2007/01/31 by Lukasz Cincio, Łukasz Cincio, Jacek Dziarmaga +2 · 3 citations
Physics and Astronomy · #Physics of Superconductivity and Magnetism #Quantum and electron transport phenomena #Quantum many-body systems #cond-mat.stat-mech #cond-mat.str-el #hep-th #quant-ph
paper · pdf · doi:10.1103/physreva.75.052321
published as Phys.Rev.A75:052321.2007 · 12 pages, 3 figures; updated references and minor cosmetic changes
arxiv created 2007/03/13 · openalex publication_date 2007/05/17 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \ensuremathτQ 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 \stackrel\ifmmode \else \\fi\ensuremathξ proportional to the square root of the quench time \ensuremathτQ: \stackrel\ifmmode \else \\fi\ensuremathξ=√\ensuremathτQ. 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 \stackrel\ifmmode \else \\fi\ensuremathξ 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)\phantom\rule0.2em0exlog2\phantom\rule0.2em0ex\stackrel\ifmmode \else \\fi\ensuremathξ, as might have been expected from the Kibble-Zurek mechanism. Close to the critical point, the entropy saturates when the block size L\ensuremath≈\stackrel\ifmmode \else \\fi\ensuremathξ, but---in the subsequent evolution in the ferromagnetic phase---a somewhat larger length scale l=√\ensuremathτQ\phantom\rule0.2em0exln\phantom\rule0.2em0ex\ensuremathτQ 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\ensuremath≈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 \stackrel\ifmmode \else \\fi\ensuremathξ, 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 \stackrel\ifmmode \else \\fi\ensuremathξ. We also derive probability distribution for the number of kinks in a finite spin chain after the transition.