Entropy of entanglement and correlations induced by a quench: Dynamics of a quantum phase transition in the quantum Ising model

Abstract

Quantum Ising model in one dimension is an exactly solvable example of a quantum phase transition. We investigate its behavior during a quench caused by a gradual turning off of the transverse bias field. The system is then driven at a fixed rate characterized by the quench time τQ across the critical point from a paramagnetic to ferromagnetic phase. In agreement with Kibble-Zurek mechanism (which recognizes that evolution is approximately adiabatic far away, but becomes approximately impulse sufficiently near the critical point), quantum state of the system after the transition exhibits a characteristic correlation lengthξ proportional to the square root of the quench time τQ: ξ = √ τQ. The inverse of this correlation length is known to determine average density of defects (e.g. kinks) after the transition. In this paper, we show that this sameξ controls the entropy of entanglement, e.g. entropy of a block of L spins that are entangled with the rest of the system after the transition from the paramagnetic ground state induced by the quench. For large L, this entropy saturates at 1 6 log 2ξ , as might have been expected from the Kibble-Zurek mechanism. Close to the critical point, the entropy saturates when the block size L ≈ξ, but -in the subsequent evolution in the ferromagnetic phase -a somewhat larger length scale l = √ τQ ln τQ develops as a result of a dephasing process that can be regarded as a quantum analogue of phase ordering, and the entropy saturates when L ≈ l. We also study the spin-spin correlation using both analytic methods and real time simulations with the Vidal algorithm. We find that at an instant when quench is crossing the critical point, ferromagnetic correlations decay exponentially with the dynamical correlation lengtĥ ξ, but (as for entropy of entanglement) in the following evolution length scale l gradually develops. The correlation function becomes oscillatory at distances less than this scale. However, both the wavelength and the correlation length of these oscillations are still determined byξ. We also derive probability distribution for the number of kinks in a finite spin chain after the transition.

Dynamics of entanglement entropy and entanglement spectrum crossing a quantum phase transition

“…However, at least in the static case, the behaviour of entanglement (and more specifically of entanglement entropy) has an universal character so that it can be used as an estimator of quantum correlations 10 and to detect as well as to classify quantum phase transitions also in fully interacting models [11][12][13][14][15][16] . Thus, it is natural to ask whether the dynamical behaviour of a closed quantum system, especially when crossing a phase transition, can be described by looking at the dynamics of entanglement entropy and entanglement spectrum, a topic on which there are only a few general results [17][18][19][20] .…”

Entanglement in many-body systems

“…We finally mention that the entropy in the case of a finite time quench has been considered in Cherng andLevitov, 2006 andby Cincio et al , 2007. Very recently rigorous bounds for the time evolution of the block entropy were obtained by Osborne and Eister (Eisert and Osborne, 2006) and by Verstraete et al (Bravyi et al, 2006).…”

“…Dynamics of two-site entanglement was discussed also in the context of spin-boson like models. In (Ciancio and Zanardi, 2006) the negativity for a two-modes Jaynes-Cummings was analyzed with particular emphasis on the entanglement between the two bosonic modes mediated by the qubit. The relaxation dynamics of the entanglement, quantified trough the concurrence, in the spin-boson model was discussed in (Lim et al, 2006) We finally mention a study where it was noticed that the entanglement encoded in the states caused by the splitting of the degeneracy determined by the transverse field in the quantum XY model is not preserved by an adiabatic perturbation.…”

Holographic entanglement entropy of a 1 + 1 dimensional p-wave superconductor

“…In particular we would like to understand possible universal scaling of the entropy as a function of the quench rate. While some results about such scaling behavior are known in solvable and integrable field theories [28][29][30][31][32] , very little is known in strongly coupled systems. In the past, holographic methods have been useful to understand scaling of one point functions in various regimes…”