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arXiv:cond-mat/0701768 · Physical Review A

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

Łukasz Cincio, Jacek Dziarmaga, Marek M. Rams, Wojciech Hubert Zurek

Published May 17, 2007• 182 citations
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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 ${\ensuremath{\tau}}_{Q}$ 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 $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$ proportional to the square root of the quench time ${\ensuremath{\tau}}_{Q}$: $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}=\sqrt{{\ensuremath{\tau}}_{Q}}$. 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 $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$ 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 $\frac{1}{6}\phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{2}\phantom{\rule{0.2em}{0ex}}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$, as might have been expected from the Kibble-Zurek mechanism. Close to the critical point, the entropy saturates when the block size $L\ensuremath{\approx}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$, but---in the subsequent evolution in the ferromagnetic phase---a somewhat larger length scale $l=\sqrt{{\ensuremath{\tau}}_{Q}}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}{\ensuremath{\tau}}_{Q}$ develops as a result of a dephasing process that can be regarded as a quantum analog of phase ordering, and the entropy saturates when $L\ensuremath{\approx}l$. 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 $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$, 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 $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$. We also derive probability distribution for the number of kinks in a finite spin chain after the transition.

DOI
10.1103/physreva.75.052321

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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{\ensuremath{\tau}}_{Q}\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 \ensuremathξ\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}\ensuremathξ\ifmmode^\elseˆ\fi​ proportional to the square root of the quench time \ensuremathτQ{\ensuremath{\tau}}_{Q}\ensuremathτQ​: \ensuremathξ\ifmmode^\elseˆ\fi=\ensuremathτQ\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}=\sqrt{{\ensuremath{\tau}}_{Q}}\ensuremathξ\ifmmode^\elseˆ\fi​=\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 \ensuremathξ\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}\ensuremathξ\ifmmode^\elseˆ\fi​ controls the entropy of entanglement, e.g., entropy of a block of LLL spins that are entangled with the rest of the system after the transition from the paramagnetic ground state induced by the quench. For large LLL, this entropy saturates at 16log2\ensuremathξ\ifmmode^\elseˆ\fi\frac{1}{6}\phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{2}\phantom{\rule{0.2em}{0ex}}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}61​log2​\ensuremathξ\ifmmode^\elseˆ\fi​, as might have been expected from the Kibble-Zurek mechanism. Close to the critical point, the entropy saturates when the block size L\ensuremath≈\ensuremathξ\ifmmode^\elseˆ\fiL\ensuremath{\approx}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}L\ensuremath≈\ensuremathξ\ifmmode^\elseˆ\fi​, but---in the subsequent evolution in the ferromagnetic phase---a somewhat larger length scale l=\ensuremathτQln\ensuremathτQl=\sqrt{{\ensuremath{\tau}}_{Q}}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}{\ensuremath{\tau}}_{Q}l=\ensuremathτQ​​ln\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≈lL\ensuremath{\approx}lL\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 \ensuremathξ\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}\ensuremathξ\ifmmode^\elseˆ\fi​, but (as for entropy of entanglement) in the following evolution length scale lll 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 \ensuremathξ\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}\ensuremathξ\ifmmode^\elseˆ\fi​. We also derive probability distribution for the number of kinks in a finite spin chain after the transition.
We consider a linear quench from the paramagnetic to ferromagnetic phase in the quantum Ising chain interacting with a static spin environment. Both decoherence from the environment and nonadiabaticity of the evolution near a critical point excite the system from the final ferromagnetic ground state. For weak decoherence and relatively fast quenches the excitation energy, proportional to the number of kinks in the final state, decays like an inverse square root of a quench time, but slow transitions or strong decoherence makes it decay in a much slower logarithmic way. We also find that fidelity between the final ferromagnetic ground state and a final state after a quench decays exponentially with a size of a chain, with a decay rate proportional to average density of excited kinks and a proportionality factor evolving from 1.3 for weak decoherence and fast quenches to approximately 1 for slow transitions or strong decoherence. Simultaneously, correlations between kinks randomly distributed along the chain evolve from a near-crystalline antibunching to a Poissonian distribution of kinks in a number of isolated Anderson localization centers randomly scattered along the chain.
The transverse field in the quantum Ising chain is linearly ramped from the para- to the ferromagnetic phase across the quantum critical point at a rate characterized by a quench time ${\ensuremath{\tau}}_{Q}$. We calculate a connected kink-kink correlator in the final state at zero transverse field. The correlator is a sum of two terms: a negative (antibunching) Gaussian that depends on the Kibble-Zurek correlation length only and a positive term that depends on a second longer scale of length. The second length is made longer by dephasing of the state excited near the critical point during the following ramp across the ferromagnetic phase. This interpretation is corroborated by considering a linear ramp that is halted in the ferromagnetic phase for a finite waiting time and then continued at the same rate as before the halt. The extra time available for dephasing increases the second scale of length that asymptotically grows linearly with the waiting time. The dephasing also suppresses magnitude of the second term, making it negligible for waiting times much longer than ${\ensuremath{\tau}}_{Q}$. The same dephasing can be obtained with a smooth ramp that slows down in the ferromagnetic phase. Assuming sufficient dephasing, we also obtain higher order kink correlators and the ferromagnetic correlation function.
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 LLL spins with the rest of the system scales logarithmically with LLL 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 LLL but the prefactor depends on the quench time. This puts a constraint on the block size LLL 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.
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 the critical region, the entanglement entropy of a block of LLL spins with the rest of the system is also estimated which is found to increase logarithmically with LLL. The dependence on the quench time puts a constraint on the block size LLL. It is also pointed out that the Lipkin-Meshkov-Glick model in point-splitting regularized form appears as a combination of the XXX\mathit{XXX}XXX model and Ising model with magnetic field in the negative zzz 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.
When a quantum system exhibiting a second order phase transition is quenched across the critical point in large but finite time, the dynamics are not adiabatic in the critical region and the Kibble-Zurek (KZ) mechanism provides a framework to determine local observables such as the mean defect density. However, to find higher-point functions, one has to go beyond the KZ paradigm asshown in recent works on one-dimensional transverse field Ising model (TFIM) following a linear quench. It has been found that (i) besides the KZ scale, the quench dynamics depend on another length scale that arises due to the finite phase difference between the low energy modes, and (ii) contrary to the expectations based on the KZ mechanism, in general, the correlation functions do not decay exponentially with distance. Motivated by these results for the linear quench, we are interested in understanding if these properties are universal, and consider the 1D TFIM when the transverse field varies algebraically in the vicinity of the critical field. We focus on the equal-time,longitudinal kink-kink correlation function at the end of the quench from the paramagnetic to the ferromagnetic phase, and find that (i) the correlator depends only on the KZ length for superlinear quenches, otherwise an additional dephasing length is required to describe it, and (ii) the dephased correlator decays as a compressed exponential with an exponent that changes continuously with the quench exponent. Our results are obtained using an adiabatic perturbation theory, analytical arguments and exact numerical integration of the relevant equations.
We have analysed here the role of geometric phase in the dynamical mechanism of quantum phase transition in the transverse Ising model. We have investigated the system when it is driven at a fixed rate characterized by a quench time τ q across the critical point from a paramagnetic to ferromagnetic phase. Our argument is based on the fact that the spin fluctuation occurring during the critical slowing down causes a random fluctuation in the ground-state geometric phase at the critical regime. The correlation function of the random geometric phase determines the excitation probability of the quasiparticles, which are excited during the transition from the inital paramagnetic to the ferromagnetic phase. This helps us to evaluate the number density of the kinks formed during the transition, which is found to scale as . In addition, we have also estimated the spin–spin correlation at criticality.