Many-Body Entanglement Properties of Finite Interacting Fermionic Hamiltonians

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.

Figures (7)

• Figure 1: a) Illustration of the non-existence theorem: When the ground state is maximally $M$-body entangled, the ground state energy $E_{\text{GS}}$ will coincide with the mean $\mu_1$, cf. Eq. (\ref{['eq:MeanEVDist']}), implying that the Hamiltonian is trivial. b) $M$-body entanglement entropy for the ground state of extended complex SYK Hamiltonian with random one, two and three-body interaction terms. We fix $\lambda_1=1, \lambda_2=10$ and change $\lambda_3$, see Eq. (\ref{['eq: SYK extended Hamiltonian']}). At $\lambda_3=200$, $S^{(1)} \approx 0.98S^{(1)}_{\text{max}}$, $S^{(2)} \approx 0.973S^{(2)}_{\text{max}}$ and $S^{(3)} \approx 0.953S^{(3)}_{\text{max}}$. See SM for the same plot with multiple random realizations, which turn out to have a very small spread at the saturation points. c) $M$-body entanglement entropy of the half-filled triangular lattice Hubbard model [Eq. (\ref{['eq: hubbard model Hamiltonian']})] ground state as a function of corresponding coupling parameters. Here, two lattice directions have equal hopping strength $\tau=1$ and the third has $\tau'=2$, and a small magnetic field is applied to break the GS degeneracy. Left sub-panel: Changing two-body interaction $U$, without three-body interaction. Right sub-panel: At $U = 100$, where entropy is saturated but not maximal for $M = 2,3$, we turn on the random three-body interaction term. At $\lambda_3=100$, $S^{(1)} \approx 0.991S^{(1)}_{\text{max}}$, $S^{(2)} \approx 0.979S^{(2)}_{\text{max}}$ and $S^{(3)} \approx 0.942S^{(3)}_{\text{max}}$.
• Figure 2: Time evolution of the second Rényi entropy at half filling for the two-body SYK and 1D Hubbard model dynamics. Initial states are Slater determinants. a) Early time evolution under two-body SYK Hamiltonian with interaction strength $\lambda_2$, showing $\propto t^2$ scaling for $t\ll \min \{\lambda_2^{-1}, \lambda_1^{-1}\}$. b) Entropy growth and saturation for two-body SYK model. The vertical dashed green line, obtained from Eq. (\ref{['eq: Entropy saturation time approximation SYK 2-body']}) by extrapolating the short-time behavior, lower bounds the saturation time. The saturation time estimate $t_\text{sat}=0.0428/\lambda_2$ is valid when $t_{\text{sat}} \ll \lambda_1^{-1}$. c) Early time evolution under 1D Hubbard Hamiltonian (hopping $\tau=1$) with on-site interaction $U$, showing $\propto t^6$ scaling at short times SM.
• Figure 3: Hubbard dynamics and entropy saturation constrained by symmetries. a) Time evolution of the one-body entanglement entropy for a 9-site 1D lattice at half-filling, starting from SD initial states in the spin-projection sectors $|S_z|$ of 1/2 and 7/2 (in units of $\hbar$). Symmetry refined upper bounds to the entanglement entropy of Eq. (\ref{['eq: Refined upper bound to entropy']}) are given by red dashed lines. Plots for other initial SD states and different $M$ values are provided in SM. b) Numerically obtained entropy saturation values (circles), similar to panels a), and theoretical refined bounds (crosses) obtained from Eq. (\ref{['eq: Refined upper bound to entropy']}).
• Figure S1: The 1-body entanglement entropy for an ensemble of three-body random terms in the SYK (a) and Hubbard (b) models corresponding to Figs. \ref{['fig: combined plots spectral distribution schematic and GS theorem']}(b)-(c) of the main text. The entanglement entropy close to saturation has a very small spread justifying the consideration of a single realization. We used respectively 40 and 100 realizations in (a) and (b).
• Figure S2: (a) Long time evolution of $2^{\text{nd}}$ Rényi entropy under two-body SYK dynamics for the case of $M=3$ and $D=16, N=8$. In this case, the upper bound is not saturated. (b) Long time evolution of $2^{\text{nd}}$ Rényi entropy under two-body SYK dynamics for the case $D=22,N=11$. Now, the upper bound is nearly saturated for $M=3$ as well. (c) Early time evolution at half filling of $2^{\text{nd}}$ Rényi entropy under three-body SYK dynamics, following a $t^2$ time dependence. The initial state is a Slater determinant (SD).