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 = h0 cos ωt, with |t| ≤ t f = π/ω. We have explicitly shown that the finite values of t f 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) = h0 cos ωt + 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 = h0 cos ωt, 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 √ ω 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 compo nent, results to a kink density which decreases continuously with τ while increases monotonically with ω. The entropy density also shows monotonous change with the parameters, increasing with τ and decreasing with ω, in sharp contrast to the situations studied earlier. We do also propose appropriate scaling relations for the defect density in above situations and compare the results with the numerical results obtained by integrating the Schrödinger equations.

Interconnections between equilibrium topology and dynamical quantum phase transitions in a linearly ramped Haldane model

“…Henceforth, experimental studies have been performed to probe intriguing dynamical phenomena like the real time evolution of closed quantum systems in cold atomic gases, [3], prethermalization [4][5][6], light-cone like propagation of quantum correlations [7], light-induced non-equilibrium superconductivity and topological systems [8,9] and many-body localization in disordered interacting systems [10]. In parallel, there have been a plethora of theoretical works on e.g., the growth of entanglement entropy following a quench [11], thermalization [12], light-induced topological matters [13][14][15], dynamics of topologically ordered systems [16][17][18], periodically driven closed quantum systems [19][20][21][22][23] and many body localization [24,25], to name a few. (For review, we refer to [26][27][28][29][30][31].…”

Defect Production Due to Quenching Through a Multicritical Point and Along a Gapless Line

“…For a large number of repetitions (l → ∞), the defect density saturates to the value 1/2 while the local entropy density saturates to ln 2. The effects of interference on the quenching dynamics of Hamiltonian in (1.8) when the transverse field h(t) varies sinusoidally with time as h = h 0 cos ωt, with |t| ≤ π/ω has also been studied in a recent work [32]. In this scheme of quenching, the time interval between two successive passages through the quantum critical points can be small enough for the presence of non-trivial effects of interference in the dynamics of the system.…”

“…The Hamiltonian in (1.29) then gets modified to the form 31) where the time dependence has been shifted to the diagonal terms only. Applying the LZ transition probability formula, the probability of excitations is found to be 32) where |∆ k | 2 = |1 + cosk| 2 = (π − k) 4 /4 when expanded about k = π. Integrating p k over the Brillouin zone, we find the that density of defects falls off as n ∼ 1/τ 1/3 .…”

“…The non-equilibrium dynamics of a quantum system when quenched very fast [3] or slowly across a quantum critical point [4,5] has attracted the attention of several groups recently. The possibility of experimental realizations of quantum dynamics in spin-1 Bose condensates [6] and atoms trapped in optical lattices [7,8] has led to an upsurge in studies of related theoretical models [3,4,5,9,10,11,12,13,14,15,16,17,18,19,20,21,23,24,25,26,27,28,29,30,31,32,33].…”

“…The Kibble-Zurek scaling has been verified in various exactly solvable spin models and systems of interacting bosons [4,13,16,17,11,5]; it has been generalized to quenching through a multicritical point [31], across a gapless phase [20,23], and along a gapless line [26,28], and to systems with quenched disorder [14], white noise [18], infinite-range interactions [27], and edge states [30]. Studies have also been made to estimate the defect density for quenching with a non-linear form [21], an oscillatory variation of an applied magnetic field [32] or under a reversal of the magnetic field [33]. In the article in this book by Mondal, Sengupta and Sen [36], the quenching dynamics through a gapless phase and quenching with a power-law form of the change of a parameter as well as the possibility of experimental realizations have been discussed in detail.…”