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Physical Review A

Low-energy-state dynamics of entanglement for spin systems

Rouhollah Jafari

Published Nov 17, 2010• 24 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.82.052317

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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 have composed the ideas of quantum renormalization group and quantum information by exploring the low energy states dynamic of entanglement resources of a system close to its quantum critical point. We demonstrate the low energy states dynamical quantities of the one dimensional magnetic systems could show the quantum phase transition point and shows the scaling behavior in the vicinity of the transition point. To present our idea, we study the evolution of two spins entanglement 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 evolvement of the generation of entanglement with increasing the magnetic field. We have obtained that the derivative of the time at which the entanglement reaches its maximums with respect to the transverse field, diverges at the critical point and its scaling behaviors versus the size of the system are as same as the static ground state entanglement of the the system.
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.
The dynamics of entanglement in the one-dimensional spin-1/2 anisotropic XXZ model is studied using the quantum renormalization-group method. We obtain the analytical expression of the concurrence, for two different quenching methods, it is found that initial state plays a key role in the evolution of system entanglement, i.e., the system returns completely to the initial state every other period. Our computations and analysis indicate that the first derivative of the characteristic time at which the concurrence reaches its maximum or minimum with respect to the anisotropic parameter occurs nonanalytic behaviors at the quantum critical point. Interestingly, the minimum value of the first derivative of the characteristic time versus the size of the system exhibits the scaling behavior which is the same as the scaling behavior of the system ground-state entanglement in equilibrium. In particular, the scaling behavior near the critical point is independent of the initial state.
Using a renormalization group approach, we solve the time evolution of random Ising spin chains with generic interactions starting from initial states of arbitrary energy. As a function of the Hamiltonian parameters, the system is tuned through a dynamical transition, similar to the ground-state critical point, at which the local spin correlations establish true long-range temporal order. In the state with a dominant transverse field, a spin that starts in an up state loses its orientation with time, while in the ``ordered'' state it never does. As in ground-state quantum phase transitions, the dynamical transition has unique signatures in the entanglement properties of the system. When the system is initialized in a product state, the entanglement entropy grows as log(t)\mathrm{log}(t)log(t) in the two ``phases,'' while at the critical point it grows as ${\mathrm{log}}^{\ensuremath{\alpha}}(t)$, with $\ensuremath{\alpha}$ a universal number. This universal entanglement growth requires generic (``integrability breaking'') interactions to be added to the pure transverse field Ising model.
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.
Adopting concurrence as entanglement measure, we study entanglement and quantum phase transition of the Ising spin systems on the triangular lattice and Sierpi\ifmmode \acute{n}\else \'{n}\fi{}ski fractal lattices by using the quantum renormalization-group method. It is found that the ground-state entanglement between two spins (or spin blocks) depends on the following factors: the size of system, the magnetic field, the exchange coupling, and the structure of lattice. As the size of the system becomes large, (a) the range of the magnetic field, in which the entanglement exists, contracts gradually and focuses on the critical point; and (b) the first derivative of entanglement shows singular behavior, and its maximum or minimum is approaching to the critical point gradually. The scaling behaviors of entanglement on the different lattices are similar but the scaling relations are diverse. For the triangular lattice, the space dimensionality determines the scaling relationship between the critical exponent of the entanglement and the critical exponent correlation length. However, for fractal lattices, it is the fractal dimensionality but not the space one to determine this relationship.