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arXiv:quant-ph/0405085 · International Journal of Quantum Information

SCALING OF ENTANGLEMENT IN FINITE ARRAYS OF EXCHANGE-COUPLED QUANTUM DOTS

Jiaxiang Wang, Sabre Kais

Published Sep 1, 2003• 7 citations
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Abstract

We present a finite-size scaling analysis of the entanglement in a two-dimensional arrays of quantum dots modeled by the Hubbard Hamiltonian on a triangular lattice. Using multistage block renormalization group approach, we have found that there is an abrupt jump of the entanglement when a first-order quantum phase transition occurs. At the critical point, the entanglement is constant, independent of the block size.

DOI
10.1142/s0219749903000267

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Transport properties of arrays of metallic quantum dots are governed by the distance-dependent exchange coupling between the dots. It is shown that the effective value of the exchange coupling, as measured by the charging energy per dot, depends monotonically on the size of the array. The effect saturates for hexagonal arrays of over 7^5 unit cells. The discussion uses a multi-stage block renormalization group approach applied to the Hubbard Hamiltonian. A first order phase transition occurs upon compression of the lattice and the size dependence is qualitatively different for the two phases.
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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.
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.
We study a quantum phase transition that occurs in a system composed of two impurities (or quantum dots), each coupled to a different interacting (Luttinger liquid) lead. While the impurities are coupled electrostatically, there is no tunneling between them. Using a mapping of this system onto a Kondo model, we show analytically that the system undergoes a Berezinskii-Kosterlitz-Thouless quantum phase transition as a function of the Luttinger liquid parameter in the leads and the dot-lead interaction. The phase with low values of the Luttinger liquid parameter is characterized by an abrupt switch of the population between the impurities as a function of a common applied gate voltage. However, this behavior is hard to verify numerically since one would have to study extremely long systems. Interestingly, though, at the transition the entanglement entropy drops from a finite value of ln(2)\mathrm{ln}(2)ln(2) to zero. The drop becomes sharp for infinite systems. One can employ finite-size scaling to extrapolate the transition point and the behavior in its vicinity from the behavior of the entanglement entropy in moderate size samples. We employ the density matrix renormalization-group numerical procedure to calculate the entanglement entropy of systems with lead lengths of up to 480 sites. Using finite-size scaling, we extract the transition value and show it to be in good agreement with the analytical prediction.