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Title: 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 {tau}{sub 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 {xi} proportional to the square root of the quench time {tau}{sub Q}: {xi}={radical}({tau}{sub 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 {xi} 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{sub 2} {xi}, as might have been expected from the Kibble-Zurek mechanism. Close to the critical point, the entropy saturatesmore » when the block size L{approx_equal}{xi}, but -- in the subsequent evolution in the ferromagnetic phase -- a somewhat larger length scale l={radical}({tau}{sub Q}) ln {tau}{sub 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_equal}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 {xi}, 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 {xi}. We also derive probability distribution for the number of kinks in a finite spin chain after the transition.« less

Authors:
; ;  [1];  [2];  [3]
  1. Institute of Physics and Centre for Complex Systems Research, Jagiellonian University, Reymonta 4, 30-059 Krakow, (Poland)
  2. (United States)
  3. Theory Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (United States)
Publication Date:
OSTI Identifier:
20982490
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physical Review. A; Journal Volume: 75; Journal Issue: 5; Other Information: DOI: 10.1103/PhysRevA.75.052321; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGORITHMS; CORRELATION FUNCTIONS; CORRELATIONS; DECAY; DEFECTS; DISTRIBUTION; ENTROPY; EXACT SOLUTIONS; GROUND STATES; ISING MODEL; OSCILLATIONS; PARAMAGNETISM; PHASE TRANSFORMATIONS; PROBABILITY; PULSES; QUANTUM ENTANGLEMENT; SIMULATION; SPIN; WAVELENGTHS

Citation Formats

Cincio, Lukasz, Dziarmaga, Jacek, Rams, Marek M., Theory Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, and Zurek, Wojciech H. Entropy of entanglement and correlations induced by a quench: Dynamics of a quantum phase transition in the quantum Ising model. United States: N. p., 2007. Web. doi:10.1103/PHYSREVA.75.052321.
Cincio, Lukasz, Dziarmaga, Jacek, Rams, Marek M., Theory Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, & Zurek, Wojciech H. Entropy of entanglement and correlations induced by a quench: Dynamics of a quantum phase transition in the quantum Ising model. United States. doi:10.1103/PHYSREVA.75.052321.
Cincio, Lukasz, Dziarmaga, Jacek, Rams, Marek M., Theory Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, and Zurek, Wojciech H. Tue . "Entropy of entanglement and correlations induced by a quench: Dynamics of a quantum phase transition in the quantum Ising model". United States. doi:10.1103/PHYSREVA.75.052321.
@article{osti_20982490,
title = {Entropy of entanglement and correlations induced by a quench: Dynamics of a quantum phase transition in the quantum Ising model},
author = {Cincio, Lukasz and Dziarmaga, Jacek and Rams, Marek M. and Theory Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 and Zurek, Wojciech H.},
abstractNote = {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 {tau}{sub 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 {xi} proportional to the square root of the quench time {tau}{sub Q}: {xi}={radical}({tau}{sub 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 {xi} 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{sub 2} {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_equal}{xi}, but -- in the subsequent evolution in the ferromagnetic phase -- a somewhat larger length scale l={radical}({tau}{sub Q}) ln {tau}{sub 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_equal}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 {xi}, 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 {xi}. We also derive probability distribution for the number of kinks in a finite spin chain after the transition.},
doi = {10.1103/PHYSREVA.75.052321},
journal = {Physical Review. A},
number = 5,
volume = 75,
place = {United States},
year = {Tue May 15 00:00:00 EDT 2007},
month = {Tue May 15 00:00:00 EDT 2007}
}
  • It is known that at the critical point of a zero-temperature quantum phase transition in a one-dimensional spin system the entanglement entropy of a block of L spins with the rest of the system scales logarithmically with L with a prefactor determined by the central charge of the relevant conformal field theory. When we introduce critical slowing down incorporating the Kibble-Zurek mechanism of defect formation induced by a quench, the implicit nonadiabatic transition disturbs the scaling behavior. We have shown that in this case the entanglement entropy also obeys a scaling law such that it increases logarithmically with L butmore » the prefactor depends on the quench time. This puts a constraint on the block size L so that we cannot arbitrarily choose it. Thus, the entanglement entropy obeys the scaling law only in a restrictive sense due to the formation of defects.« less
  • Cited by 1
  • The ground state of the one-dimensional Bose-Hubbard model at unit filling undergoes the Mott-superfluid quantum phase transition. It belongs to the Kosterlitz-Thouless universality class with an exponential divergence of the correlation length in place of the usual power law. Here, we present numerical simulations of a linear quench both from the Mott insulator to superfluid and back. The results satisfy the scaling hypothesis that follows from the Kibble-Zurek mechanism (KZM). In the superfluid-to-Mott quenches there is no significant excitation in the superfluid phase despite its gaplessness. And since all critical superfluid ground states are qualitatively similar, the excitation begins tomore » build up only after crossing the critical point when the ground state begins to change fundamentally. The last process falls into the KZM framework.« less
  • We have studied quantum phase transition induced by a quench in different one-dimensional spin systems. Our analysis is based on the dynamical mechanism which envisages nonadiabaticity in the vicinity of the critical point. This causes spin fluctuation which leads to the random fluctuation of the Berry phase factor acquired by a spin state when the ground state of the system evolves in a closed path. The two-point correlation of this phase factor is associated with the probability of the formation of defects. In this framework, we have estimated the density of defects produced in several one-dimensional spin chains. At themore » critical region, the entanglement entropy of a block of L spins with the rest of the system is also estimated which is found to increase logarithmically with L. The dependence on the quench time puts a constraint on the block size L. It is also pointed out that the Lipkin-Meshkov-Glick model in point-splitting regularized form appears as a combination of the XXX model and Ising model with magnetic field in the negative z axis. This unveils the underlying conformal symmetry at criticality which is lost in the sharp point limit. Our analysis shows that the density of defects as well as the scaling behavior of the entanglement entropy follows a universal behavior in all these systems.« less