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arXiv:quant-ph/0606133 · Physical Review A

Local entanglement and quantum phase transition in a one-dimensional transverse field Ising model

Shi-Quan Su, Jun-Liang Song, Shi-Jian Gu

Published Sep 12, 2006• 38 citations
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Abstract

In this paper, we study the entanglement between two neighboring sites and the rest of the system in a simple quantum phase transition of a one-dimensional transverse field Ising model. We find that the entanglement shows interesting scaling and singular behaviors around the critical point, and thus can be used as a convenient marker for the transition point.

DOI
10.1103/physreva.74.032308

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In this paper, we study the entanglement between two neighboring sites and the rest of the system in a simple quantum phase transition of a one-dimensional transverse field Ising model. We find that the entanglement shows interesting scaling and singular behaviors around the critical point, and thus can be used as a convenient marker for the transition point.
What entanglement is present in naturally occurring physical systems at thermal equilibrium? Most such systems are intractable and it is desirable to study simple but realistic systems that can be solved. An example of such a system is the one-dimensional infinite-lattice anisotropic XY\mathrm{XY}XY model. This model is exactly solvable using the Jordan-Wigner transform, and it is possible to calculate the two-site reduced density matrix for all pairs of sites. Using the two-site density matrix, the entanglement of formation between any two sites is calculated for all parameter values and temperatures. We also study the entanglement in the transverse Ising model, a special case of the XY\mathrm{XY}XY model, which exhibits a quantum phase transition. It is found that the next-nearest-neighbor entanglement (though not the nearest-neighbor entanglement) is a maximum at the critical point. Furthermore, we show that the critical point in the transverse Ising model corresponds to a transition in the behavior of the entanglement between a single site and the remainder of the lattice.
In this paper we consider the quantum phase transition in the Ising model in the presence of a transverse field in one, two, and three dimensions from a multipartite entanglement point of view. Using exact numerical solutions, we are able to study such systems up to 25 qubits. The Meyer-Wallach measure of global entanglement is used to study the critical behavior of this model. The transition we consider is between a symmetric Greenberger-Horne-Zeilinger--like state to a paramagnetic product state. We find that global entanglement serves as a good indicator of quantum phase transition with interesting scaling behavior. We use finite-size scaling to extract the critical point as well as some critical exponents for the one- and two-dimensional models. Our results indicate that such multipartite measure of global entanglement shows universal features regardless of dimension ddd. Our results also provide evidence that multipartite entanglement is better suited for the study of quantum phase transitions than the much-studied bipartite measures.
We study the evolution of nearest-neighbor entanglement in the one-dimensional Ising model with an external transverse field. The system is initialized as the so-called thermal ground state of the pure Ising model. We analyze properties of the generation of entanglement for different regions of external transverse fields. We find that the derivative of the time at which the entanglement reaches its first maximum with respect to the reciprocal transverse field has a minimum at the critical point. This is an indicator of quantum phase transition.
We studied the entangled state for a one-dimensional S=1/2S=1/2S=1/2 antiferromagnetic quantum spin chain in a transverse field. We calculate the ground state using the density matrix renormalization group and discuss how the entangled state changes around a quantum phase transition (QPT) point. By analyzing concurrence C(ρ)C(ρ)C(ρ) for two-qubit density matrix ρρρ after the Lewenstein-Sanpera decomposition, $ρ=Λρ_s + (1-Λ) ρ_e $, where ρsρ_sρs​ is a separable density matrix and ρeρ_eρe​ is a pure entangled state obtained by a linear combination of Bell states, we find singular behaviors both in C(ρe)C(ρ_e)C(ρe​) and 1−Λ1-Λ1−Λ at the QPT point. C(ρe)C(ρ_e)C(ρe​) includes the effects of quantum fluctuations, which manifest the competition between the antiferromagnetic spin fluctuation and the effect of transverse field in the transverse Ising model. The quantum fluctuation shows the singular maximum at the QPT point as expected from the general picture of phase transition. In contrast, 1−Λ1-Λ1−Λ reveals the singular minimum at QPT point.
We develop the ideas of the quantum renormalization group and quantum information by exploring the low-energy-state dynamics of entanglement resources of a system close to its quantum critical point. We demonstrate that low-energy-state dynamical quantities of one-dimensional magnetic systems can show a quantum phase transition point and show scaling behavior in the vicinity of the transition point. To present our idea, we study the evolution of two spin entanglements in the one-dimensional Ising model in the transverse field. The system is initialized as the so-called thermal ground state of the pure Ising model. We investigate the evolution of the generation of entanglement with increasing magnetic field. We obtain that the derivative of the time at which the entanglement reaches its maximum with respect to the transverse field diverges at the critical point and its scaling behaviors versus the size of the system are the same as the static ground-state entanglement of the system.
We analyze the fidelity per lattice site for two different ground states of the one-dimensional quantum Ising model in a transverse field near the critical point. It is found that, in the thermodynamic limit, the fidelity per lattice site is singular, and the derivative of its logarithmic function with respect to the transverse field strength is logarithmically divergent at the critical point. The scaling behavior is confirmed numerically by performing a finite-size scaling analysis for systems of different sizes, consistent with the conformal invariance at the critical point. This allows us to extract the correlation length critical exponent, which turns out to be universal in the sense that the correlation length critical exponent does not depend on either the anisotropic parameter or the transverse field strength.