Effects of interference in the dynamics of a spin-1/2 transverse XY chain driven periodically through quantum critical points

Abstract

We study the effects of interference on the quenching dynamics of a one-dimensional spin1/2 XY model in the presence of a transverse field (h(t)) which variessinusoidally with time as h = h0cosωt, with |t|≤tf = π/ω. We have explicitly shown that the finite values oftf make the dynamics inherently dependent on the phases of probability amplitudes,which had been hitherto unseen in all cases of linear quenching with large initialand final times. In contrast, we also consider the situation where the magneticfield consists of an oscillatory as well as a linearly varying component, i.e.,h(t) = h0cosωt+t/τ, where the interference effects lose importance in the limit of largeτ. Our purpose is to estimate the defect density and the local entropydensity in the final state if the system is initially prepared in its groundstate. For a single crossing through the quantum critical point withh = h0cosωt, the density of defects in the final state is calculated by mapping the dynamics to anequivalent Landau–Zener problem by linearizing near the crossing point, and is found tovary as in the limit of small ω. On the other hand, the local entropy density is found to attain a maximum as a function ofω near acharacteristic scale ω0. Extending to the situation of multiple crossings, we show that the role of finite initial andfinal times of quenching are manifested non-trivially in the interference effects of certainresonance modes which solely contribute to the production of defects. Kink density as wellas the diagonal entropy density show oscillatory dependence on the number of full cycles ofoscillation. Finally, the inclusion of a linear term in the transverse field on top of theoscillatory component results in a kink density which decreases continuously withτ while it increasesmonotonically with ω. The entropy density also shows monotonic change with the parameters, increasing withτ and decreasingwith ω, in sharp contrast to the situations studied earlier. We also propose appropriate scalingrelations for the defect density in the above situations and compare the results with thenumerical results obtained by integrating the Schrödinger equations.