Entanglement of indistinguishable particles in condensed-matter physics

The concept of entanglement in systems where the particles are indistinguishable has been the subject of much recent interest and controversy. In this paper we study the notion of entanglement of particles introduced by Wiseman and Vaccaro [Phys. Rev. Lett. 91, 097902 (2003)] in several specific physical systems, including some that occur in condensed-matter physics. The entanglement of particles is relevant when the identical particles are itinerant and so not distinguished by their position as in spin models. We show that entanglement of particles can behave differently than other approaches that have been used previously, such as entanglement of modes (occupation-number entanglement) and the entanglement in the two-spin reduced density matrix. We argue that the entanglement of particles is what could actually be measured in most experimental scenarios and thus its physical significance is clear. This suggests that entanglement of particles may be useful in connecting theoretical and experimental studies of entanglement in condensed-matter systems.

Figure 1

Bipartite entanglement of particles in the ground state of two bosons in four modes with Bose-Hubbard Hamiltonian and periodic boundary conditions. $A$ and $B$ control adjacent modes in the ring. $E_P$ is zero at $U=0$ (noninteracting bosons) and saturates at $(3∕4)h(1∕2+\sqrt{2}∕3)\simeq 0.1405$ for large $U∕t$ corresponding to $P_{1,1}=3∕4$ and $E_F=h(1∕2+\sqrt{2}∕3)$.

Figure 2

(Color online) (a) Bipartite entanglement of particles in the canonical-ensemble thermal state of two bosons in four modes with Bose-Hubbard Hamiltonian and periodic boundary conditions. $A$ and $B$ each control adjacent modes in the ring, as in Fig. 1. (b) Probability of finding one particle at $A$ and one at $B$, $P_{1,1}$. (c) Entanglement of formation of the a posteriori state ${\rho }_{AB}^{(1,1)}$.

Figure 3

(Color online) Bipartite entanglement of particles in the canonical-ensemble thermal state of the Hubbard dimer as a function of the scaled temperature $k_{B}T∕t$ and on-site interaction $U∕t$. $A$ controls the up and down modes of one site and $B$ the up and down modes of the other. The color represents the probability of finding one particle at each site and the height represents the entanglement of formation of the a posteriori state.

Figure 4

(a) Entanglement of particles for noninteracting fermions on a $1-D$ lattice with 30 sites as a function of inverse temperature $(t∕k_{B}T)$ and separation between the sites $\left(\mid j_A-j_B\mid \right)$. The subfigures (b) and (c) indicate the contributions to $E_P$ from the probability of finding one particle at each site, $P_{1,1}$, and the entanglement of formation in the a posteriori state, respectively. The chemical potential was $\mu \simeq -1.89$ corresponding to a $T\to 0$ filling factor of $\overline{n}=0.2$. In the $T\to 0$ limit we therefore roughly expect that $P_{1,1}\sim {\left(0.2\right)}^2=0.04$.

Figure 5

(Color online) Bipartite entanglement of formation (height) and probability $P_{1,1}$ (greyscale) as a function of separation and filling factor in the $T\to 0$ limit of the grand canonical-ensemble thermal state of the Fermi-Hubbard model with 30 sites and periodic boundary conditions. The chemical potential was varied from $-2$ to 2 to achieve different filling factors.

Figure 6

Entanglement length (crosses) $r_e$ vs inverse filling factor $1∕\overline{n}$ for noninteracting fermions on a $1-D$ lattice. The circles show what the entanglement length would have been if it were defined using the spin-correlation matrix, Eq. (40), instead of the projected matrix, Eq. (25). The line is a linear least-squares fit to the crosses. The inset shows that the projected matrix predicts a longer entanglement length for some values of the filling factor.