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arXiv:quant-ph/0405087 · Physical Review A

Scaling of entanglement at a quantum phase transition for a two-dimensional array of quantum dots

Jiaxiang Wang, Sabre Kais

Published Aug 4, 2004• 18 citations
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Abstract

Using the Hubbard model, the entanglement scaling behavior in a two-dimensional itinerant system is investigated. It has been found that, on the two sides of the critical point denoting an inherent quantum phase transition (QPT), the entanglement follows different scalings with the size, just as an order parameter does. This fact reveals the subtle role played by the entanglement in QPT as a fungible physical resource.

DOI
10.1103/physreva.70.022301

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Using the Hubbard model, the entanglement scaling behavior in a two-dimensional itinerant system is investigated. It has been found that, on the two sides of the critical point denoting an inherent quantum phase transition (QPT), the entanglement follows different scalings with the size, just as an order parameter does. This fact reveals the subtle role played by the entanglement in QPT as a fungible physical resource.
We use the quantum renormalization-group (QRG) method to study the entanglement and quantum phase transition (QPT) in the one-dimensional spin-1/2 Heisenberg—Ising model [Lieb E, Schultz T and Mattis D 1961 Ann. Phys. (N.Y.) 16 407]. We find the quantum phase boundary of this model by investigating the evolution of concurrence in terms of QRG iterations. We also investigate the scaling behavior of the system close to the quantum critical point, which shows that the minimum value of the first derivative of concurrence and the position of the minimum scale with an exponent of the system size. Also, the first derivative of concurrence between two blocks diverges at the quantum critical point, which is directly associated with the divergence of the correlation length.
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 study the entanglement of formation and quantum phase transition in a one-dimensional quantum dot system with disorder modeled by the Hubbard Hamiltonian. The entanglement for three different cases is studied: the homogeneous case, the impurity case of symmetric electron hopping, and asymmetric electron hopping. We find that the local entanglement of the system can be tuned by introducing different impurities characterized by the physical parameters of the system. In particular, for certain parameters, the entanglement is negligible up to a critical point Uc, where a quantum phase transition occurs, and is different from zero above Uc.
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The ground-state entanglement of the two-mode Bose-Einstein condensate is investigated through a quantum-phase-transition approach. The entanglement measure is taken as the order parameter and this is a nonlocal order parameter, which is different from the conventional order parameter of the Mott-insulator--superfluid-phase transitions. For this nonlocal order parameter, the scaling behavior corresponding to a continuous phase transition is obtained and a power-law divergence near the critical region follows it. This scaling behavior of quantum entanglement is analyzed by the finite-size scaling and the critical exponents are obtained as $\ensuremath{\nu}=1.01$ and $\ensuremath{\gamma}=0.86$. A close connection between quantum fluctuations and the phase transition of entanglement is also obtained.