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Journal of King Saud University - Science

Impurity effect on entanglement in an XY two-dimensional spin lattice

Gehad Sadiek

Published Aug 23, 2011• 2 citations
Publisher

Abstract

We consider a finite two dimensional XY spin model. The model consists of a set of seven localized spin- 12 particles in a two dimensional triangular lattice coupled through nearest neighbor exchange interaction in presence of an external magnetic field. We study the effect of a single impurity spin coupled to its nearest neighbor through an exchange interaction J′ on the pairwise entanglement between the different spins in the lattice. We found that when the spin is located at a border site the entanglement between the impurity and its nearest neighbors increases monotonically with J′ reaching a saturation value that depends on the magnetic field strength h. On the other hand the entanglement with the next nearest spins shows a critical behavior where the entanglement increases and then decreases rapidly until it vanishes. The critical coupling value at which the entanglement vanishes increases as h increases. Studying the entanglement between two spins excluding the impurity show that while the entanglement between two spins coupled to the impurity reach a saturation value as J′ increases, it vanishes for spins not coupled to the impurity. Furthermore studying the effect of a central impurity show that the entanglement between nearest neighbor and next nearest neighbor spins reaches an asymptotic value as J′ increases. Interestingly it was demonstrated that the impurity can be used as a switch to control the entanglement between different spins in the lattice turning it on and off.

DOI
10.1016/j.jksus.2011.08.002

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We consider the entanglement in a two-dimensional XYXYXY model in an external magnetic field hhh. The model consists of a set of seven localized spin-12\frac{1}{2}21​ particles in a two-dimensional triangular lattice coupled through nearest-neighbor exchange interaction JJJ. We examine the effect of single and double impurities in the system as well as the degree of anisotropy on the nearest-neighbor entanglement and ergodicity of the system. We have found that the entanglement of the system at the different degrees of anisotropy mimics that of the one-dimensional spin systems at the extremely small and large values of the parameter $\ensuremath{\lambda}=h/J$. The entanglement of the Ising and partially anisotropic systems shows phase transition in the vicinity of $\ensuremath{\lambda}=2$, whereas, the entanglement of the isotropic system suddenly vanishes there. Also, we investigate the dynamic response of the system containing single and double impurities to an external exponential magnetic field at different degrees of anisotropy. We have demonstrated that the ergodicity of the system can be controlled by varying the strength and location of the impurities as well as the degree of anisotropy of the coupling.
The entanglement in one-dimensional random XY spin systems where the impurities of exchange couplings and the external magnetic fields are considered as random variables is investigated by solving the different spin–spin correlation functions and the average magnetization per spin. The entanglement dynamics near particular locations of the system is also studied when the exchange couplings (or the external magnetic fields) satisfy three different distributions (the Gaussian distribution, double-Gaussian distribution, and bimodal distribution). We find that the entanglement can be controlled by varying the strength of external magnetic field and the distributions of impurities. Moreover, the entanglement of some nearest-neighbouring qubits can be increased for certain parameter values of the three different distributions.
We consider a finite two-dimensional Heisenberg triangular spin lattice coupled to a dissipative Markovian environment at finite temperature in the presence of an external uniform magnetic field. We show how inserting a magnetic impurity in the spin system can be used to effectively control the dynamics and asymptotic state of the system. A strong impurity, at a border or central site, was found to enhance its entanglement with the other spins and their thermal robustness to the dissipative environment reaching an asymptotic state that is independent of the system initial state. However, it reduces the entanglement among the other spins in the lattice and their thermal robustness and may diminish them depending on its strength and the environment temperature. Moreover, the effect of the impurity increases significantly as the degree of anisotropy of the spin system increases. Therefore, the entanglement distribution over the different sites of the lattice can be modulated by tuning the impurity strength, system anisotropy and environment temperature. The impurity can be used as a switch that turns the entanglement on among specific spins and off among others simultaneously.
We consider the time evolution of entanglement in a finite two-dimensional transverse Ising model. The model consists of a set of seven localized spin-12\frac{1}{2}21​ particles in a two-dimensional triangular lattice coupled through nearest-neighbor exchange interaction in the presence of an external time-dependent magnetic field. The magnetic field is applied in different function forms: step, exponential, hyperbolic, and periodic. We found that the time evolution of the entanglement shows an ergodic behavior under the effect of the time-dependent magnetic fields. Also, we found that while the step magnetic field causes great disturbance to the system, creating rapid oscillations, the system shows great controllability under the effects of the other magnetic fields where the entanglement profile follows closely the shape of the applied field even with the same frequency for periodic fields. This follow-up trend breaks down as the strength of the field, the transition constant for the exponential and hyperbolic forms, or the frequency for periodic field increase leading to rapid oscillations. We observed that the entanglement is very sensitive to the initial value of the applied periodic field: the smaller the initial value is, the less distorted the entanglement profile is. Furthermore, the effect of thermal fluctuations is very devastating to the entanglement, which decays very rapidly as the temperature increases. Interestingly, although a large value of the magnetic field strength may yield a small entanglement, the magnetic field strength was found to be more persistent against thermal fluctuations than the small field strengths.
We study the dynamics of entanglement for the XY-model, one-dimensional spin systems coupled through the nearest neighbor exchange interaction and subject to an external time-dependent magnetic field. Using the two-site density matrix, we calculate the time-dependent entanglement of formation between nearest neighbor qubits. We investigate the effect of varying the temperature, the anisotropy parameter and the external time-dependent magnetic field on the entanglement. We have found that the entanglement can be localized between nearest neighbor qubits for certain values of the external time-dependent magnetic field. Moreover, as known for the magnetization of this model, the entanglement shows nonergodic behavior, it does not approach its equilibrium value at the infinite time limit.
We study the ground-state entanglement properties of an XX spin 1/2 chain in transverse field, in its quasi-long-ranged ordered phase, with a magnetic impurity, represented in terms of an additional transverse magnetic field located at one precise site. For such a system, we show that a control of the ground state entanglement can be achieved by acting on the impurity field. To demonstrate this possibility, we evaluate exactly the nearest neighbor and next-nearest neighbor concurrence in the presence of the impurity. It turns out that either an enhancement or a quenching of entanglement between selected spin pairs can be obtained by acting on the intensity of the impurity. For specific values of the magnetic field a spatial modulation of concurrence along the chain is also obtained.