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arXiv:cond-mat/0611091 · Physical Review B

Entanglement in extended Hubbard models and quantum phase transitions

Alberto Anfossi, Paolo Giorda, Arianna Montorsi

Published Apr 11, 2007• 40 citations
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Abstract

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.

DOI
10.1103/physrevb.75.165106

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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=0T=0T=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.
We analyze correlations between subsystems for an extended Hubbard model exactly solvable in one dimension, which exhibits a rich structure of quantum phase transitions (QPTs). The T = 0 phase diagram is exactly reproduced by studying singularities of single-site entanglement. It is shown how comparison of the latter quantity and quantum mutual information allows one to recognize whether two-point or shared quantum correlations are responsible for each of the occurring QPTs. The method works in principle for any number D of degrees of freedom per site. As a by-product, we are providing a benchmark for direct measures of bipartite entanglement; in particular, here we discuss the role of negativity at the transition.
We investigate entanglement properties at quantum phase transitions (QPTs) of a one-dimensional integrable extended Hubbard model in the momentum-space representation. Two elementary subsystems are recognized: the single mode of an electron and the pair of modes (electrons coupled through the $\ensuremath{\eta}$-pairing mechanism). We first detect the two or multipartite nature of each quantum phase transition by a comparative study of the singularities of Von Neumann entropy and quantum mutual information. We establish the existing relations between the quantum correlations in the momentum representation and the relevant (superconducting) ones exhibited in the complementary picture: the direct-lattice representation. We then investigate the presence of multipartite entanglement in momentum space by means of a generalization of the Meyer-Wallach measure and we relate its behavior to the nature of the various QPTs.
We study the quantum discord (QD) and the classical correlations (CC) in a reference model for strongly correlated electrons, the one-dimensional bond-charge extended Hubbard model. We show that the comparison of QD and CC and of their derivatives in the direct and reciprocal lattice allows one to efficiently inspect the structure of two-points driven quantum phase transitions, discriminating those at which off diagonal long-range order (ODLRO) is involved. Moreover, we observe that QD between pair of sites is a monotonic function of ODLRO, thus establishing a direct relation between the latter and two point quantum correlations that differ from the entanglement. The study of the ground-state properties allows us to show that for a whole class of permutation invariant ($\ensuremath{\eta}$-pair) states quantum discord can violate the monogamy property, both in presence and in absence of bipartite entanglement. In the thermodynamic limit, due to the presence of ODLRO, the violation for $\ensuremath{\eta}$-pair states is maximal, while, for the purely fermionic ground state, it is finite. From a general perspective, all our results validate the importance of the concepts of QD and CC for the study of critical condensed-matter systems.
We study the quantum discord (QD) and the classical correlations (CC) in a reference model for strongly correlated electrons, the one-dimensional bond-charge extended Hubbard model.We show that the comparison of QD and CC and of their derivatives in the direct and reciprocal lattice allows one to efficiently inspect the structure of two-points driven quantum phase transitions, discriminating those at which off diagonal long-range order (ODLRO) is involved.Moreover, we observe that QD between pair of sites is a monotonic function of ODLRO, thus establishing a direct relation between the latter and two point quantum correlations that differ from the entanglement.The study of the ground-state properties allows us to show that for a whole class of permutation invariant (η-pair) states quantum discord can violate the monogamy property, both in presence and in absence of bipartite entanglement.In the thermodynamic limit, due to the presence of ODLRO, the violation for η-pair states is maximal, while, for the purely fermionic ground state, it is finite.From a general perspective, all our results validate the importance of the concepts of QD and CC for the study of critical condensed-matter systems.
Entanglement has been widely investigated in condensed-matter systems since they are considered good candidates for developing quantum technologies. Additionally, entanglement is a powerful tool to explore quantum phase transitions in strongly correlated systems, with the von Neumann entropy being considered a proper measure of quantum entanglement for pure bipartite systems. For lattice systems, in particular, the single-site entanglement quantifies how much information about the quantum state of the remaining sites can be obtained by a measurement at a single site. Here, we use quantum Monte Carlo calculations to obtain the average single-site entanglement for the two-dimensional Hubbard model in different geometries, probing the effects of varying temperature and interaction strength. We find that the average single-site entanglement signals the quantum phase transitions in such systems, allowing us to identify and characterize signatures of quantum phase transitions even at finite temperatures. We also analyze the relation between entanglement and magnetic susceptibility: in all the geometries considered, we find regimes in which the quantities are roughly linearly connected, revealing their link to the Mott transition.
In this work, we study the two-point entanglement S ( i , j ), which measures the entanglement between two separated degrees of freedom ( ij ) and the rest of system, near a quantum phase transition. Away from the critical point, S ( i , j ) saturates with a characteristic length scale ξ E , as the distance | i − j | increases. The entanglement length ξ E agrees with the correlation length. The universality and finite size scaling of entanglement are demonstrated in a class of exactly solvable one-dimensional spin model. By connecting the two-point entanglement to correlation functions in the long range limit, we argue that the prediction power of a two-point entanglement is universal as long as the two involved points are separated far enough.