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Pramana

Quenching dynamics of a quantum XY spin-1/2 chain in the presence of transverse field by the application of a generalized Landau-Zener formula

Victor Mukherjee, Uma Divakaran, Amit Dutta, Diptiman Sen

Published Aug 1, 2008• 1 citation
Publisher
DOI
10.1007/s12043-008-0174-1

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We study analytically and numerically quench dynamics and defects formation in the quantum Ising model in the presence of a time-dependent transverse magnetic field. We generalize the Landau-Ziner formula to the case of non-adiabatic evolution of the quantum system. For a quasi-static magnetic field, with a slow dependence on time, our outcomes are similar to the results predicted by the Landau-Zener formula. However, a quench dynamics under a shock-wave load is more complicated. The final state of the system depends on the amplitude and pulse velocity, resulting in the mixture of ground and excited states and significant density of defects.
We study the quantum dynamics of a one-dimensional spin-12\frac{1}{2}21​ anisotropic XYXYXY model in a transverse field when the transverse field or the anisotropic interaction is quenched at a slow but uniform rate. The two quenching schemes are called transverse and anisotropic quenching, respectively. Our emphasis in this paper is on the anisotropic quenching scheme, and we compare the results with those of the other scheme. In the process of anisotropic quenching, the system crosses all the quantum critical lines of the phase diagram where the relaxation time diverges. The evolution is nonadiabatic in the time interval when the parameters are close to their critical values, and is adiabatic otherwise. The density of defects produced due to nonadiabatic transitions is calculated by mapping the many-particle system to an equivalent Landau-Zener problem and is generally found to vary as $1∕\sqrt{\ensuremath{\tau}}$, where $\ensuremath{\tau}$ is the characteristic time scale of quenching, a scenario that supports the Kibble-Zurek mechanism. Interestingly, in the case of anisotropic quenching, there exists an additional nonadiabatic transition, in comparison to the transverse quenching case, with the corresponding probability peaking at an incommensurate value of the wave vector. In the special case in which the system passes through a multicritical point, the defect density is found to vary as $1∕{\ensuremath{\tau}}^{1∕6}$. The von Neumann entropy of the final state is shown to maximize at a quenching rate around which the ordering of the final state changes from antiferromagnetic to ferromagnetic.
We study the quenching dynamics of a one-dimensional spin-1/2 XYXYXY model in a transverse field when the transverse field $h(=t/\ensuremath{\tau})$ is quenched repeatedly between $\ensuremath{-}\ensuremath{\infty}$ and $+\ensuremath{\infty}$. A single passage from $h\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$ to $h\ensuremath{\rightarrow}+\ensuremath{\infty}$ or the other way around is referred to as a half period of quenching. For an even number of half periods, the transverse field is brought back to the initial value of $\ensuremath{-}\ensuremath{\infty}$; in the case of an odd number of half periods, the dynamics is stopped at $h\ensuremath{\rightarrow}+\ensuremath{\infty}$. The density of defects produced due to the nonadiabatic transitions is calculated by mapping the many-particle system to an equivalent Landau-Zener problem and is generally found to vary as $1/\sqrt{\ensuremath{\tau}}$ for large $\ensuremath{\tau}$; however, the magnitude is found to depend on the number of half periods of quenching. For two successive half periods, the defect density is found to decrease in comparison to a single half period, suggesting the existence of a corrective mechanism in the reverse path. A similar behavior of the density of defects and the local entropy is observed for repeated quenching. The defect density decays as $1/\sqrt{\ensuremath{\tau}}$ for large $\ensuremath{\tau}$ for any number of half periods, and shows an increase in kink density for small $\ensuremath{\tau}$ for an even number; the entropy shows qualitatively the same behavior for any number of half periods. The probability of nonadiabatic transitions and the local entropy saturate to 1/2 and ln 2\text{ln}\text{ }2ln 2, respectively, for a large number of repeated quenching.