Particle partitioning entanglement in itinerant many-particle systems

For itinerant fermionic and bosonic systems, we study “particle entanglement,” defined as the entanglement between two subsets of particles making up the system. We formulate the general structure of particle entanglement in many-fermion ground states, analogous to the “area law” for the more usually studied entanglement between spatial regions. Basic properties of particle entanglement are uncovered by considering relatively simple itinerant models.

Figure 1

(Color online) Entanglement between two bosons, in the ground state of a two-site lattice model with on-site repulsion. The solid curve is for the basic Bose-Hubbard model. Dashed $(ϵ=0.1)$ and dash-dotted $(ϵ=0.1U)$ curves illustrate the fragility of the “cat” state via an $ϵ{\widehat{b}}_1^{†}{\widehat{b}}_1$ term.

Figure 2

(Color online) One- and two-particle entanglement entropies for $N$ bosons in two sites.

Figure 3

(Color online) $n=1$, 2, and 3 entanglement entropy in the half-filled $t\text{−}V$ model $(N=L∕2)$. The free-fermion contribution $\ln \left[C(N,n)\right]$ has been subtracted off. The $n=1$ plot also displays the effect of a symmetry-breaking $ϵc_1^{†}{c}_1$ term, with $ϵ=0.1$. Inset: position of the maximum as a function of $ϵ$.

Figure 4

(Color online) (a) $S_n$ for $N=7$, $L=12\ne 2N$. Horizontal lines are corresponding maximal bounds $\ln \left[C(L,n)\right]$. (b) Negative $V$, half-filling. Free-fermion contribution $\ln \left[C(N,n)\right]$ has been subtracted off in each case.