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 from a paramagnetic to ferromagnetic phase caused by a gradual turning off of the transverse field at a fixed rate characterized by the quench time $\tau_Q$. In agreement with Kibble-Zurek mechanism, quantum state of the system after the transition exhibits a characteristic correlation length $\hat\xi$ proportional to the square root of the quench time $\tau_Q$. The inverse of this correlation length determines average density of defects after the transition. In this paper, we show that $\hat\xi$ also controls the entropy of entanglement of a block of $L$ spins with the rest of the system. For large $L$, this entropy saturates at $\frac16\log_2\hat\xi$, as might have been expected from the Kibble-Zurek mechanism. Close to the critical point, the entropy saturates when the block size $L\approx\hat\xi$, but -- in the subsequent evolution in the ferromagnetic phase -- a somewhat larger length scale $l=\sqrt{\tau_Q}\ln\tau_Q$ develops as a result of quantum dephasing, and the entropy saturates when $L\approx l$. We also study the spin-spin correlation. We find that close to the critical point ferromagnetic correlations decay exponentially with the dynamical correlation length $\hat\xi$, but in the following evolution this correlation function becomes oscillatory at distances less than this scale. However, both the wavelength and the correlation length of these oscillations are still determined by $\hat\xi$. We also derive probability distribution for the number of kinks in a finite spin chain after the transition.