Many-Body Entanglement Properties of Finite Interacting Fermionic Hamiltonians

TL;DR

This paper investigates the entanglement properties of many-body fermionic systems, establishing bounds based on the Hamiltonian's interaction order and analyzing entanglement dynamics and constraints in specific models.

Contribution

It proves that ground states of M-body interaction Hamiltonians cannot be maximally M-body entangled and explores entanglement growth and bounds in fermionic models.

Findings

Ground states of M-body Hamiltonians are not maximally M-body entangled.

Derived bounds on entanglement saturation times in fermionic models.

Demonstrated constraints on entanglement based on Hamiltonian symmetry and interaction order.

Abstract

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.