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Quantum Information and Computation

Renormalization of quantum deficit and monogamy relation in the Heisenberg XXZ model

Meng Qin, Xin Zhang, Zhongzhou Ren

Published Jul 1, 2016• 2 citations
Publisher

Abstract

In this study, the dynamical behavior of quantum deficit and monogamy relation in the Heisenberg XXZ model is investigated by implementing quantum renormalization group theory. The results demonstrate that the quantum deficit can be used to capture the quantum phase transitions point and show scaling behavior with the spin chain size increasing. It was also found that the critical exponent has no change when varying measure from entanglement to quantum correlation. The monogamy relation is influenced by the steps of quantum renormalization group and the ways of splitting the block states. Furthermore, the monogamy relation of generalized W state also is given by means of quantum deficit.

DOI
10.26421/qic16.9-10-6

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In this study, the dynamical behavior of quantum deficit and monogamy relation in the Heisenberg XXZ model is investigated by implementing quantum renormalization group theory. The results demonstrate that the quantum deficit can be used to capture the quantum phase transitions point and show scaling behavior with the spin chain size increasing. It was also found that the critical exponent has no change when varying measure from entanglement to quantum correlation. The monogamy relation is influenced by the steps of quantum renormalization group and the ways of splitting the block states. Furthermore, the monogamy relation of generalized W state also is given by means of quantum deficit.
The purpose of the paper is mainly to investigate the quantum critical behavior of two-dimensional XY spin system by calculating quantum correlation and monogamy relation through implementation of quantum renormalization group theory. Numerical analysis indicates that quantum correlation as well as quantum nonlocality can be used to efficiently detect the quantum critical property in two-dimensional XY spin system. The nonanalytic behavior of the first derivative of quantum correlation approaches infinity and the critical point is reached as the size of the model increases. Furthermore, we discuss the quantum correlation distribution in this model based on square of concurrence (SC) and square of quantum discord (SQD). The monogamous properties of SC and SQD are obtained for the present system. We finally reveal that the monogamy score can be used to capture the quantum critical point.
Abstract Using the quantum renormalization group approach, we have investigated the entanglement and critical behavior of an anisotropic XXZ Heisenberg spin chain. The resolution of the quantum renormalization equations yields stable and unstable fixed points, corresponding respectively to the phases and critical point of the system. It is shown that a quantum phase transition exists at a critical anisotropy parameter , separating the Néel phase from the spin-fluid phase. Then, the concurrence between two blocks of sites and the quantum coherence have been calculated and presented for various renormalization group iterations. In the thermodynamic limit, i.e. , for higher renormalization steps, the concurrence and coherence exhibit two different saturated values in the two regions separated by Δ c . The first derivative of concurrence diverges at the critical point as the size of the chain increases, showing nonanalytic behavior. Moreover, the scaling behavior has been examined near the critical point with some universal exponential scaling laws.
The purpose of the paper is mainly to investigate the quantum critical behavior of two-dimensional XY spin system by calculating quantum correlation and monogamy relation through implementation of quantum renormalization group theory. Numerical analysis indicates that quantum correlation as well as quantum nonlocality can be used to efficiently detect the quantum critical property in two-dimensional XY spin system. The nonanalytic behavior of the first derivative of quantum correlation approaches infinity and the critical point is reached as the size of the model increases. Furthermore, we discuss the quantum correlation distribution in this model based on square of concurrence (SC) and square of quantum discord (SQD). The monogamous properties of SC and SQD are obtained for the present system. We finally reveal that the monogamy score can be used to capture the quantum critical point.
We have applied our recent approach [M. Kargarian et al., Phys. Rev. A 76, 060304(R) (2007)] to study the quantum-information properties of the anisotropic s=1/2s=1/2s=1/2 Heisenberg chain. We have investigated the underlying quantum-information properties such as the evolution of concurrence, entanglement entropy, nonanalytic behaviors, and the scaling close to the quantum critical point of the model. Both the concurrence and the entanglement entropy develop two saturated values after enough iterations of the renormalization of coupling constants. These values are associated with the two different phases, i.e., N\'eel and spin liquid phases. The nonanalytic behavior comes from the divergence of the first derivative of both measures of entanglement as the size of the system becomes large. The renormalization scheme demonstrates how the minimum value of the first derivative and its position scales with an exponent of the system size. It is shown that this exponent is directly related to the critical properties of the model, i.e., the exponent governing the divergence of the correlation length close to the quantum critical point. We also use a renormalization method based on the quantum group concept in order to get more insight about the critical properties of the model and the renormalization of entanglement.
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.
In this paper, we investigate the crossover behaviors of entanglement and return probability at the critical point in one-dimensional XXZ model by quantum renormalization group method. Two types of quench protocols (A) adding the interaction term in z spin direction suddenly and (B) rotating all the even spins by π about the z spin direction suddenly are considered to drive the dynamics of the system, respectively. The dynamical behaviors are closely related to the initial state under two types of quench protocols, especially at N→∞. For the left of critical point, return probabilities eventually approach one under both types of quench protocols. For the right of critical point, return probability oscillates between the maximum value and zero for quench A and tends to one for quench B, respectively. In addition, we discuss the evolution period of entanglement (return probability), find that it is positively and negatively correlated with N in the spin-fluid and Néel phases, respectively. By further analyzing the behaviors of entanglement, return probability and evolution period under both types of quench protocols near the critical point, we observe that they exist the same scaling behaviors, in which the exponents are identical to entanglement exponent of the stationary system, which is related to the results of renormalization group.