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arXiv:0711.2358 · Physical Review A

Renormalization of entanglement in the anisotropic Heisenberg (XXZ) model

Mehdi Kargarian, Rouhollah Jafari, A. Langari

Published Mar 27, 2008• 122 citations
arXivHTMLPDF

Abstract

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 $s=1/2$ 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.

DOI
10.1103/physreva.77.032346

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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.
In this paper the entanglement and quantum phase transition of the anisotropic spin-1/21/21/2 XY\mathit{XY}XY model are studied by using the quantum renormalization-group method. By solving the renormalization equations, we get the trivial and nontrivial fixed points, which correspond to the phase of the system and the critical point, respectively. The concurrence between two blocks are calculated and it is found that when the number of iterations of the renormalization tends to infinity, the concurrence develops two saturated values that are associated with two different phases, i.e., Ising-like and spin-fluid phases. We also investigate the first derivative of the concurrence and find that there exists nonanalytic behaviors at the quantum critical point, which are directly associated with the divergence of the correlation length. To gain further insight, the scaling behaviors of the system are analyzed and it is shown that the maximum value of the first derivative of the concurrence reaches infinity and the critical point is approached as the size of the system increases.
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.
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 dynamics of entanglement in the one-dimensional spin-1/2 anisotropic XXZ model is studied using the quantum renormalization-group method. We obtain the analytical expression of the concurrence, for two different quenching methods, it is found that initial state plays a key role in the evolution of system entanglement, i.e., the system returns completely to the initial state every other period. Our computations and analysis indicate that the first derivative of the characteristic time at which the concurrence reaches its maximum or minimum with respect to the anisotropic parameter occurs nonanalytic behaviors at the quantum critical point. Interestingly, the minimum value of the first derivative of the characteristic time versus the size of the system exhibits the scaling behavior which is the same as the scaling behavior of the system ground-state entanglement in equilibrium. In particular, the scaling behavior near the critical point is independent of the initial state.
We have combined the idea of renormalization group and quantum-information theory. We have shown how the entanglement or concurrence evolve as the size of the system becomes large, i.e., the finite size scaling is obtained. Moreover, we introduce how the renormalization-group approach can be implemented to obtain the quantum-information properties of a many-body system. We have obtained the concurrence as a measure of entanglement, its derivatives and their scaling behavior versus the size of system for the one-dimensional Ising model in transverse field. We have found that the derivative of concurrence between two blocks each containing half of the system size diverges at the critical point with the exponent, which is directly associated with the divergence of the correlation length.
We investigate entanglement and quantum phase transition in a two-dimensional Heisenberg anisotropic spin-1/2XY1/2\mathit{XY}1/2XY model, using the quantum renormalization-group method (QRG) on a square lattice of $N\ifmmode\times\else\texttimes\fi{}N$ sites. The entanglement through geometric average of concurrences is calculated after each step of the QRG. We show that the concurrence achieves a nonzero value at the critical point more rapidly as compared to the one-dimensional case. The relationship between the entanglement and the quantum phase transition is studied. The evolution of entanglement develops two saturated values corresponding to two different phases. We compute the derivative of the concurrence, which is found to be discontinuous at the critical point $\ensuremath{\gamma}=0$, and indicates a second-order phase transition in the spin system. Further, the scaling behavior of the system is investigated by computing the derivative of the concurrence in terms of the system size.