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Lecture notes in physics

Quench Dynamics of Quantum and Classical Ising Chains: From the Viewpoint of the Kibble–Zurek Mechanism

Sei Suzuki

Published Jan 1, 2010• 1 citation
Publisher
DOI
10.1007/978-3-642-11470-0_5

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According to the Kibble-Zurek mechanism, there is a universal power-law relationship between the defect density and the quench rate during a slow linear quench through a critical point. It is generally accepted that a fast quench results in a deviation from the Kibble-Zurek scaling law and leads to the formation of a saturated plateau in the defect density. By adjusting the quench rate from slow to very fast limits, we observe the varying quench dynamics and identify a presaturated regime that lies between the saturated and Kibble-Zurek regimes. This significant result is first elucidated through the adiabatic-impulse approximation, then verified by a rigorous analysis on the transverse Ising chain as well. As we approach the turning point from the saturated to presaturated regimes, we notice a change in scaling laws and, with an increase in the initial transverse field, a shrinking of the saturated regime until it disappears. During another turning point from the Kibble-Zurek to presaturated regimes, we observe an attenuation of the dephasing effect and a change in the behavior of the kink-kink correlation function from a Gaussian decay to an exponential decay. Finally, the coherent many-body oscillation after quench exhibits different behaviors in the three regimes and shows a significant change of scaling behavior between the S and PS regimes.

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Dynamics of Quantum Ising Systems

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Lecture notes in physics•Dec 12, 2012•0 citations
According to the Kibble-Zurek mechanism, there is a universal power-law relationship between the defect density and the quench rate during a slow linear quench through a critical point. It is generally accepted that a fast quench results in a deviation from the Kibble-Zurek scaling law and leads to the formation of a saturated plateau in the defect density. By adjusting the quench rate from slow to very fast limits, we observe the varying quench dynamics and identify a presaturated regime that lies between the saturated and Kibble-Zurek regimes. This significant result is first elucidated through the adiabatic-impulse approximation, then verified by a rigorous analysis on the transverse Ising chain as well. As we approach the turning point from the saturated to presaturated regimes, we notice a change in scaling laws and, with an increase in the initial transverse field, a shrinking of the saturated regime until it disappears. During another turning point from the Kibble-Zurek to presaturated regimes, we observe an attenuation of the dephasing effect and a change in the behavior of the kink-kink correlation function from a Gaussian decay to an exponential decay. Finally, the coherent many-body oscillation after quench exhibits different behaviors in the three regimes and shows a significant change of scaling behavior between the S and PS regimes.
Dynamics of quenching temperature is studied in pure and random Ising chains. Using the Kibble-Zurek argument, we obtain for the pure Ising model that the density of kinks after quenching decays as 1/\sqrtτ with the quench rate of temperature 1/τfor large τ. For the random Ising model, we show that decay rates of the density of kinks and the residual energy are 1/\lnτand 1/(\lnτ)^2 for large τrespectively. Analytic results for the random Ising model are confirmed by the Monte-Carlo simulation. Our results reveal a clear difference between classical and quantum quenches in the random Ising chain.
We study the driven dynamics across the critical points of the Yang-Lee edge singularities (YLESs) in a finite-size quantum Ising chain with an imaginary symmetry-breaking field. In contrast to the conventional classical or quantum phase transitions, these phase transitions are induced by tuning the strength of the dissipation in a non-Hermitian system and can occur even at finite size. For conventional phase transitions, universal behaviors in driven dynamics across critical points are usually described by the Kibble-Zurek mechanism, which states that the scaling in dynamics is dictated by the critical exponents associated with one critical point and topological defects will emerge after the quench. While the mechanism leading to topological defects breaks down in the YLES, we find that for small lattice size, the driven dynamics can still be described by the Kibble-Zurek scaling with the exponents determined by the (0+1)-dimensional YLES. For medium finite size, however, the driven dynamics can be described by the Kibble-Zurek scaling with two sets of critical exponents determined by both the (0+1)-dimensional and the (1+1)-dimensional YLESs.