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arXiv:quant-ph/0505168

Singular behavior of an entangled state for a one-dimensional quantum spin system

Kawaguchi, Akira, Kaoru Shimizu

Published May 23, 2005• 0 citations
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Abstract

We studied the entangled state for a one-dimensional $S=1/2$ 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 $C(ρ)$ for two-qubit density matrix $ρ$ after the Lewenstein-Sanpera decomposition, $ρ=Λρ_s + (1-Λ) ρ_e $, where $ρ_s$ is a separable density matrix and $ρ_e$ is a pure entangled state obtained by a linear combination of Bell states, we find singular behaviors both in $C(ρ_e)$ and $1-Λ$ at the QPT point. $C(ρ_e)$ 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, $1-Λ$ reveals the singular minimum at QPT point.

DOI
10.48550/arxiv.quant-ph/0505168

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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=Λρs+(1−Λ)ρeρ=Λρ_s + (1-Λ) ρ_e ρ=Λρs​+(1−Λ)ρe​, 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 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.
For a system at absolute zero temperature, all thermal fluctuations are frozen out, while quantum fluctuations prevail.These microscopic quantum fluctuations can induce a quantum phase transition (QPT) in the ground state of a many-body system when the relative strength of two competing energy terms is varied across a critical value.This paper presents a detailed and quantitative study of the magnetic properties of Ising two-spin system at zero temperature in the presence of longitudinal, transverse and mixed fields, and also at finite temperatures.At the critical longitudinal magnetic field, the system is found to undergo a QPT to a ferromagnetic state.For this system, an infinitesimal departure from zero transverse field is found to trigger off QPT and also remove the degeneracies that persisted even in the presence of longitudinal field.These quantum fluctuations, leading to QPT are more evident in the mixed field than the separate effect of longitudinal and transverse fields.The observed quantum effects at zero temperature for this system are found to be completely wiped out at finite temperatures due to thermal fluctuations.A temperature increase favours antiferromagnetic alignment of spins, while a decrease in temperature favours ferromagnetic alignment.
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 study the ground state quantum phase transition by means of entanglement in the one-dimensional asymmetric Hubbard model with open boundary conditions. The local entanglement between the middle two sites and the rest of the system, and the block entanglement between the left and right portions of the system, are calculated using the density-matrix renormalization group (DMRG) method. We find that the entanglement shows interesting scaling and singular behavior around the phase transition line.
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Abstract We investigate the quantum coherence behavior of the ground states of 2D Heisenberg XY model and 2D Ising model with a transverse field on square lattices using the Quantum Renormalization Group (QRG) method. Our analysis focused on the ground state density matrix and its marginal states, revealing non-analytic behavior of quantum coherence (especially two-site coherence) near the critical point. This behavior allowed us to detect quantum phase transitions (QPT) in these models. By examining the scaling behavior of the maximum derivative of quantum coherence with system size, we determined the critical exponent of coherence for both models and the length exponent of the Ising model. Additionally, we investigated the time evolution of coherence in both models. Our results closely align with those obtained from entanglement analysis, that is while quantum coherence requires fewer computational calculations compared to discord and entanglement approaches.