Entanglement in extended Hubbard models and quantum phase transitions

The role of two-point and multipartite entanglement at quantum phase transitions (QPTs) in correlated electron systems is investigated. We consider a bond-charge extended Hubbard model exactly solvable in one dimension which displays various QPTs—with two (qubit) as well as more (qudit) on-site degrees of freedom involved. The analysis is carried out by means of appropriate measures of bipartite/multipartite quantum correlations. It is found that all transitions ascribed to two-point correlations are characterized by an entanglement range which diverges at the transition points. The exponent coincides with that of the correlation length at the transitions. We introduce the correlation ratio, namely, the ratio of quantum mutual information and single-site entanglement. We show that at $T=0$, it captures the relative role of two-point and multipartite quantum correlations at transition points, generalizing to qudit systems the entanglement ratio. Moreover, a finite value of quantum mutual information between infinitely distant sites is seen to quantify the presence of off-diagonal long-range order induced by multipartite entanglement.

Figure 1

(Color online) Ground-state phase diagram of $H$. (a) $n-u$ plane; empty circles stand for empty sites, slashed and full circles stand for singly and doubly occupied sites, respectively. (b) $\mu -u$ plane.

Figure 2

(Color online) Region I, $u=4$. The curves $a$ and $c$ are the concurrencies ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$, respectively, while the curves $b$ and $d$ are the negativities ${\mathcal{N}}_1$ and ${\mathcal{N}}_2$, respectively.

Figure 3

(Color online) Region I, $u=4$. (a) ${\mathcal{C}}_{ij}$ and (b) ${\mathcal{N}}_{ij}$ for $r=1,..,8$.

Figure 4

(Color online) Region I, $u=4$. Entanglement ratio $E_R$ and normalized correlation ratio $C_R∕L$ with $L=1000$.

Figure 5

Region II, $n=1∕2u∊[-4,0]$. Normalized correlation ratio $C_R∕L$ with $L=1000$. Inset: zoom of the transition $\mathrm{II}\to \mathrm{I}$.

Figure 6

Region II, $n=1$, $u∊[-4,4]$. Normalized correlation ratio $C_R∕L$ with $L=1000$. Inset: zoom of the transition $\mathrm{II}\to \mathrm{IV}$.

Figure 7

(Color online) Region I, $u=4$. Mutual information ${\mathcal{I}}_{ij}$, $r=1,\dots ,5$.

Figure 8

(Color online) Region II, $n=1u∊[-4,4]$. Mutual information ${\mathcal{I}}_{ij}$, $r=1,\dots ,5$.