the Hamiltonian on the corresponding $N$-particle subspace. We further demonstrate that the many-body entanglement structure of a ground state can place quantitative constraint on the interaction strength of its parent Hamiltonian. We illustrate the theorem and its implications in Hubbard and extended SYK models. Going beyond ground states, we analyze entanglement generation under unitary dynamics from Slater-determinant initial states in these models. We determine early-time growth and estimate entanglement saturation times. Finally, we derive explicit symmetry-refined saturation upper bounds for $M$-body entanglement in the presence of an Abelian symmetry.