Defect generation in a spin-1/2 transverse XY chain under repeated   quenching of the transverse field
  We study the quenching dynamics of a one-dimensional spin-1/2 $XY$ model in a transverse field when the transverse field $h(=t/\tau)$ is quenched repeatedly between $-\infty$ and $+\infty$. A single passage from $h \to - \infty$ to $h \to +\infty$ 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 $-\infty$; in the case of an odd number of half-periods, the dynamics is stopped at $h \to +\infty$. The density of defects produced due to the non-adiabatic transitions is calculated by mapping the many-particle system to an equivalent Landau-Zener problem and is generally found to vary as $1/\sqrt{\tau}$ for large $\tau$; 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 $1/{\sqrt\tau}$ for large $\tau$ for any number of half-periods, and shows a increase in kink density for small $\tau$ for an even number; the entropy shows qualitatively the same behavior for any number of half-periods. The probability of non-adiabatic transitions and the local entropy saturate to 1/2 and $\ln 2$, respectively, for a large number of repeated quenching. 
Victor Mukherjee, Amit Dutta, Diptiman Sen