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

Abstract

We study the effects of interference on the quenching dynamics of a one-dimensional spin 1/2 XY model in the presence of a transverse field (h(t)) which varies sinusoidally with time as h = h0cosωt, with |t|≤tf = π/ω. We have explicitly shown that the finite values of tf make the dynamics inherently dependent on the phases of probability amplitudes, which had been hitherto unseen in all cases of linear quenching with large initial and final times. In contrast, we also consider the situation where the magnetic field 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 large τ. Our purpose is to estimate the defect density and the local entropy density in the final state if the system is initially prepared in its ground state. For a single crossing through the quantum critical point with h = h0cosωt, the density of defects in the final state is calculated by mapping the dynamics to an equivalent Landau–Zener problem by linearizing near the crossing point, and is found to vary as in the limit of small ω. On the other hand, the local entropy density is found to attain a maximum as a function of ω near a characteristic scale ω0. Extending to the situation of multiple crossings, we show that the role of finite initial and final times of quenching are manifested non-trivially in the interference effects of certain resonance modes which solely contribute to the production of defects. Kink density as well as the diagonal entropy density show oscillatory dependence on the number of full cycles of oscillation. Finally, the inclusion of a linear term in the transverse field on top of the oscillatory component results in a kink density which decreases continuously with τ while it increases monotonically with ω. The entropy density also shows monotonic change with the parameters, increasing with τ and decreasing with ω, in sharp contrast to the situations studied earlier. We also propose appropriate scaling relations for the defect density in the above situations and compare the results with the numerical results obtained by integrating the Schrödinger equations.