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European Journal of Physics

Concurrence of finite Ising chains with the Dzyaloshinsky–Moriya interaction

C.T. Gonzalez-H, Roberto Franco, Jereson Silva-Valencia

Published Apr 28, 2010• 7 citations
Publisher

Abstract

Due to the great impact that quantum information theory has and its applicability for condensed matter systems, we study the concurrence measure of entanglement for the finite Ising chain with the Dzyaloshinsky–Moriya interaction. The explicit calculation of the concurrence and the reduced density matrix necessary to calculate it is presented, in order to use them as a pedagogical tool to make such important concepts clearer. A growth of the concurrence was observed as the parameter of the interaction increased, reaching a maximum at D = 1, which indicates the phase transition of the system from an antiferromagnetic phase to a chiral one.

DOI
10.1088/0143-0807/31/3/026

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Due to the great impact that quantum information theory has and its applicability for condensed matter systems, we study the concurrence measure of entanglement for the finite Ising chain with the Dzyaloshinsky–Moriya interaction. The explicit calculation of the concurrence and the reduced density matrix necessary to calculate it is presented, in order to use them as a pedagogical tool to make such important concepts clearer. A growth of the concurrence was observed as the parameter of the interaction increased, reaching a maximum at D = 1, which indicates the phase transition of the system from an antiferromagnetic phase to a chiral one.
"We have studied the quantum phase transition of the Ising model with Dzyaloshinsky-Moriya (DM) interaction, by means of measures of entanglement as concurrence and von Neumann entropy. The study was performed using the Density Matrix Renormalization Group (DMRG). The parameter which drives the phase transition in this model is the D, which accounts for the strength of the DM interaction. The model has two phases, an antiferromagnetic phase and a chiral saturated phase, with critical point D = 1. By calculating the entanglement measures for both, open and periodic boundary conditions it was found that the critical point is signaled by an extreme behavior of the entanglement measure, maximum or minimum, it was also found these measures seem to tend to an asymptotic value when D → ∞ representing the chiral saturated phase."
We have studied the phase diagram and entanglement of the one-dimensional Ising model with Dzyaloshinskii-Moriya (DM) interaction. We have applied the quantum renormalization-group (QRG) approach to get the stable fixed points, critical point, and the scaling of coupling constants. This model has two phases: antiferromagnetic and saturated chiral ones. We have shown that the staggered magnetization is the order parameter of the system and DM interaction produces the chiral order in both phases. We have also implemented the exact diagonalization (Lanczos) method to calculate the static structure factors. The divergence of structure factor at the ordering momentum as the size of systems goes to infinity defines the critical point of the model. Moreover, we have analyzed the relevance of the entanglement in the model which allows us to shed insight on how the critical point is touched as the size of the system becomes large. Nonanalytic behavior of entanglement and finite-size scaling have been analyzed which is tightly connected to the critical properties of the model. It is also suggested that a spin-fluid phase has a chiral order in terms of spin operators which are defined by a nonlocal transformation.
We study the entanglement in the one-dimensional Kondo necklace model with exact diagonalization, calculating the concurrence as a function of the Kondo coupling J and an anisotropy η in the interaction between conduction spins, and we review some results previously obtained in the limiting cases η = 0 and 1. We observe that as J increases, localized and conduction spins get more entangled, while neighboring conduction spins diminish their concurrence; localized spins require a minimum concurrence between conduction spins to be entangled. The anisotropy η diminishes the entanglement for neighboring spins when it increases, driving the system to the Ising limit η = 1 where conduction spins are not entangled. We observe that the concurrence does not give information about the quantum phase transition in the anisotropic Kondo necklace model (between a Kondo singlet and an antiferromagnetic state), but calculating the von Neumann block entropy with the density matrix renormalization group in a chain of 100 sites for the Ising limit indicates that this quantity is useful for locating the quantum critical point.
We study the effects of the Dzyaloshinski-Moriya (DM) anisotropic interaction on the ground-state properties of the Heisenberg XY spin chain by means of the fidelity susceptibility, order parameter, and entanglement entropy. Our results show that the DM interaction could influence the distribution of the regions of quantum phase transitions and cause different critical regions in the XY spin model. Meanwhile, the DM interaction has effective influence on the degree of entanglement of the system and could be used to increase the entanglement of the spin system.
A systematic analysis is performed for quantum phase transitions in a bond-alternative one-dimensional Ising model with a Dzyaloshinskii–Moriya (DM) interaction by using the fidelity of ground state wavefunctions based on the infinite matrix product states algorithm. For an antiferromagnetic phase, the fidelity per lattice site exhibits a bifurcation, which shows spontaneous symmetry breaking in the system. A critical DM interaction is inversely proportional to an alternating exchange coupling strength for a quantum phase transition. Further, a finite-entanglement scaling of von Neumann entropy with respect to truncation dimensions gives a central charge c ≃ 0.5 at the critical point.
We study quantum phase transition of the [Formula: see text] spin model with Dzyaloshinsky–Moriya interaction, by using quantum correlation measures, i.e. quantum deficit and measurement-induced disturbance. It is shown that as the Dzyaloshinsky–Moriya coupling parameter [Formula: see text] increases, the behaviors of quantum phase transition can be suppressed. We also investigate quantum phase transition for the Ising and [Formula: see text] spin models at finite temperature. It is found that quantum phase transition characterized by measurement-induced disturbance is greater than or equal to that characterized by quantum deficit. Other interesting analytical results and numerical results on quantum phase transition for the proposed spin models are also presented by applying the two measures. Furthermore, we also compare quantum deficit and measurement-induced disturbance with quantum entanglement, quantum discord and quantum coherence.