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arXiv:0909.4764 · Physical Review A

Exact calculation of entanglement in a 19 -site two-dimensional spin system

Qing Yang Xu, Sabre Kais, Maxim Naumov, Ahmed Sameh

Published Feb 23, 2010• 23 citations
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Abstract

Using the trace minimization algorithm, we carried out an exact calculation of entanglement in a 19-site two-dimensional transverse Ising model. This model consists of a set of localized spin-$\frac{1}{2}$ particles in a two-dimensional triangular lattice coupled through exchange interaction $J$ and subject to an external magnetic field of strength $h$. We demonstrate, for such a class of two-dimensional magnetic systems, that entanglement can be controlled and tuned by varying the parameter $\ensuremath{\lambda}=h/J$ in the Hamiltonian and by introducing impurities into the systems. Examining the derivative of the concurrence as a function of $\ensuremath{\lambda}$ shows that the system exhibits a quantum phase transition at about ${\ensuremath{\lambda}}_{c}=3.01$, a transition induced by quantum fluctuations at the absolute zero of temperature.

DOI
10.1103/physreva.81.022324

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Using the trace minimization algorithm, we carried out an exact calculation of entanglement in a 19-site two-dimensional transverse Ising model. This model consists of a set of localized spin-12\frac{1}{2}21​ particles in a two-dimensional triangular lattice coupled through exchange interaction JJJ and subject to an external magnetic field of strength hhh. We demonstrate, for such a class of two-dimensional magnetic systems, that entanglement can be controlled and tuned by varying the parameter \ensuremathλ=h/J\ensuremath{\lambda}=h/J\ensuremathλ=h/J in the Hamiltonian and by introducing impurities into the systems. Examining the derivative of the concurrence as a function of \ensuremathλ\ensuremath{\lambda}\ensuremathλ shows that the system exhibits a quantum phase transition at about \ensuremathλc=3.01{\ensuremath{\lambda}}_{c}=3.01\ensuremathλc​=3.01, a transition induced by quantum fluctuations at the absolute zero of temperature.
We study a set of localized spins coupled through exchange interaction and subject to an external magnetic field. We demonstrate, for such a class of one-dimensional magnetic systems, that entanglement can be controlled and tuned by varying the anisotropy parameter in the Hamiltonian and by introducing impurities into the systems. In particular, for certain parameters, the entanglement is zero up to a critical point ${\ensuremath{\lambda}}_{c},$ where a quantum phase transition occurs, and is different from zero above ${\ensuremath{\lambda}}_{c}.$
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.
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.
A 2D Ising model with transverse field on a triangular lattice is studied using exact diagonalization. The quantum entanglement of the system is quantified by the entanglement of formation. The ground state property of the system is studied and the quantified entanglement is shown to be closely related to the ground state wavefunction while the singularity in the entanglement as a function of the transverse field is a reasonable indicator of the quantum phase transition. In order to tune the entanglement, one can either include an impurity in the otherwise homogeneous system whose strength is tunable, or one can vary the external transverse field as a tuner. The latter kind of tuning involves complicated dynamical properties of the system. From the study of the dynamics on a comparatively smaller system, we provide ways to tune the entanglement without triggering any decoherence. The finite temperature effect is also discussed. Besides showing above physical results, the realization of the trace-minimization method in our system is provided; the scalability of such method to larger systems is argued.
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.
We study the dynamics of entanglement for a one-dimensional spin system, where spins are coupled through nearest-neighbor exchange interaction and subject to different external magnetic fields. First we examine the system size effect on the entanglement with three different external magnetic fields changing with time ttt: an exponential function ${e}^{[\ensuremath{-}Kt]}$ and two periodic sin[Kt]\mathrm{sin}[Kt]sin[Kt] and cos[Kt]\mathrm{cos}[Kt]cos[Kt] functions, where KKK is a control parameter. We have found that the entanglement fluctuates shortly after a disturbance by an external magnetic field when the system size is small. For larger system size, the entanglement reaches a stable state for a long time before it fluctuates. However, this fluctuation of entanglement disappears when the system size goes to infinity. We also show that in a periodic external magnetic field, the nearest-neighbor entanglement displays a periodic structure with a period related to that of the magnetic field. Moreover, changing the direction of the magnetic field will destroy the concurrence in the system.