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

Lukasz Cincio, Jacek Dziarmaga, Marek M. Rams, and Wojciech H. Zurek

Phys. Rev. A 75, 052321 – Published 17 May 2007

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 $τ_{Q}$ 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 length $\hat{ξ}$ proportional to the square root of the quench time $τ_{Q}$ : $\hat{ξ} = \sqrt{τ_{Q}}$ . 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 same $\hat{ξ}$ 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 $\frac{1}{6} \log_{2} \hat{ξ}$ , as might have been expected from the Kibble-Zurek mechanism. Close to the critical point, the entropy saturates when the block size $L \approx \hat{ξ}$ , but—in the subsequent evolution in the ferromagnetic phase—a somewhat larger length scale $l = \sqrt{τ_{Q}} \ln τ_{Q}$ develops as a result of a dephasing process that can be regarded as a quantum analog of phase ordering, and the entropy saturates when $L \approx l$ . 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 length $\hat{ξ}$ , 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 by $\hat{ξ}$ . We also derive probability distribution for the number of kinks in a finite spin chain after the transition.

Received 7 February 2007

DOI:https://doi.org/10.1103/PhysRevA.75.052321

Authors & Affiliations

Lukasz Cincio^1,2, Jacek Dziarmaga^1,2, Marek M. Rams^1,2, and Wojciech H. Zurek²

¹Institute of Physics and Centre for Complex Systems Research, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland
²Theory Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

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Vol. 75, Iss. 5 — May 2007

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Figure 1
(Color) (a) The entropy of a block of $L$ spins after the dynamical phase transition as a function of the block size $L$ . The multiple plots correspond to different values of the quench time $τ_{Q}$ . For all $τ_{Q}$ , the entropy grows with the block size $L$ and saturates at a finite value $S_{\infty} (τ_{Q})$ for large enough $L$ . (b) Fit of this asymptotic value of entropy with the function $S_{\infty} (τ_{Q}) = A + B \ln τ_{Q}$ . The best fit has $B = 0.128 \pm 0.004$ and $A = 1.80 \pm 0.05$ . This $B$ is in reasonably good agreement with the expected value of $B = \ln 2 ∕ 12 = 0.120$ . (c) and (d), we rescale values of entropy $S (L, τ_{Q})$ by the best fit to its asymptotic value $S_{\infty} (τ_{Q}) = A + B \ln τ_{Q}$ . With this rescaling we can focus on how the entropy depends on the block size $L$ . (c) Rescale of the block size by $\hat{ξ} = \sqrt{τ_{Q}}$ and the rescaled plots do not overlap exactly. However, as shown in (d) rescaling the block size $L$ by the second scale $l = \sqrt{τ_{Q}} \ln τ_{Q}$ makes the six plots overlap quite well.Reuse & Permissions
Figure 2
(Color) (a) the entropy of a block of $L$ spins during the dynamical phase transition at the critical point $g = 1$ as a function of the block size $L$ . The multiple plots correspond to different values of the quench time $τ_{Q}$ . For all the quench times, the entropy grows with the block size $L$ and saturates at a finite value $S_{\infty} (τ_{Q})$ for large enough $L$ . (b) Fit of this asymptotic value of entropy with the function $S_{\infty} (τ_{Q}) = A + B \ln τ_{Q}$ . The best fit has $B = 0.126 \pm 0.005$ and $A = 0.80 \pm 0.03$ . This $B$ is in reasonably good agreement with the expected value of $B = \ln 2 ∕ 12 = 0.120$ . (c) and (d), we rescale the entropy $S (L, τ_{Q})$ by the best fit to its asymptotic value $S_{\infty} (τ_{Q}) \approx A + B \ln τ_{Q}$ . With this rescaling we can focus on how the entropy depends on the block size $L$ . (d) Rescaling the block size $L$ by $\hat{ξ} = \sqrt{τ_{Q}}$ makes the six plots overlap quite well.Reuse & Permissions
Figure 3
(Color) (a) The dynamical transverse correlation $C_{1}^{x x}$ as a function of magnetic field $g$ in the linear quench. For each $τ_{Q}$ , we show both numerical (dashed) and analytical (solid) result. The plots overlap near the critical point at $g = 1$ but diverge in the ferromagnetic phase when $g < 1$ , indicating a breakdown of our numerical simulations in this regime. (b) The analytic and numerical results for the dynamical transverse correlation function at the moment when the quench crosses the critical point at $g = 1$ . The transverse correlators overlap well confirming that our numerical simulations are still accurate at the critical point. Finally, in (c), we show the dynamical ferromagnetic correlation function $C_{R}^{z z}$ at $g = 1$ and in (d), we show the same correlation function after rescaling $R ∕ \sqrt{τ_{Q}}$ . The rescaled plots overlap quite well supporting the idea that near the critical point the KZ correlation length $\hat{ξ} = \sqrt{τ_{Q}}$ is the only relevant scale of length.Reuse & Permissions

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