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Chinese Journal of Physics

Quantum Renormalization Group for Ground State Quantum Discord

Javad Vahedi

Published Jun 1, 2015• 0 citations
Publisher

Abstract

Using the quantum renormalization group, we have studied a quantum phase transition via studying the ground state quantum discord. Our results show that the characterization of quantum criticality, as well as the scaling analysis of relevant properties with system size for the Ising model in the transverse field based on the quantum renormalization group, have a similar behavior to the concurrence, which was studied in Ref. [Phys. Rev. A 76, 060304(R) (2007)].

DOI
10.6122/cjp.20150128

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Using the quantum renormalization group, we have studied a quantum phase transition via studying the ground state quantum discord. Our results show that the characterization of quantum criticality, as well as the scaling analysis of relevant properties with system size for the Ising model in the transverse field based on the quantum renormalization group, have a similar behavior to the concurrence, which was studied in Ref. [Phys. Rev. A 76, 060304(R) (2007)].
Using the quantum renormalization group (QRG) method, we investigate the ground state geometric measure of quantum discord (GQD) of the transverse Ising model. The analytical expression of GQD together with the RG recurrence relations are obtained. The critical and scaling behaviors of GQD are investigated. Our results show that with the increasing of RG steps, GQD (first derivative of GQD) exhibits sudden change (dip) in behavior at the critical point, which indicates that the system undergoes phase transitions between the Ising and the paramagnetic phases. The critical exponent that governs the divergences of the first derivative of GQD is calculated. How the critical point of the model is approached as the increasing of the system size is revealed.
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 perform an extensive study of the properties of global quantum correlations in finite-size one-dimensional quantum spin models at finite temperature. By adopting a recently proposed measure for global quantum correlations (Rulli and Sarandy 2011 Phys. Rev. A 84 042109), called global discord , we show that critical points can be neatly detected even for many-body systems that are not in their ground state. We consider the transverse Ising model, the cluster-Ising model where three-body couplings compete with an Ising-like interaction, and the nearest-neighbor XX Hamiltonian in transverse magnetic field. These models embody our canonical examples showing the sensitivity of global quantum discord close to criticality. For the Ising model, we find a universal scaling of global discord with the critical exponents pertaining to the Ising universality class.
We have investigated the quantum phase transition employing the quantum renormalization-group method while, in most of the previous literature, entanglement (concurrence) has been barely demonstrated. However, it is now well known that entanglement is not the only signature of quantum correlations and a variety of computable measures have been developed to characterize quantum correlations in the composite systems. As an illustration, two cases are elaborated: a one-dimensional anisotropic (i) XXZXXZXXZ model and (ii) an XYXYXY model, with various measures of quantum correlations, including quantum discord, geometric discord, measurement-induced disturbance, measurement-induced nonlocality, and violation of Bell inequalities [e.g., Clauser-Horne-Shimony-Holt (CHSH) inequality]. We have proved that all of these correlation measures can effectively detect the quantum critical points associated with quantum phase transitions after several iterations of the renormalization in both cases. Nonetheless, it is shown that some of their dynamical behaviors are not totally similar with entanglement and, even when concurrence vanishes, there still exists some kind of quantum correlation which is not captured by entanglement. Intriguingly, CHSH inequality can never be violated in the whole iteration procedure, which indicates that block-block entanglement cannot revealed by the CHSH inequality. Moreover, the nonanalytic and scaling behaviors of Bell violation have also been discussed in detail. As a by-product, we verify that measurement-induced disturbance is exactly equal to the quantum discord measured by ${\ensuremath{\sigma}}_{z}$ for general X\mathsf{X}X-structured states.
Ground-state fidelity (GSF) and quantum renormalization group theory (QRG) have proven useful tools in the study of quantum critical systems. Here we lay out a general, unified formalism of GSF and QRG; specifically, we propose a method to calculate GSF through QRG, obviating the need for calculating or approximating ground states. This method thus enhances characterization of quantum criticality as well as scaling analysis of relevant properties with system size. We illustrate the formalism in the one-dimensional Ising model in a transverse field and the anisotropic spin-1/2 Heisenberg model.
Ground-state fidelity (GSF) and quantum renormalization group (QRG) theory have proven to be useful tools in the study of quantum critical systems. Here we lay out a general, unified formalism of GSF and QRG; specifically, we propose a method for calculating GSF through QRG, obviating the need for calculating or approximating ground states. This method thus enhances the characterization of quantum criticality as well as scaling analysis of relevant properties with system size. We illustrate the formalism in the one-dimensional Ising model in a transverse field (ITF) and the anisotropic spin-1/2 Heisenberg ( XXZ ) model. Explicitly, we find the scaling behavior of the GSF for the ITF model in both small- and large-size limits, the corresponding critical exponents, the exact value of the GSF in the thermodynamic limit and a closed form for the GSF for arbitrary size and system parameters. In the case of the XXZ model, we also present an analytic expression for the GSF, which captures well the criticality of the model, hence excluding doubts that GSF might be an insufficient tool for signaling criticality in this model.